5. Basic API tutorial


In this chapter the reader will learn how to build a simple application that uses MOSEK.

A number of examples is provided to demonstrate the functionality required for solving linear, quadratic, and conic problems as well as mixed integer problems.

Please note that the section on linear optimization also describes most of the basic functionality that is not specific to linear problems. Hence, it is recommended to read Section 5.2 before reading the rest of this chapter.

5.1. The basics

A typical program using the MOSEK .NET interface can be described shortly:

  1. Create an environment (mosek.Env) object.
  2. Set up some environment specific data and initialize the environment object.
  3. Create a task (mosek.Task) object.
  4. Load a problem into the task object.
  5. Optimize the problem.
  6. Fetch the result.
  7. Dispose of the environment and task.

5.1.1. The environment and the task

The first MOSEK related step in any program that employs MOSEK is to create an environment (mosek.Env) object. The environment contains environment specific data such as information about the license file, streams for environment messages etc. Before creating any task objects, the environment must be initialized using mosek.Env.initenv. When this is done one or more task (mosek.Task) objects can be created. Each task is associated with a single environment and defines a complete optimization problem as well as task message streams and optimization parameters.

When done, all task and environments created must be explicitly disposed of using the Dispose method. As tasks depend on their environment, a task must be disposed of before its environment; not doing so will cause memory leaks or fatal errors.

In .NET creation of an environment and a task would look something like this:

...
mosek.Env env = new mosek.Env ();
// input environment data here
env.Init ();
 
mosek.Task task = new mosek.Task (env, taskid, num_con, num_var);
...
// input some task data, optimize etc.
...
task.dispose ()
env.dispose ()

Please note that an environment should, if possible, be shared between multiple tasks.

5.1.2. A simple working example

The following simple example shows a working .NET program which

  • creates an environment and a task,
  • reads a problem from a file,
  • optimizes the problem, and
  • writes the solution to a file.

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: simple.cs Purpose: Demonstrates a very simple example using MOSEK by reading a problem file, solving the problem and writing the solution to a file. */ using System; public class simple { public static void Main (string[] args) { mosek.Task task = null; mosek.Env env = null; if (args.Length == 0) { Console.WriteLine ("Missing argument. The syntax is:"); Console.WriteLine (" simple inputfile [ solutionfile ]"); } else { try { // Make mosek environment. env = new mosek.Env (); // Initialize the environment. env.init (); // Create a task object linked with the environment env. // We create it initially with 0 variables and 0 columns, // since we don't know the size of the problem. task = new mosek.Task (env, 0,0); // We assume that a problem file was given as the first command // line argument (received in `args') task.readdata (args[0]); // Solve the problem task.optimize(); // Print a summary of the solution task.solutionsummary(mosek.streamtype.log); // If an output file was specified, write a solution if (args.Length > 1) { // We define the output format to be OPF, and tell MOSEK to // leave out parameters and problem data from the output file. task.putintparam (mosek.iparam.write_data_format, mosek.dataformat.op); task.putintparam (mosek.iparam.opf_write_solutions, mosek.onoffkey.on); task.putintparam (mosek.iparam.opf_write_hints, mosek.onoffkey.off); task.putintparam (mosek.iparam.opf_write_parameters, mosek.onoffkey.off); task.putintparam (mosek.iparam.opf_write_problem, mosek.onoffkey.off); task.writedata(args[1]); } } finally { // Dispose of task end environment if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } } }

5.1.2.1. Writing a problem to a file

It is frequently beneficial to write a problem to a file that can be stored for later use or inspected visually. The mosek.Task.writedata function is used write a problem to a file as follows

task.writedata(args[1]);

By default the extension of the filename is the format written. I.e. the filename somename.opf implies the file is written in the OPF format.

Similarly, the function mosek.Task.readdata reads a problem from a file:

task.readdata (args[0]);

5.1.2.2. Inputting and outputting problem data

An optimization problem consists of several components; objective, objective sense, constraints, variable bounds etc. Therefore, the task (mosek.Task) provides a number of methods to operate on the task specific data, all of which are listed in Section 15.9.

5.1.2.3. Setting parameters

Apart from the problem data, the task contains a number of parameters defining the behavior of MOSEK. For example the mosek.iparam.optimizer parameter defines which optimizer to use. A complete list of all parameters are listed in Chapter 16.

5.1.3. Compiling and running examples

All examples presented in this chapter are distributed with MOSEK and are available in the directory

 mosek/6/tools/examples/ 

in the MOSEK installation. Chapter 4 describes how to compile and run the examples.

It is recommended to copy examples to a different directory before modifying and compiling them.

5.2. Linear optimization

The simplest optimization problem is a purely linear problem. A linear optimization problem is a problem of the following form:

Minimize or maximize the objective function

\begin{math}\nonumber{}\sum _{{j=0}}^{{n-1}}c_{j}x_{j}+c^{f}\end{math} (5.2.1)

subject to the linear constraints

\begin{math}\nonumber{}l_{k}^{c}\leq{}\sum _{{j=0}}^{{n-1}}a_{{kj}}x_{j}\leq{}u_{k}^{c},~k=0,\ldots ,m-1,\end{math} (5.2.2)

and the bounds

\begin{math}\nonumber{}l_{j}^{x}\leq{}x_{j}\leq{}u_{j}^{x},~j=0,\ldots ,n-1,\end{math} (5.2.3)

where we have used the problem elements

m and n, which are the number of constraints and variables respectively,

x, which is the variable vector of length n,

c, which is a coefficient vector of size n

\begin{displaymath}\nonumber{}c=\left[\begin{array}{c}\nonumber{}c_{0}\\\nonumber{}\vdots \\\nonumber{}c_{{n-1}}\end{array}\right],\end{displaymath}

[[MathCmd 5]], which is a constant,

A, which is a [[MathCmd 6]] matrix of coefficients is given by

\begin{displaymath}\nonumber{}A=\left[\begin{array}{ccc}\nonumber{}a_{{0,0}} & \cdots  & a_{{0,(n-1)}}\\\nonumber{}\vdots  & \cdots  & \vdots \\\nonumber{}a_{{(m-1),0}} & \cdots  & a_{{(m-1),(n-1)}}\end{array}\right],\end{displaymath}

[[MathCmd 8]] and [[MathCmd 9]], which specify the lower and upper bounds on constraints respectively, and

[[MathCmd 10]] and [[MathCmd 11]], which specifies the lower and upper bounds on variables respectively.

Please note the unconventional notation using 0 as the first index rather than 1. Hence, [[MathCmd 12]] is the first element in variable vector x. This convention has been adapted from .NET arrays which are indexed from 0.

5.2.1. Linear optimization example: lo1

The following is an example of a linear optimization problem:

\begin{math}\nonumber{}\begin{array}{lccccccccl}\nonumber{}\mbox{maximize} & 3x_{0} & + & 1x_{1} & + & 5x_{2} & + & 1x_{3} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 1x_{1} & + & 2x_{2} &  &  & = & 30,\\\nonumber{} & 2x_{0} & + & 1x_{1} & + & 3x_{2} & + & 1x_{3} & \geq{} & 15,\\\nonumber{} &  &  & 2x_{1} &  &  & + & 3x_{3} & \leq{} & 25,\end{array}\end{math} (5.2.4)

having the bounds

\begin{math}\nonumber{}\begin{array}{ccccc}\nonumber{}0 & \leq{} & x_{0} & \leq{} & \infty ,\\\nonumber{}0 & \leq{} & x_{1} & \leq{} & 10,\\\nonumber{}0 & \leq{} & x_{2} & \leq{} & \infty ,\\\nonumber{}0 & \leq{} & x_{3} & \leq{} & \infty .\end{array}\end{math} (5.2.5)

5.2.1.1. Solving the problem

To solve the problem above we go through the following steps:

  1. Create an environment.
  2. Create an optimization task.
  3. Load a problem into the task object.
  4. Optimization.
  5. Extracting the solution.

Below we explain each of these steps. For the complete source code see section 5.2.1.2. The code can also be found in:

    mosek\6\tools\examples\csharp\lo1.cs
Create an environment.

Before setting up the optimization problem, a MOSEK environment must be created and initialized. This is done in the lines:

// Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init ();

We connect a call-back function to the environment log stream. In this case the call-back function simply prints messages to the standard output stream.

Create an optimization task.

Next, an empty task object is created:

// Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass (""));

We also connect a call-back function to the task log stream. Messages related to the task are passed to the call-back function. In this case the stream call-back function writes its messages to the standard output stream.

Load a problem into the task object.

First an estimate of the size of the input data is set. This is done to increase the speed of inputting data and is optional.

task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ);

Before any problem data can be set, variables and constraints must be added to the problem via calls to the function mosek.Task.append.

/* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR);

New variables can now be referenced from other functions with indexes in [[MathCmd 15]] and new constraints can be referenced with indexes in [[MathCmd 16]]. More variables / constraints can be appended later as needed, these will be assigned indexes from [[MathCmd 17]] / [[MathCmd 18]] and up.

Next step is to set the problem data. We loop over each variable index [[MathCmd 19]] calling functions to set problem data. We first set the objective coefficient [[MathCmd 20]] by calling the function mosek.Task.putcj.

/* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]);

The bounds on variables are stored in the arrays

mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.ra, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0, 0.0}; double[] bux = {+infinity, 10.0, +infinity, +infinity};

and are set with calls to mosek.Task.putbound.

/* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]);

The Bound key stored in bkx specify the type of the bound according to Table 5.1.

Bound key Type of bound Lower bound Upper bound
mosek.boundkey.fx [[MathCmd 21]] Finite Identical to the lower bound
mosek.boundkey.fr Free Minus infinity Plus infinity
mosek.boundkey.lo [[MathCmd 22]] Finite Plus infinity
mosek.boundkey.ra [[MathCmd 23]] Finite Finite
mosek.boundkey.up [[MathCmd 24]] Minus infinity Finite
Table 5.1: Interpretation of the bound keys.

For instance bkx[0]= mosek.boundkey.lo means that [[MathCmd 25]]. Finally, the numerical values of the bounds on variables are given by

\begin{math}\nonumber{}l_{j}^{x}=\mathtt{blx[j]}\end{math} (5.2.6)

and

\begin{math}\nonumber{}u_{j}^{x}=\mathtt{bux[j]}.\end{math} (5.2.7)

Recall that in our example the A matrix is given by

\begin{displaymath}\nonumber{}A=\left[\begin{array}{cccc}\nonumber{}3 & 1 & 2 & 0\\\nonumber{}2 & 1 & 3 & 1\\\nonumber{}0 & 2 & 0 & 3\end{array}\right].\end{displaymath}

This matrix is stored in sparse format in the arrays:

int[][] asub = { new int[] {0, 1}, new int[] {0, 1, 2}, new int[] {0, 1}, new int[] {1, 2}}; double[][] aval = { new double[] {3.0, 2.0}, new double[] {1.0, 1.0, 2.0}, new double[] {2.0, 3.0}, new double[] {1.0, 3.0}};

The array aval[j] contains the non-zero values of column j and asub[j] contains the row index of these non-zeros.

Using the function mosek.Task.putavec we set column j of A

/* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */

Alternatively, the same A matrix can be set one row at a time; please see section 5.2.2 for an example.

Finally, the bounds on each constraint are set by looping over each constraint index [[MathCmd 29]]

/* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]);
Optimization:

After the problem is set-up the task can be optimized by calling the function mosek.Task.optimizetrm.

task.optimize();
Extracting the solution.

After optimizing the status of the solution is examined with a call to mosek.Task.getsolutionstatus. If the solution status is reported as mosek.solsta.optimal or mosek.solsta.near_optimal the solution is extracted in the lines below:

task.getsolutionslice(mosek.soltype.bas, // Basic solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx);

The mosek.Task.getsolutionslice function obtains a “slice” of the solution. MOSEK may compute several solutions depending on the optimizer employed. In this example the basic solution is requested by setting the first argument to mosek.soltype.bas. The second argument mosek.solitem.xx specifies that we want the variable values of the solution. The two following arguments 0 and NUMVAR specifies the range of variable values we want.

The range specified is the first index (here “0”) up to but not including the second index (here “NUMVAR).

Catching exceptions:

We cache any exceptions thrown by mosek in the lines:

catch (mosek.Exception e) { Console.WriteLine (e.Code); Console.WriteLine (e); }

The types of exceptions that MOSEK can throw can be seen in 15.5 and 15.10.

5.2.1.2. Source code for lo1

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: lo1.cs Purpose: Demonstrates how to solve small linear optimization problem using the MOSEK C# API. */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class lo1 { public static void Main () { const int NUMCON = 3; const int NUMVAR = 4; const int NUMANZ = 9; // Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; double[] c = {3.0, 1.0, 5.0, 1.0}; int[][] asub = { new int[] {0, 1}, new int[] {0, 1, 2}, new int[] {0, 1}, new int[] {1, 2}}; double[][] aval = { new double[] {3.0, 2.0}, new double[] {1.0, 1.0, 2.0}, new double[] {2.0, 3.0}, new double[] {1.0, 3.0}}; mosek.boundkey[] bkc = {mosek.boundkey.fx, mosek.boundkey.lo, mosek.boundkey.up}; double[] blc = {30.0, 15.0, -infinity}; double[] buc = {30.0, +infinity, 25.0}; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.ra, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0, 0.0}; double[] bux = {+infinity, 10.0, +infinity, +infinity}; mosek.Task task = null; mosek.Env env = null; double[] xx = new double[NUMVAR]; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("")); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); /* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */ } /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); task.putobjsense(mosek.objsense.maximize); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.bas, out prosta, out solsta); task.getsolutionslice(mosek.soltype.bas, // Basic solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e.Code); Console.WriteLine (e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } }

5.2.2. Row-wise input

In the previous example the A matrix is set one column at a time. Alternatively the same matrix can be set one row at a time or the two methods can be mixed as in the example in section 5.6. The following example show how to set the A matrix by rows.

The source code for this example can be found in:

  mosek\6\tools\examples\dotnet\lo2.cs

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: lo2.cs Purpose: Demonstrates how to solve small linear optimization problem using the MOSEK C# API. */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class lo2 { public static void Main () { const int NUMCON = 3; const int NUMVAR = 4; const int NUMANZ = 9; // Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; double[] c = {3.0, 1.0, 5.0, 1.0}; int[][] asub = { new int[] {0,1,2}, new int[] {0,1,2,3}, new int[] {1,3} }; double[][] aval = { new double[] {3.0,1.0,2.0}, new double[] {2.0,1.0,3.0,1.0}, new double[] {2.0,3.0} }; mosek.boundkey[] bkc = {mosek.boundkey.fx, mosek.boundkey.lo, mosek.boundkey.up}; double[] blc = {30.0, 15.0, -infinity}; double[] buc = {30.0, +infinity, 25.0}; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.ra, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0, 0.0}; double[] bux = {+infinity, 10.0, +infinity, +infinity}; mosek.Task task = null; mosek.Env env = null; double[] xx = new double[NUMVAR]; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("")); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); } /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) { task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); /* Input row i of A */ task.putavec(mosek.accmode.con, /* Input row of A.*/ i, /* Row index.*/ asub[i], /* Column indexes of non-zeros in row i.*/ aval[i]); /* Non-zero Values of row i. */ } task.putobjsense(mosek.objsense.maximize); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.bas, out prosta, out solsta); task.getsolutionslice(mosek.soltype.bas, // Basic solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e.Code); Console.WriteLine (e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } }

5.3. Quadratic optimization

MOSEK can solve quadratic and quadratically constrained convex problems. This class of problems can be formulated as follows:

\begin{math}\nonumber{}\begin{array}{lrcccll}\nonumber{}\mbox{minimize} &  &  & \frac{1}{2}x^{T}Q^{o}x+c^{T}x+c^{f} &  &  & \\\nonumber{}\mbox{subject to} & l_{k}^{c} & \leq{} & \frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j} & \leq{} & u_{k}^{c}, & k=0,\ldots ,m-1,\\\nonumber{} & l^{x} & \leq{} & x & \leq{} & u^{x}, & j=0,\ldots ,n-1.\end{array}\end{math} (5.3.1)

Without loss of generality it is assumed that [[MathCmd 31]] and [[MathCmd 32]] are all symmetric because

\begin{displaymath}\nonumber{}x^{T}Qx=0.5x^{T}(Q+Q^{T})x.\end{displaymath}

This implies that a non-symmetric Q can be replaced by the symmetric matrix [[MathCmd 34]].

The problem is required to be convex. More precisely, the matrix [[MathCmd 31]] must be positive semi-definite and the kth constraint must be of the form

\begin{math}\nonumber{}l_{k}^{c}\leq{}\frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j}\end{math} (5.3.2)

with a negative semi-definite [[MathCmd 32]] or of the form

\begin{math}\nonumber{}\frac{1}{2}x^{T}Q^{k}x+\sum \limits _{{j=0}}^{{n-1}}a_{{k,j}}x_{j}\leq{}u_{k}^{c}.\end{math} (5.3.3)

with a positive semi-definite [[MathCmd 32]]. This implies that quadratic equalities are not allowed. Specifying a non-convex problem will result in an error when the optimizer is called.

5.3.1. Example: Quadratic objective

The following is an example if a quadratic, linearly constrained problem:

\begin{math}\nonumber{}\begin{array}{lcccl}\nonumber{}\mbox{minimize} &  &  & x_{1}^{2}+0.1x_{2}^{2}+x_{3}^{2}-x_{1}x_{3}-x_{2} & \\\nonumber{}\mbox{subject to} & 1 & \leq{} & x_{1}+x_{2}+x_{3} & \\\nonumber{} &  &  & x\geq{}0 &\end{array}\end{math} (5.3.4)

This can be written equivalently as

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{minimize} & 1/2x^{T}Q^{o}x+c^{T}x &  & \\\nonumber{}\mbox{subject to} & Ax & \geq{} & b\\\nonumber{} & x & \geq{} & 0,\end{array}\end{math} (5.3.5)

where

\begin{math}\nonumber{}Q^{o}=\left[\begin{array}{ccc}\nonumber{}2 & 0 & -1\\\nonumber{}0 & 0.2 & 0\\\nonumber{}-1 & 0 & 2\end{array}\right],\quad{}c=\left[\begin{array}{c}\nonumber{}0\\\nonumber{}-1\\\nonumber{}0\end{array}\right],\quad{}A=\left[\begin{array}{ccc}\nonumber{}1 & 1 & 1\end{array}\right],\mbox{ and }b=1.\end{math} (5.3.6)

Please note that MOSEK always assumes that there is a 1/2 in front of the [[MathCmd 43]] term in the objective. Therefore, the 1 in front of [[MathCmd 44]] becomes 2 in Q, i.e. [[MathCmd 45]].

5.3.1.1. Source code

/* File: qo1.cs Purpose: Demonstrate how to solve a quadratic optimization problem using the MOSEK .NET API. */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class qo1 { public static void Main () { // Since the value infinity is never used, we define // 'infinity' symbolic purposes only const double infinity = 0; const int NUMCON = 1; /* Number of constraints. */ const int NUMVAR = 3; /* Number of variables. */ const int NUMANZ = 9; /* Number of numzeros in A. */ double[] c = {0.0,-1.0,0.0}; mosek.boundkey[] bkc = {mosek.boundkey.lo}; double[] blc = {1.0}; double[] buc = {infinity}; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0}; double[] bux = {+infinity, +infinity, +infinity}; int[][] asub = { new int[] {0}, new int[] {0}, new int[] {0}}; double[][] aval = { new double[] {1.0}, new double[] {1.0}, new double[] {1.0}}; mosek.Task task = null; mosek.Env env = null; double[] xx = new double[NUMVAR]; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("")); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); /* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */ } /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); /* * The lower triangular part of the Q * matrix in the objective is specified. */ int[] qsubi = {0, 1, 2, 2 }; int[] qsubj = {0, 1, 0, 2 }; double[] qval = {2.0, 0.2, -1.0, 2.0}; /* Input the Q for the objective. */ task.putobjsense(mosek.objsense.minimize); task.putqobj(qsubi,qsubj,qval); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.itr, out prosta, out solsta); task.getsolutionslice(mosek.soltype.itr, // Interior point solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } /* Main */ }

5.3.1.2. Example code comments

Most of the functionality in this example has already been explained for the linear optimization example in Section 5.2 and it will not be repeated here.

This example introduces one new function, mosek.Task.putqobj, which is used to input the quadratic terms of the objective function.

Since [[MathCmd 31]] is symmetric only the lower triangular part of [[MathCmd 31]] is inputted. The upper part of [[MathCmd 31]] is computed by MOSEK using the relation

\begin{displaymath}\nonumber{}Q^{o}_{{ij}}=Q^{o}_{{ji}}.\end{displaymath}

Entries from the upper part may not appear in the input.

The lower triangular part of the matrix [[MathCmd 31]] is specified using an unordered sparse triplet format (for details, see Section 5.8.3):

int[] qsubi = {0, 1, 2, 2 }; int[] qsubj = {0, 1, 0, 2 }; double[] qval = {2.0, 0.2, -1.0, 2.0};

Please note that

  • only non-zero elements are specified (any element not specified is 0 by definition),
  • the order of the non-zero elements is insignificant, and
  • only the lower triangular part should be specified.

Finally, the matrix [[MathCmd 31]] is loaded into the task:

task.putqobj(qsubi,qsubj,qval);

5.3.2. Example: Quadratic constraints

In this section describes how to solve a problem with quadratic constraints. Please note that quadratic constraints are subject to the convexity requirement (5.3.2).

Consider the problem:

\begin{math}\nonumber{}\begin{array}{lcccl}\nonumber{}\mbox{minimize} &  &  & x_{1}^{2}+0.1x_{2}^{2}+x_{3}^{2}-x_{1}x_{3}-x_{2} & \\\nonumber{}\mbox{subject to} & 1 & \leq{} & x_{1}+x_{2}+x_{3}-x_{1}^{2}-x_{2}^{2}-0.1x_{3}^{2}+0.2x_{1}x_{3}, & \\\nonumber{} &  &  & x\geq{}0. &\end{array}\end{math} (5.3.7)

This is equivalent to

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{minimize} & 1/2x^{T}Q^{o}x+c^{T}x &  & \\\nonumber{}\mbox{subject to} & 1/2x^{T}Q^{0}x+Ax & \geq{} & b,\end{array}\end{math} (5.3.8)

where

\begin{math}\nonumber{}Q^{o}=\left[\begin{array}{ccc}\nonumber{}2 & 0 & -1\\\nonumber{}0 & 0.2 & 0\\\nonumber{}-1 & 0 & 2\end{array}\right],\quad{}c=\left[\begin{array}{c}\nonumber{}0\\\nonumber{}-1\\\nonumber{}0\end{array}\right],\quad{}A=\left[\begin{array}{ccc}\nonumber{}1 & 1 & 1\end{array}\right],\quad{}b=1.\end{math} (5.3.9)
\begin{math}\nonumber{}Q^{0}=\left[\begin{array}{ccc}\nonumber{}-2 & 0 & 0.2\\\nonumber{}0 & -2 & 0\\\nonumber{}0.2 & 0 & -0.2\end{array}\right].\end{math} (5.3.10)

5.3.2.1. Source code

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: qcqo1.cs Purpose: Demonstrate how to solve a quadratic optimization problem using the MOSEK API. minimize x0^2 + 0.1 x1^2 + x2^2 - x0 x2 - x1 s.t 1 <= x0 + x1 + x2 - x0^2 - x1^2 - 0.1 x2^2 + 0.2 x0 x2 x >= 0 */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class qcqo1 { public static void Main () { const double inf = 0.0; /* We don't actually need any value for infinity */ const int NUMCON = 1; /* Number of constraints. */ const int NUMVAR = 3; /* Number of variables. */ const int NUMANZ = 3; /* Number of numzeros in A. */ const int NUMQNZ = 4; /* Number of nonzeros in Q. */ mosek.boundkey[] bkc = { mosek.boundkey.lo }, bkx = { mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo }; int[][] asub = { new int[] {0}, new int[] {0}, new int[] {0} }; double[][] aval = { new double[]{1.0}, new double[]{1.0}, new double[]{1.0} }; double[] blc = { 1.0 }, buc = { inf }, c = { 0.0, -1.0, 0.0 }, blx = { 0.0, 0.0, 0.0 }, bux = { inf, inf, inf }, xx = new double[NUMVAR]; mosek.Task task = null; mosek.Env env = null; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("")); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); /* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */ } /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); /* * The lower triangular part of the Q * matrix in the objective is specified. */ { int[] qsubi = { 0, 1, 2, 2 }, qsubj = { 0, 1, 0, 2 }; double[] qval = { 2.0, 0.2, -1.0, 2.0 }; /* Input the Q for the objective. */ task.putqobj(qsubi,qsubj,qval); } /* * The lower triangular part of the Q^0 * matrix in the first constraint is specified. * This corresponds to adding the term * - x0^2 - x1^2 - 0.1 x2^2 + 0.2 x0 x2 */ { int[] qsubi = { 0, 1, 2, 2 }, qsubj = { 0, 1, 2, 0 }; double[] qval = { -2.0, -2.0, -0.2, 0.2 }; /* put Q^0 in constraint with index 0. */ task.putqconk (0, qsubi, qsubj, qval); } task.putobjsense(mosek.objsense.minimize); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.itr, out prosta, out solsta); task.getsolutionslice(mosek.soltype.itr, // Basic solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } /* Main */ }

The only new function introduced in this example is mosek.Task.putqconk, which is used to add quadratic terms to the constraints. While mosek.Task.putqconk add quadratic terms to a specific constraint, it is also possible to input all quadratic terms in all constraints in one chunk using the mosek.Task.putqcon function.

5.4. Conic optimization

Conic problems are a generalization of linear problems, allowing constraints of the type

\begin{displaymath}\nonumber{}x\in{}\mathcal{C}\end{displaymath}

where [[MathCmd 57]] is a convex cone.

MOSEK can solve conic optimization problems of the following form

\begin{math}\nonumber{}\begin{array}{lccccl}\nonumber{}\mbox{minimize} &  &  & c^{T}x+c^{f} &  & \\\nonumber{}\mbox{subject to} & l^{c} & \leq{} & Ax & \leq{} & u^{c},\\\nonumber{} & l^{x} & \leq{} & x & \leq{} & u^{x},\\\nonumber{} &  &  & x\in{}\mathcal{C} &  &\end{array}\end{math} (5.4.1)

where [[MathCmd 57]] is a cone. [[MathCmd 57]] can be a product of cones, i.e.

\begin{displaymath}\nonumber{}\mathcal{C}=\mathcal{C}_{0}\times \cdots \times \mathcal{C}_{{p-1}}\end{displaymath}

in which case [[MathCmd 62]] means [[MathCmd 63]]. Please note that the set of real numbers [[MathCmd 64]] is itself a cone, so linear variables are still allowed.

MOSEK supports two specific cones apart from the real numbers:

When creating a conic problem in MOSEK, each cone is defined by a cone type (quadratic or rotated quadratic cone) and a list of variable indexes. To summarize:

5.4.1. Example: cqo1

The problem

\begin{math}\nonumber{}\begin{array}{lccccc}\nonumber{}\mbox{minimize} & x_{4}+x_{5} &  & \\\nonumber{}\mbox{subject to} & x_{0}+x_{1}+x_{2}+x_{3} & = & 1,\\\nonumber{} & x_{0},x_{1},x_{2},x_{3} & \geq{} & 0,\\\nonumber{} & x_{4}\geq{}\sqrt{x_{0}^{2} + x_{2}^{2}}, &  & \\\nonumber{} & x_{5}\geq{}\sqrt{x_{1}^{2} + x_{3}^{2}} &  &\end{array}\end{math} (5.4.2)

is an example of a conic quadratic optimization problem. The problem includes a set of linear constraints and two quadratic cones.

5.4.1.1. Source code

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: cqo1.cs Purpose: Demonstrates how to solve a small conic qaudratic optimization problem using the MOSEK API. */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class cqo1 { public static void Main () { const int NUMCON = 1; const int NUMVAR = 6; const int NUMANZ = 4; // Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; mosek.boundkey[] bkc = { mosek.boundkey.fx }; double[] blc = { 1.0 }; double[] buc = { 1.0 }; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.fr, mosek.boundkey.fr}; double[] blx = { 0.0, 0.0, 0.0, 0.0, -infinity, -infinity}; double[] bux = { +infinity, +infinity, +infinity, +infinity, +infinity, +infinity}; double[] c = { 0.0, 0.0, 0.0, 0.0, 1.0, 1.0}; double[][] aval = {new double[] {1.0}, new double[] {1.0}, new double[] {1.0}, new double[] {1.0}}; int[][] asub = {new int[] {0}, new int[] {0}, new int[] {0}, new int[] {0}}; int[] csub = new int[3]; double[] xx = new double[NUMVAR]; mosek.Env env = null; mosek.Task task = null; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, 0,0); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("")); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); } for(int j=0; j<aval.Length; ++j) /* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */ /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); csub[0] = 4; csub[1] = 0; csub[2] = 2; task.appendcone(mosek.conetype.quad, 0.0, /* For future use only, can be set to 0.0 */ csub); csub[0] = 5; csub[1] = 1; csub[2] = 3; task.appendcone(mosek.conetype.quad,0.0,csub); task.putobjsense(mosek.objsense.minimize); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.itr, out prosta, out solsta); task.getsolutionslice(mosek.soltype.itr, // Basic solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e); throw(e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } }

5.4.1.2. Source code comments

The only new function introduced in the example is mosek.Task.appendcone, which is called here:

task.appendcone(mosek.conetype.quad, 0.0, /* For future use only, can be set to 0.0 */ csub);

Here mosek.conetype.quad defines the cone type, in this case it is a quadratic cone. The cone parameter 0.0 is currently not used by MOSEK — simply passing 0.0 will work.

The last argument is a list of indexes of the variables in the cone.

5.5. Integer optimization

An optimization problem where one or more of the variables are constrained to integer values is denoted an integer optimization problem.

5.5.1. Example: milo1

In this section the example

\begin{math}\nonumber{}\begin{array}{lccl}\nonumber{}\mbox{maximize} & x_{0}+0.64x_{1} &  & \\\nonumber{}\mbox{subject to} & 50x_{0}+31x_{1} & \leq{} & 250,\\\nonumber{} & 3x_{0}-2x_{1} & \geq{} & -4,\\\nonumber{} & x_{0},x_{1}\geq{}0 &  & \mbox{and integer}\end{array}\end{math} (5.5.1)

is used to demonstrate how to solve a problem with integer variables.

5.5.1.1. Source code

The example (5.5.1) is almost identical to a linear optimization problem except for some variables being integer constrained. Therefore, only the specification of the integer constraints requires something new compared to the linear optimization problem discussed previously. In MOSEK these constraints are specified using the function mosek.Task.putvartype as shown in the code:

for(int j=0; j<NUMVAR; ++j) task.putvartype(j,mosek.variabletype.type_int);

The complete source for the example is listed below.

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: milo1.cs Purpose: Demonstrates how to solve a small mixed integer linear optimization problem using the MOSEK C# API. */ using System; public class MsgClass : mosek.Stream { public MsgClass () { /* Construct the object */ } public override void streamCB (string msg) { Console.Write ("{0}",msg); } } public class milo1 { public static void Main () { const int NUMCON = 2; const int NUMVAR = 2; const int NUMANZ = 4; // Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; mosek.boundkey[] bkc = { mosek.boundkey.up, mosek.boundkey.lo }; double[] blc = { -infinity, -4.0 }; double[] buc = { 250.0, infinity }; mosek.boundkey[] bkx = { mosek.boundkey.lo, mosek.boundkey.lo }; double[] blx = { 0.0, 0.0 }; double[] bux = { infinity, infinity }; double[] c = {1.0, 0.64 }; int[][] asub = { new int[] {0, 1}, new int[] {0, 1} }; double[][] aval = { new double[] {50.0, 3.0},new double[] {31.0, -2.0} }; double[] xx = new double[NUMVAR]; mosek.Env env = null; mosek.Task task = null; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB MsgClass env_msg_obj = new MsgClass (); env.set_Stream (mosek.streamtype.log,env_msg_obj); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, NUMCON,NUMVAR); // Directs the log task stream to the user specified // method task_msg_obj.streamCB MsgClass task_msg_obj = new MsgClass (); task.set_Stream (mosek.streamtype.log,task_msg_obj); /* Give MOSEK an estimate of the size of the input data. This is done to increase the speed of inputting data. However, it is optional. */ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append 'NUMCON' empty constraints. The constraints will initially have no bounds. */ task.append(mosek.accmode.con,NUMCON); /* Append 'NUMVAR' variables. The variables will initially be fixed at zero (x=0). */ task.append(mosek.accmode.var,NUMVAR); /* Optionally add a constant term to the objective. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) { /* Set the linear term c_j in the objective.*/ task.putcj(j,c[j]); /* Set the bounds on variable j. blx[j] <= x_j <= bux[j] */ task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); /* Input column j of A */ task.putavec(mosek.accmode.var, /* Input columns of A.*/ j, /* Variable (column) index.*/ asub[j], /* Row index of non-zeros in column j.*/ aval[j]); /* Non-zero Values of column j. */ } /* Set the bounds on constraints. for i=1, ...,NUMCON : blc[i] <= constraint i <= buc[i] */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); /* Specify integer variables. */ for(int j=0; j<NUMVAR; ++j) task.putvartype(j,mosek.variabletype.type_int); task.putobjsense(mosek.objsense.maximize); task.optimize(); // Print a summary containing information // about the solution for debugging purposes task.solutionsummary(mosek.streamtype.msg); mosek.solsta solsta; mosek.prosta prosta; /* Get status information about the solution */ task.getsolutionstatus(mosek.soltype.itg, out prosta, out solsta); task.getsolutionslice(mosek.soltype.itg, // Integer solution. mosek.solitem.xx, // Which part of solution. 0, // Index of first variable. NUMVAR, // Index of last variable+1 xx); switch(solsta) { case mosek.solsta.optimal: case mosek.solsta.near_optimal: Console.WriteLine ("Optimal primal solution\n"); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:",xx[j]); break; case mosek.solsta.dual_infeas_cer: case mosek.solsta.prim_infeas_cer: case mosek.solsta.near_dual_infeas_cer: case mosek.solsta.near_prim_infeas_cer: Console.WriteLine("Primal or dual infeasibility.\n"); break; case mosek.solsta.unknown: Console.WriteLine("Unknown solution status.\n"); break; default: Console.WriteLine("Other solution status"); break; } } catch (mosek.Exception e) { Console.WriteLine (e.Code); Console.WriteLine (e); } finally { if (task != null) task.Dispose (); if (env != null) env.Dispose (); } } }

5.5.1.2. Code comments

Please note that when mosek.Task.getsolutionslice is called, the integer solution is requested by using mosek.soltype.itg. No dual solution is defined for integer optimization problems.

5.5.2. Specifying an initial solution

Integer optimization problems are generally hard to solve, but the solution time can often be reduced by providing an initial solution for the solver. Solution values can be set using mosek.Task.putsolution (for inputting a whole solution) or mosek.Task.putsolutioni (for inputting solution values related to a single variable or constraint).

It is not necessary to specify the whole solution. By setting the mosek.iparam.mio_construct_sol parameter to mosek.onoffkey.on and inputting values for the integer variables only, will force MOSEK to compute the remaining continuous variable values.

If the specified integer solution is infeasible or incomplete, MOSEK will simply ignore it.

5.5.3. Example: Specifying an integer solution

Consider the problem

\begin{math}\nonumber{}\begin{array}{ll}\nonumber{}\mbox{maximize} & 7x_{0}+10x_{1}+x_{2}+5x_{3}\\\nonumber{}\mbox{subject to} & x_{0}+x_{1}+x_{2}+x_{3}\leq{}2.5\\\nonumber{} & x_{0},x_{1},x_{2}\mathrm{integer},\quad{}x_{0},x_{1},x_{2},x_{3}\geq{}0\end{array}\end{math} (5.5.2)

The following example demonstrates how to optimize the problem using a feasible starting solution generated by selecting the integer values as [[MathCmd 71]].

/* Copyright: Copyright (c) 1998-2011 MOSEK ApS, Denmark. All rights reserved. File: mioinitsol.c Purpose: Demonstrates how to solve a MIP with a start guess. Syntax: mioinitsol mioinitsol.lp */ using System; class msgclass : mosek.Stream { string prefix; public msgclass (string prfx) { prefix = prfx; } public override void streamCB (string msg) { Console.Write ("{0}{1}", prefix,msg); } } public class mioinitsol { public static void Main () { mosek.Env env = null; mosek.Task task = null; // Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; int NUMVAR = 4; int NUMCON = 1; int NUMINTVAR = 3; double[] c = { 7.0, 10.0, 1.0, 5.0 }; mosek.boundkey[] bkc = {mosek.boundkey.up}; double[] blc = {-infinity}; double[] buc = {2.5}; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0, 0.0}; double[] bux = {infinity, infinity, infinity, infinity}; int[] ptrb = {0, 1, 2, 3}; int[] ptre = {1, 2, 3, 4}; double[] aval = {1.0, 1.0, 1.0, 1.0}; int[] asub = {0, 0, 0, 0 }; int[] intsub = {0, 1, 2}; double[] xx = new double[NUMVAR]; try { // Make mosek environment. env = new mosek.Env (); // Direct the env log stream to the user specified // method env_msg_obj.streamCB env.set_Stream (mosek.streamtype.log, new msgclass ("[env]")); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, NUMCON,NUMVAR); // Directs the log task stream to the user specified // method task_msg_obj.streamCB task.set_Stream (mosek.streamtype.log, new msgclass ("[task]")); task.inputdata(NUMCON,NUMVAR, c, 0.0, ptrb, ptre, asub, aval, bkc, blc, buc, bkx, blx, bux); for(int j=0 ; j<NUMINTVAR ; ++j) task.putvartype(intsub[j],mosek.variabletype.type_int); task.putobjsense(mosek.objsense.maximize); // Construct an initial feasible solution from the // values of the integer valuse specified task.putintparam(mosek.iparam.mio_construct_sol, mosek.onoffkey.on); // Set status of all variables to unknown task.makesolutionstatusunknown(mosek.soltype.itg); // Assign values 1,1,0 to integer variables task.putsolutioni ( mosek.accmode.var, 0, mosek.soltype.itg, mosek.stakey.supbas, 0.0, 0.0, 0.0, 0.0); task.putsolutioni ( mosek.accmode.var, 1, mosek.soltype.itg, mosek.stakey.supbas, 2.0, 0.0, 0.0, 0.0); task.putsolutioni ( mosek.accmode.var, 2, mosek.soltype.itg, mosek.stakey.supbas, 0.0, 0.0, 0.0, 0.0); try { task.optimize(); } catch (mosek.Warning w) { Console.WriteLine("Mosek warning:"); Console.WriteLine (w.Code); Console.WriteLine (w); } task.getsolutionslice(mosek.soltype.itg, /* Basic solution. */ mosek.solitem.xx, /* Which part of solution. */ 0, /* Index of first variable. */ NUMVAR, /* Index of last variable+1 */ xx); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:{1}", j,xx[j]); } catch (mosek.Exception e) { Console.WriteLine (e.Code); Console.WriteLine (e); } if (task != null) task.Dispose (); if (env != null) env.Dispose (); } }

5.6. Problem modification and reoptimization

Often one might want to solve not just a single optimization problem, but a sequence of problem, each differing only slightly from the previous one. This section demonstrates how to modify and reoptimize an existing problem. The example we study is a simple production planning model.

5.6.1. A production planning problem

A company manufactures three types of products. Suppose the stages of manufacturing can be split into three parts, namely Assembly, Polishing and Packing. In the table below we show the time required for each stage as well as the profit associated with each product.

Product no. Assembly (minutes) Polishing (minutes) Packing (minutes) Profit ($)
0 2 3 2 1.50
1 4 2 3 2.50
2 3 3 2 3.00

With the current resources available, the company has 100,000 minutes of assembly time, 50,000 minutes of polishing time and 60,000 minutes of packing time available per year.

Now the question is how many items of each product the company should produce each year in order to maximize profit?

Denoting the number of items of each type by [[MathCmd 72]] and [[MathCmd 73]], this problem can be formulated as the linear optimization problem:

\begin{math}\nonumber{}\begin{array}{lccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} &  & \\\nonumber{}\mbox{subject to} & 2x_{0} & + & 4x_{1} & + & 3x_{2} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & \leq{} & 60000,\end{array}\end{math} (5.6.1)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2}\geq{}0.\end{math} (5.6.2)

The following code loads this problem into the optimization task.

// Since the value infinity is never used, we define // 'infinity' symbolic purposes only double infinity = 0; const int NUMCON = 3; const int NUMVAR = 3; const int NUMANZ = 9; double[] c = {1.5, 2.5, 3.0}; mosek.boundkey[] bkc = {mosek.boundkey.up, mosek.boundkey.up, mosek.boundkey.up}; double[] blc = {-infinity, -infinity, -infinity}; double[] buc = {100000, 50000, 60000}; mosek.boundkey[] bkx = {mosek.boundkey.lo, mosek.boundkey.lo, mosek.boundkey.lo}; double[] blx = {0.0, 0.0, 0.0}; double[] bux = {+infinity, +infinity, +infinity}; int[][] asub = new int[NUMVAR][]; asub[0] = new int[] {0, 1, 2}; asub[1] = new int[] {0, 1, 2}; asub[2] = new int[] {0, 1, 2}; double[][] aval = new double[NUMVAR][]; aval[0] = new double[] { 2.0, 3.0, 2.0 }; aval[1] = new double[] { 4.0, 2.0, 3.0 }; aval[2] = new double[] { 3.0, 3.0, 2.0 }; double[] xx = new double[NUMVAR]; mosek.Task task = null; mosek.Env env = null; try { // Create mosek environment. env = new mosek.Env (); // Initialize the environment. env.init (); // Create a task object linked with the environment env. task = new mosek.Task (env, NUMCON,NUMVAR); /* Give MOSEK an estimate on the size of the data to input. This is done to increase the speed of inputting data and is optional.*/ task.putmaxnumvar(NUMVAR); task.putmaxnumcon(NUMCON); task.putmaxnumanz(NUMANZ); /* Append the constraints. */ task.append(mosek.accmode.con,NUMCON); /* Append the variables. */ task.append(mosek.accmode.var,NUMVAR); /* Put C. */ task.putcfix(0.0); for(int j=0; j<NUMVAR; ++j) task.putcj(j,c[j]); /* Put constraint bounds. */ for(int i=0; i<NUMCON; ++i) task.putbound(mosek.accmode.con,i,bkc[i],blc[i],buc[i]); /* Put variable bounds. */ for(int j=0; j<NUMVAR; ++j) task.putbound(mosek.accmode.var,j,bkx[j],blx[j],bux[j]); /* Put A. */ if ( NUMCON>0 ) { for(int j=0; j<NUMVAR; ++j) task.putavec(mosek.accmode.var, j, asub[j], aval[j]); } task.putobjsense(mosek.objsense.maximize); try { task.optimize(); } catch (mosek.Warning w) { Console.WriteLine("Mosek warning:"); Console.WriteLine (w.Code); Console.WriteLine (w); } task.getsolutionslice(mosek.soltype.bas, /* Basic solution. */ mosek.solitem.xx, /* Which part of solution. */ 0, /* Index of first variable. */ NUMVAR, /* Index of last variable+1 */ xx); for(int j = 0; j < NUMVAR; ++j) Console.WriteLine ("x[{0}]:{1}", j,xx[j]);

5.6.2. Changing the A matrix

Suppose we want to change the time required for assembly of product 0 to 3 minutes. This corresponds to setting [[MathCmd 76]], which is done by calling the function mosek.Task.putaij as shown below.

task.putaij(0, 0, 3.0);

The problem now has the form:

\begin{math}\nonumber{}\begin{array}{lccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 4x_{1} & + & 3x_{2} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & \leq{} & 60000,\end{array}\end{math} (5.6.3)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2}\geq{}0.\end{math} (5.6.4)

After changing the A matrix we can find the new optimal solution by calling

mosek.Task.optimizetrm

again

5.6.3. Appending variables

We now want to add a new product with the following data:

Product no. Assembly (minutes) Polishing (minutes) Packing (minutes) Profit ($)
3 4 0 1 1.00

This corresponds to creating a new variable [[MathCmd 79]], appending a new column to the A matrix and setting a new value in the objective. We do this in the following code.

/* Append a new varaible x_3 to the problem */ task.append(mosek.accmode.var,1); /* Get index of new variable, this should be 3 */ int numvar; task.getnumvar(out numvar); /* Set bounds on new varaible */ task.putbound(mosek.accmode.var, numvar-1, mosek.boundkey.lo, 0, +infinity); /* Change objective */ task.putcj(numvar-1,1.0); /* Put new values in the A matrix */ int[] acolsub = new int[] {0, 2}; double[] acolval = new double[] {4.0, 1.0}; task.putavec(mosek.accmode.var, numvar-1, /* column index */ acolsub, acolval);

After this operation the problem looks this way:

\begin{math}\nonumber{}\begin{array}{lccccccccl}\nonumber{}\mbox{maximize} & 1.5x_{0} & + & 2.5x_{1} & + & 3.0x_{2} & + & 1.0x_{3} &  & \\\nonumber{}\mbox{subject to} & 3x_{0} & + & 4x_{1} & + & 3x_{2} & + & 4x_{3} & \leq{} & 100000,\\\nonumber{} & 3x_{0} & + & 2x_{1} & + & 3x_{2} &  &  & \leq{} & 50000,\\\nonumber{} & 2x_{0} & + & 3x_{1} & + & 2x_{2} & + & 1x_{3} & \leq{} & 60000,\end{array}\end{math} (5.6.5)

and

\begin{math}\nonumber{}x_{0},x_{1},x_{2},x_{3}\geq{}0.\end{math} (5.6.6)

5.6.4. Reoptimization

When

mosek.Task.optimizetrm

is called MOSEK will store the optimal solution internally. After a task has been modified and

mosek.Task.optimizetrm

is called again the solution will automatically be used to reduce solution time of the new problem, if possible.

In this case an optimal solution to problem (5.6.3) was found and then added a column was added to get (5.6.5). The simplex optimizer is well suited for exploiting an existing primal or dual feasible solution. Hence, the subsequent code instructs MOSEK to choose the simplex optimizer freely when optimizing.

/* Change optimizer to simplex free and reoptimize */ task.putintparam(mosek.iparam.optimizer,mosek.optimizertype.free_simplex); task.optimize();

5.6.5. Appending constraints

Now suppose we want to add a new stage to the production called “Quality control” for which 30000 minutes are available. The time requirement for this stage is shown below:

Product no. Quality control (minutes)
0 1
1 2
2 1
3 1

This corresponds to adding the constraint

\begin{math}\nonumber{}x_{0}+2x_{1}+x_{2}+x_{3}\leq{}30000\end{math} (5.6.7)

to the problem which is done in the following code:

/* Append a new constraint */ task.append(mosek.accmode.con,1); /* Get index of new constraint, this should be 4 */ int numcon; task.getnumcon(out numcon); /* Set bounds on new constraint */ task.putbound( mosek.accmode.con, numcon-1, mosek.boundkey.up, -infinity, 30000); /* Put new values in the A matrix */ int[] arowsub = new int[] {0, 1, 2, 3 }; double[] arowval = new double[] {1.0, 2.0, 1.0, 1.0}; task.putavec(mosek.accmode.con, numcon-1, /* row index */ arowsub, arowval);

5.7. Efficiency considerations

Although MOSEK is implemented to handle memory efficiently, the user may have valuable knowledge about a problem, which could be used to improve the performance of MOSEK. This section discusses some tricks and general advice that hopefully make MOSEK process your problem faster.

Avoid memory fragmentation:

MOSEK stores the optimization problem in internal data structures in the memory. Initially MOSEK will allocate structures of a certain size, and as more items are added to the problem the structures are reallocated. For large problems the same structures may be reallocated many times causing memory fragmentation. One way to avoid this is to give MOSEK an estimated size of your problem using the functions:

None of these functions change the problem, they only give hints to the eventual dimension of the problem. If the problem ends up growing larger than this, the estimates are automatically increased.

Tune the reallocation process:

It is possible to obtain information about how often MOSEK reallocates storage for the A matrix by inspecting mosek.iinfitem.sto_num_a_realloc. A large value indicates that maxnumanz has been reestimated many times and that the initial estimate should be increased.

Do not mix put- and get- functions:

For instance, the functions mosek.Task.putavec and mosek.Task.getavec. MOSEK will queue put- commands internally until a get- function is called. If every put- function call is followed by a get- function call, the queue will have to be flushed often, decreasing efficiency.

In general get- commands should not be called often during problem setup.

Use the LIFO principle when removing constraints and variables:

MOSEK can more efficiently remove constraints and variables with a high index than a small index.

An alternative to removing a constraint or a variable is to fix it at 0, and set all relevant coefficients to 0. Generally this will not have any impact on the optimization speed.

Add more constraints and variables than you need (now):

The cost of adding one constraint or one variable is about the same as adding many of them. Therefore, it may be worthwhile to add many variables instead of one. Initially fix the unused variable at zero, and then later unfix them as needed. Similarly, you can add multiple free constraints and then use them as needed.

Use one environment (env) only:

If possible share the environment (env) between several tasks. For most applications you need to create only a single env.

Do not remove basic variables:

When doing reoptimizations, instead of removing a basic variable it may be more efficient to fix the variable at zero and then remove it when the problem is reoptimized and it has left the basis. This makes it easier for MOSEK to restart the simplex optimizer.

5.7.1. API overhead

The .NET interface is a thin wrapper around a native MOSEK library. The layer between the .NET application and the native MOSEK library is made as thin as possible to minimize the overhead from function calls.

A call to a method in a MOSEK class will result in a call to a public .NET method, which in turn calls the native function, converting data and types as necessary. As data and processes in .NET are kept rigidly apart from the native code, converting data at least implies that a complete copy of the data is created, and calling of native functions (in this case) means calling into an unsafe (relative to the .NET environment) execution context. For larger problems this may mean, that fetching or inputting large chunks of data is less expensive than fetching/inputting the same data as single values.

5.8. Conventions employed in the API

5.8.1. Naming conventions for arguments

In the definition of the MOSEK .NET API a consistent naming convention has been used. This implies that whenever for example numcon is an argument in a function definition it indicates the number of constraints.

In Table 5.2 the variable names used to specify the problem parameters are listed.

.NET name .NET type Dimension Related problem
  parameter
numcon int m
numvar int n
numcone int t
numqonz int [[MathCmd 83]]
qosubi int[] numqonz [[MathCmd 83]]
qosubj int[] numqonz [[MathCmd 83]]
qoval out double numqonz [[MathCmd 83]]
c double[] numvar [[MathCmd 87]]
cfix double [[MathCmd 5]]
numqcnz int [[MathCmd 89]]
qcsubk int[] qcnz [[MathCmd 89]]
qcsubi int[] qcnz [[MathCmd 89]]
qcsubj int[] qcnz [[MathCmd 89]]
qcval out double qcnz [[MathCmd 89]]
aptrb int[] numvar [[MathCmd 94]]
aptre int[] numvar [[MathCmd 94]]
asub int[] aptre[numvar-1] [[MathCmd 94]]
aval double[] aptre[numvar-1] [[MathCmd 94]]
bkc boundkey[] numcon [[MathCmd 98]] and [[MathCmd 99]]
blc double[] numcon [[MathCmd 98]]
buc double[] numcon [[MathCmd 99]]
bkx boundkey[] numvar [[MathCmd 102]] and [[MathCmd 103]]
blx double[] numvar [[MathCmd 102]]
bux double[] numvar [[MathCmd 103]]
Table 5.2: Naming convention used in MOSEK

The relation between the variable names and the problem parameters is as follows:

  • The quadratic terms in the objective:

    \begin{math}\nonumber{}q_{{\mathtt{qosubi[t]},\mathtt{qosubj[t]}}}^{o}=\mathtt{qoval[t]},~t=0,\ldots ,\mathtt{numqonz}-1.\end{math} (5.8.1)
  • The linear terms in the objective:

    \begin{math}\nonumber{}c_{j}=\mathtt{c[j]},~j=0,\ldots ,\mathtt{numvar}-1\end{math} (5.8.2)
  • The fixed term in the objective:

    \begin{math}\nonumber{}c^{f}=\mathtt{cfix}.\end{math} (5.8.3)
  • The quadratic terms in the constraints:

    \begin{math}\nonumber{}q_{{\mathtt{qcsubi[t]},\mathtt{qcsubj[t]}}}^{\mathtt{qcsubk[t]}}=\mathtt{qcval[t]},~t=0,\ldots ,\mathtt{numqcnz}-1.\end{math} (5.8.4)
  • The linear terms in the constraints:

    \begin{math}\nonumber{}\begin{array}{rl}\nonumber{}a_{{\mathtt{asub[t],j}}}=\mathtt{aval[t]}, & t=\mathtt{ptrb[j]},\ldots ,\mathtt{ptre[j]}-1,\\\nonumber{} & j=0,\ldots ,\mathtt{numvar}-1.\end{array}\end{math} (5.8.5)
  • The bounds on the constraints are specified using the variables bkc, blc, and buc. The components of the integer array bkc specify the bound type according to Table 5.3.

    Symbolic constant Lower bound Upper bound
    mosek.boundkey.fx finite identical to the lower bound
    mosek.boundkey.fr minus infinity plus infinity
    mosek.boundkey.lo finite plus infinity
    mosek.boundkey.ra finite finite
    mosek.boundkey.up minus infinity finite
    Table 5.3: Interpretation of the bound keys.

    For instance bkc[2]=mosek.boundkey.lo means that [[MathCmd 111]] and [[MathCmd 112]]. Finally, the numerical values of the bounds are given by

    \begin{math}\nonumber{}l_{k}^{c}=\mathtt{blc[k]},~k=0,\ldots ,\mathtt{numcon}-1\end{math} (5.8.6)

    and

    \begin{math}\nonumber{}u_{k}^{c}=\mathtt{buc[k]},~k=0,\ldots ,\mathtt{numcon}-1.\end{math} (5.8.7)
  • The bounds on the variables are specified using the variables bkx, blx, and bux. The components in the integer array bkx specify the bound type according to Table 5.3. The numerical values for the lower bounds on the variables are given by

    \begin{math}\nonumber{}l_{j}^{x}=\mathtt{blx[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.8.8)

    The numerical values for the upper bounds on the variables are given by

    \begin{math}\nonumber{}u_{j}^{x}=\mathtt{bux[j]},~j=0,\ldots ,\mathtt{numvar}-1.\end{math} (5.8.9)

5.8.1.1. Bounds

A bound on a variable or on a constraint in MOSEK consists of a bound key, as defined in Table 5.3, a lower bound value and an upper bound value. Even if a variable or constraint is bounded only from below, e.g. x0, both bounds are inputted or extracted; the value inputted as upper bound for (x0) is ignored.

5.8.2. Vector formats

Three different vector formats are used in the MOSEK API:

Full vector:

This is simply an array where the first element corresponds to the first item, the second element to the second item etc. For example to get the linear coefficients of the objective in task, one would write

double[] c = new double[numvar]; task.getc(c);

where numvar is the number of variables in the problem.

Vector slice:

A vector slice is a range of values. For example, to get the bounds associated constraint 3 through 10 (both inclusive) one would write

double[] upper_bound = new double[8]; double[] lower_bound = new double[8]; mosek.boundkey[] bound_key = new mosek.boundkey[8]; task.getboundslice(mosek.accmode.con, 2,10, bound_key,lower_bound,upper_bound);

Please note that items in MOSEK are numbered from 0, so that the index of the first item is 0, and the index of the n'th item is n-1.

Sparse vector:

A sparse vector is given as an array of indexes and an array of values. For example, to input a set of bounds associated with constraints number 1, 6, 3, and 9, one might write

int[] bound_index = { 1, 6, 3, 9 }; mosek.boundkey[] bound_key = { mosek.boundkey.fr, mosek.boundkey.lo, mosek.boundkey.up, mosek.boundkey.fx }; double[] lower_bound = { 0.0, -10.0, 0.0, 5.0 }; double[] upper_bound = { 0.0, 0.0, 6.0, 5.0 }; task.putboundlist(mosek.accmode.con, bound_index, bound_key,lower_bound,upper_bound);

Note that the list of indexes need not be ordered.

5.8.3. Matrix formats

The coefficient matrices in a problem are inputted and extracted in a sparse format, either as complete or a partial matrices. Basically there are two different formats for this.

5.8.3.1. Unordered triplets

In unordered triplet format each entry is defined as a row index, a column index and a coefficient. For example, to input the A matrix coefficients for [[MathCmd 117]], [[MathCmd 118]], and [[MathCmd 119]], one would write as follows:

int[] subi = { 1, 3, 5 }; int[] subj = { 2, 3, 4 }; double[] cof = { 1.1, 4.3, 0.2 }; task.putaijlist(subi,subj,cof);

Please note that in some cases (like mosek.Task.putaijlist) only the specified indexes remain modified — all other are unchanged. In other cases (such as mosek.Task.putqconk) the triplet format is used to modify all entries — entries that are not specified are set to 0.

5.8.3.2. Row or column ordered sparse matrix

In a sparse matrix format only the non-zero entries of the matrix are stored. MOSEK uses a sparse matrix format ordered either by rows or columns. In the column-wise format the position of the non-zeros are given as a list of row indexes. In the row-wise format the position of the non-zeros are given as a list of column indexes. Values of the non-zero entries are given in column or row order.

A sparse matrix in column ordered format consists of:

asub:

List of row indexes.

aval:

List of non-zero entries of A ordered by columns.

ptrb:

Where ptrb[j] is the position of the first value/index in aval / asub for column j.

ptre:

Where ptre[j] is the position of the last value/index plus one in aval / asub for column j.

The values of a matrix A with numcol columns are assigned so that for

\begin{displaymath}\nonumber{}j=0,\ldots ,\mathtt{numcol}-1.\end{displaymath}

We define

\begin{math}\nonumber{}\begin{array}{rcl}\nonumber{}a_{{\mathtt{asub}[k],j}}=\mathtt{aval}[k],\quad{}k=\mathtt{ptrb}[j],\ldots ,\mathtt{ptre}[j]-1.\end{array}\end{math} (5.8.10)

Figure 5.1: The matrix A (5.8.11) represented in column ordered sparse matrix format.

As an example consider the matrix

\begin{math}\nonumber{}A=\left[\begin{array}{ccccc}\nonumber{}1.1 &  & 1.3 & 1.4 & \\\nonumber{} & 2.2 &  &  & 2.5\\\nonumber{}3.1 &  &  & 3.4 & \\\nonumber{} &  & 4.4 &  &\end{array}\right].\end{math} (5.8.11)

which can be represented in the column ordered sparse matrix format as

\begin{displaymath}\nonumber{}\begin{array}{lcl}\nonumber{}\mathtt{ptrb} & = & [0,2,3,5,7],\\\nonumber{}\mathtt{ptre} & = & [2,3,5,7,8],\\\nonumber{}\mathtt{asub} & = & [0,2,1,0,3,0,2,1],\\\nonumber{}\mathtt{aval} & = & [1.1,3.1,2.2,1.3,4.4,1.4,3.4,2.5].\end{array}\end{displaymath}

Fig. 5.1 illustrates how the matrix A (5.8.11) is represented in column ordered sparse matrix format.

5.8.3.3. Row ordered sparse matrix

The matrix A (5.8.11) can also be represented in the row ordered sparse matrix format as:

\begin{displaymath}\nonumber{}\begin{array}{lcl}\nonumber{}\mathtt{ptrb} & = & [0,3,5,7],\\\nonumber{}\mathtt{ptre} & = & [3,5,7,8],\\\nonumber{}\mathtt{asub} & = & [0,2,3,1,4,0,3,2],\\\nonumber{}\mathtt{aval} & = & [1.1,1.3,1.4,2.2,2.5,3.1,3.4,4.4].\end{array}\end{displaymath}

5.9. The license system

By default a license token is checked out when mosek.Task.optimizetrm is first called and is returned when the MOSEK environment is deleted. Calling mosek.Task.optimizetrm from different threads using the same MOSEK environment only consumes one license token.

To change the license systems behavior to returning the license token after each call to mosek.Task.optimizetrm set the parameter mosek.iparam.cache_license to mosek.onoffkey.off. Please note that there is a small overhead associated with setting this parameter, since checking out a license token from the license server can take a small amount of time.

Additionally license checkout and checkin can be controlled manually with the functions mosek.Env.checkinlicense and mosek.Env.checkoutlicense.

5.9.1. Waiting for a free license

By default an error will be returned if no license token is available. By setting the parameter mosek.iparam.license_wait MOSEK can be instructed to wait until a license token is available.

Wed Feb 29 16:04:36 2012