13. Feasibility repair


Section 12.2.2 discusses how MOSEK treats infeasible problems. In particular, it is discussed which information MOSEK returns when a problem is infeasible and how this information can be used to pinpoint the elements causing the infeasibility.

In this section we will discuss a method for repairing a primal infeasible problem by relaxing the constraints in a controlled way. For the sake of simplicity we discuss the method in the context of linear optimization. MOSEK can also repair infeasibilities in quadratic and conic optimization problems possibly having integer constrained variables. Please note that infeasibilities in nonlinear optimization problems can't be repaired using the method described below.

13.1. The main idea

Consider the linear optimization problem with m constraints and n variables

\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},\end{array}\end{math} (13.1.1)

which we assume is infeasible. Moreover, we assume that

\begin{math}\nonumber{}(l^{c})_{i}\leq{}(u^{c})_{i},~\forall i\end{math} (13.1.2)

and

\begin{math}\nonumber{}(l^{x})_{j}\leq{}(u^{x})_{j},~\forall j\end{math} (13.1.3)

because otherwise the problem (13.1.1) is trivially infeasible.

One way of making the problem feasible is to reduce the lower bounds and increase the upper bounds. If the change is sufficiently large the problem becomes feasible.

One obvious question is: What is the smallest change to the bounds that will make the problem feasible?

We associate a weight with each bound:

Now, the problem

\begin{math}\nonumber{}\begin{array}{lccccl}\nonumber{}\mbox{minimize} &  &  & p &  & \\\nonumber{}\mbox{subject to} & l^{c} & \leq{} & Ax+v_{l}^{c}-v_{u}^{c} & \leq{} & u^{c},\\\nonumber{} & l^{x} & \leq{} & x+v_{l}^{x}-v_{u}^{x} & \leq{} & u^{x},\\\nonumber{} &  &  & (w_{l}^{c})^{T}v_{l}^{c}+(w_{u}^{c})^{T}v_{u}^{c}+(w_{l}^{x})^{T}v_{l}^{x}+(w_{u}^{x})^{T}v_{u}^{x}-p & \leq{} & 0,\\\nonumber{} &  &  & v_{l}^{c},v_{u}^{c},v_{l}^{x},v_{u}^{x}\geq{}0 &  &\end{array}\end{math} (13.1.4)

minimizes the weighted sum of changes to the bounds that makes the problem feasible. The variables [[MathCmd 401]], [[MathCmd 402]], [[MathCmd 403]] and [[MathCmd 402]] are elasticity variables because they allow a constraint to be violated and hence add some elasticity to the problem. For instance, the elasticity variable [[MathCmd 401]] shows how much the lower bound [[MathCmd 406]] should be relaxed to make the problem feasible. Since p is minimized and

\begin{math}\nonumber{}(w_{l}^{c})^{T}v_{l}^{c}+(w_{u}^{c})^{T}v_{u}^{c}+(w_{l}^{x})^{T}v_{l}^{x}+(w_{u}^{x})^{T}v_{u}^{x}-p\leq{}0,\end{math} (13.1.5)

a large [[MathCmd 408]] tends to imply that the elasticity variable [[MathCmd 401]] will be small in an optimal solution.

The reader may want to verify that the problem (13.1.4) is always feasible given the assumptions (13.1.2) and (13.1.3).

Please note that if a weight is negative then the resulting problem (13.1.4) is unbounded.

The weights [[MathCmd 410]], [[MathCmd 411]], [[MathCmd 412]], and [[MathCmd 413]] can be regarded as a costs (penalties) for violating the associated constraints. Thus a higher weight implies that higher priority is given to the satisfaction of the associated constraint.

The main idea can now be presented as follows. If you have an infeasible problem, then form the problem (13.1.4) and optimize it. Next inspect the optimal solution [[MathCmd 414]], and [[MathCmd 415]] to problem (13.1.4). This solution provides a suggested relaxation of the bounds that will make the problem feasible.

Assume that [[MathCmd 416]] is an optimal objective value to (13.1.4). An extension of the idea presented above is to solve the problem

\begin{math}\nonumber{}\begin{array}{lccccl}\nonumber{}\mbox{minimize} &  &  & c^{T}x &  & \\\nonumber{}\mbox{subject to} & l^{c} & \leq{} & Ax+v_{l}^{c}-v_{u}^{c} & \leq{} & u^{c},\\\nonumber{} & l^{x} & \leq{} & x+v_{l}^{x}-v_{u}^{x} & \leq{} & u^{x},\\\nonumber{} &  &  & (w_{l}^{c})^{T}v_{l}^{c}+(w_{u}^{c})^{T}v_{u}^{c}+(w_{l}^{x})^{T}v_{l}^{x}+(w_{u}^{x})^{T}v_{u}^{x}-p & \leq{} & 0,\\\nonumber{} &  &  & p & = & p^{*},\\\nonumber{} &  &  & v_{l}^{c},v_{u}^{c},v_{l}^{x},v_{u}^{x}\geq{}0 &  &\end{array}\end{math} (13.1.6)

which minimizes the true objective while making sure that total weighted violations of the bounds is minimal, i.e. equals to [[MathCmd 416]].

13.2. Feasibility repair in MOSEK

MOSEK includes functionality that help you construct the problem (13.1.4) simply by passing a set of weights to MOSEK. This can be used for linear, quadratic, and conic optimization problems, possibly having integer constrained variables.

13.2.1. Usage of negative weights

As the problem (13.1.4) is presented it does not make sense to use negative weights since that makes the problem unbounded. Therefore, if the value of a weight is negative MOSEK fixes the associated elasticity variable to zero, e.g. if

\begin{displaymath}\nonumber{}(w_{l}^{c})_{i}<0\end{displaymath}

then MOSEK imposes the bound

\begin{displaymath}\nonumber{}(v_{l}^{c})_{i}\leq{}0.\end{displaymath}

This implies that the lower bound on the ith constraint will not be violated. (Clearly, this could also imply that the problem is infeasible so negative weight should be used with care). Associating a negative weights with a constraint tells MOSEK that the constraint should not be relaxed.

13.2.2. Automatical naming

MOSEK can automatically create a new problem of the form (13.1.4) starting from an existing problem by adding the elasticity variables and the extra constraints. Specificly, the variables [[MathCmd 421]], [[MathCmd 422]], [[MathCmd 423]], [[MathCmd 424]], and p are appended to existing variable vector x in their natural order. Moreover, the constraint (13.1.5) is appended to the constraints.

The new variables and constraints are automatically given names as follows:

  • The names of the variables [[MathCmd 401]] and [[MathCmd 402]] are constructed from the name of the ith constraint. For instance, if the 9th original constraint is named c9, then by default [[MathCmd 427]] and [[MathCmd 428]] are given the names LO*c9 and UP*c9 respectively. If necessary, the character “*” can be replaced by a different string by changing the
    MSK_SPAR_FEASREPAIR_NAME_SEPARATOR
    parameter.
  • The additional constraints

    \begin{displaymath}\nonumber{}l^{x}\leq{}x+v_{l}^{x}-v_{u}^{x}\leq{}u^{x}\end{displaymath}

    are given names as follows. There is exactly one constraint per variable in the original problem, and thus the ith of these constraints is named after the ith variable in the original problem. For instance, if the first original variable is named “x0”, then the first of the above constraints is named “MSK-x1”. If necessary, the prefix “MSK-” can be replaced by a different string by changing the
    MSK_SPAR_FEASREPAIR_NAME_PREFIX
    parameter.

  • The variable p is by default given the name WSUMVIOLVAR, and the constraint (13.1.5) is given the name WSUMVIOLCON.

    The substring “WSUMVIOL” can be replaced by a different string by changing the
    MSK_SPAR_FEASREPAIR_NAME_WSUMVIOL
    parameter.

13.2.3. An example

Consider the example linear optimization

\begin{math}\nonumber{}\begin{array}{lccccc}\nonumber{}\mbox{minimize} & -10x_{1} &  & -9x_{2}, &  & \\\nonumber{}\mbox{subject to} & 7/10x_{1} & + & 1x_{2} & \leq{} & 630,\\\nonumber{} & 1/2x_{1} & + & 5/6x_{2} & \leq{} & 600,\\\nonumber{} & 1x_{1} & + & 2/3x_{2} & \leq{} & 708,\\\nonumber{} & 1/10x_{1} & + & 1/4x_{2} & \leq{} & 135,\\\nonumber{} & x_{1}, &  & x_{2} & \geq{} & 0.\\\nonumber{} &  & x_{2}\geq{}650 &  &  &\end{array}\end{math} (13.2.1)

This is an infeasible problem. Now suppose we wish to use MOSEK to suggest a modification to the bounds that makes the problem feasible.

The command

  mosek -d MSK_IPAR_FEASREPAIR_OPTIMIZE
  MSK_FEASREPAIR_OPTIMIZE_PENALTY -d
  MSK_IPAR_OPF_WRITE_SOLUTIONS MSK_ON  feasrepair.lp
  -infrepo minv.opf

writes the problem (13.1.4) and it's solution to an OPF formatted file. In this case the file minv.opf.

The parameter

MSK_IPAR_FEASREPAIR_OPTIMIZE

controls whether the function returns the problem (13.1.4) or the problem (13.1.6). In the case

MSK_IPAR_FEASREPAIR_OPTIMIZE

is equal to

MSK_FEASREPAIR_OPTIMIZE_NONE

then (13.1.4) is returned, but the problem is not solved. For MSK_FEASREPAIR_OPTIMIZE_PENALTY the problem (13.1.4) is returned and solved. Finally for MSK_FEASREPAIR_OPTIMIZE_COMBINED (13.1.6) is returned and solved.

Wed Feb 29 16:20:49 2012