problem.repairinfeas

Purpose

Provides a simplified interface for problem.repairweightedinfeas. The returned value is as follows:

0: relaxed optimum found;
1: relaxed problem is infeasible;
2: relaxed problem is unbounded;
3: solution of the relaxed problem regarding the original objective is nonoptimal;
4: error (when return code is nonzero);
5: numerical instability;
6: analysis of an infeasible relaxation was performed, but the relaxation is feasible.

Synopsis

status_code = problem.repairinfeas(pflags, oflags, gflags, lrp, grp, lbp, ubp, delta)

Arguments

pflags

The type of penalties created from the preferences:

c	each penalty is the reciprocal of the preference (default);
s	the penalties are placed in the scaled problem.

oflags

Controls the second phase of optimization:

o	use the objective sense of the original problem (default);
x	maximize the relaxed problem using the original objective;
f	skip optimization regarding the original objective;
n	minimize the relaxed problem using the original objective;
i	if the relaxation is infeasible, generate an irreducible infeasible subset for the analys of the problem;
a	if the relaxation is infeasible, generate all irreducible infeasible subsets for the analys of the problem.

gflags

Specifies if the global search should be done:

g	do the global search (default);
l	solve as a linear model ignoring the discreteness of variables.

lrp

Preference for relaxing the less or equal side of row.

grp

Preference for relaxing the greater or equal side of a row.

lbp

Preferences for relaxing lower bounds.

ubp

Preferences for relaxing upper bounds.

delta

The relaxation multiplier in the second phase -1. A positive value means a relative relaxation by multiplying the first phase objective with ( delta-1), while a negative value means an absolute relaxation, by adding abs(delta) to the first phase objective.

Further information

1. A row or bound is relaxed by introducing a new nonnegative variable that will contain the infeasibility of the row or bound. Suppose for example that row a ^Tx = b is relaxed from below. Then a new variable (infeasibility breaker) s>=0 is added to the row, which becomes a ^Tx +s = b. Observe that a ^Tx may now take smaller values than b. To minimize such violations, the weighted sum of these new variables is minimized.

2. A preference of 0 results in the row or bound not being relaxed.

3. A negative preference indicates that a quadratic penalty cost should be applied. This can specified on a per constraint side or bound basis.

4. Note that the set of preferences are scaling independent.

5. If a feasible solution is identified for the relaxed problem, with a sum of violations p, then the sum of violations is restricted to be no greater than ( 1+delta) p, and the problem is optimized with respect to the original objective function. A nonzero delta increases the freedom of the original problem.

6. Note that on some problems, slight modifications of delta may affect the value of the original objective drastically.

7. Note that because of their special associated modeling properties, binary and semi-continuous variables are not relaxed.

8. The default algorithm for the first phase is the simplex algorithm, since the primal problem can be efficiently warm started in case of the extended problem. These may be altered by setting the value of control DEFAULTALG.

9. If pflags is set such that each penalty is the reciprocal of the preference, the following rules are applied while introducing the auxiliary variables:


Preference	Affects	Relaxation	Cost if pref.>0	Cost if pref.<0
lrp	= rows	a^Tx - aux_var = b	1/lrp*aux_var	1/lrp*aux_var²
lrp	<= rows	a^Tx - aux_var <= b	1/lrp*aux_var	1/lrp*aux_var²
grp	= rows	a^Tx + aux_var = b	1/grp*aux_var	1/grp*aux_var²
grp	>= rows	a^Tx + aux_var >= b	1/grp*aux_var	1/grp*aux_var²
ubp	upper bounds	x_i - aux_var <= u	1/ubp*aux_var	1/ubp*aux_var²
lbp	lower bounds	x_i + aux_var >= l	1/lbp*aux_var	1/lbp*aux_var²

10. If an irreducible infeasible set (IIS) has been identified, the generated IIS(s) are accesible through the IIS retrieval functions, see NUMIIS and problem.getiisdata.

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problem.repairinfeas

problem.repairinfeas