problem.repairweightedinfeas

Purpose

By relaxing a set of selected constraints and bounds of an infeasible problem, it attempts to identify a 'solution' that violates the selected set of constraints and bounds minimally, while satisfying all other constraints and bounds. Among such solution candidates, it selects one that is optimal regarding to the original objective function. Similar to repairinfeas, the returned value is as follows:

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.repairweightedinfeas(lrp_array, grp_array, lbp_array, ubp_array, phase2, delta, optflags)

Arguments

lrp_array

Array of size ROWS containing the preferences for relaxing the less or equal side of row.

grp_array

Array of size ROWS containing the preferences for relaxing the greater or equal side of a row.

lbp_array

Array of size COLS containing the preferences for relaxing lower bounds.

ubp_array

Array of size COLS containing preferences for relaxing upper bounds.

phase2

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.

delta

The relaxation multiplier in the second phase -1.

optflags

Specifies flags to be passed to the Optimizer.

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. The higher the preference, the more willing the modeller is to relax a given row or bound.

3. The weight of each infeasibility breaker in the objective minimizing the violations is 1/p, where p is the preference associated with the infeasibility breaker. Thus the higher the preference is, the lower a penalty is associated with the infeasibility breaker while minimizing the violations.

4. 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.

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

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

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


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

8. 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.repairweightedinfeas

problem.repairweightedinfeas