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# problem.repairWeightedInfeasBounds

Purpose
An extended version of problem.repairWeightedInfeas that allows for bounding the level of relaxation allowed.
Topic areas
Infeasibility, Optimizer
Synopsis
`status = problem.repairWeightedInfeasBounds(lepref, gepref, lbpref, ubpref, lerelax, gerelax, lbrelax, ubrelax, phase2, delta, flags)`
Arguments

lepref: list[float] | Array of size ROWS containing the preferences for relaxing the less or equal side of row. For the console use -lrp value.
gepref: list[float] | Array of size ROWS containing the preferences for relaxing the greater or equal side of a row. For the console use -grp value.
lbpref: list[float] | Array of size COLS containing the preferences for relaxing lower bounds. For the console use -lbp value.
ubpref: list[float] | Array of size COLS containing preferences for relaxing upper bounds. For the console use -ubp value.
lerelax: list[float] | Array of size ROWS containing the upper bounds on the amount the less or equal side of a row can be relaxed. For the console use -lrb value.
gerelax: list[float] | Array of size ROWS containing the upper bounds on the amount the greater or equal side of a row can be relaxed. For the console use -grb value.
lbrelax: list[float] | Array of size COLS containing the upper bounds on the amount the lower bounds can be relaxed. For the console use -lbb value.
ubrelax: list[float] | Array of size COLS containing the upper bounds on the amount the upper bounds can be relaxed. For the console use -ubb value.
phase2: str | 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 analysis of the problem;
a | if the relaxation is infeasible, generate all irreducible infeasible subsets for the analysis of the problem.
delta: float | The relaxation multiplier in the second phase -1.
flags: str | Specifies flags to be passed to problem.optimize.

Return value

status: int | The status after the relaxation:
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.

Further information
1. This function applies to all problem types.
2. 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.
3. 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.
4. A negative preference indicates that a quadratic penalty cost should be applied. This can specified on a per constraint side or bound basis.
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. The weighted sum of violations will be available in PENALTYVALUE.
6. Note that on some problems, slight modifications of delta may affect the value of the original objective drastically.
7. The default value for delta in the console is 0.001.
8. Note that because of their special associated modeling properties, binary and semi-continuous variables are not relaxed.
9. Given any row j with preferences lrp=lepref[j] and grp=gepref[j], or variable i with bound preferences ubp=ubpref[i] and lbp=lbpref[i], the following rules are applied while introducing the auxiliary variables:

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

10. Only positive bounds are applied; a zero or negative bound is ignored and the amount of relaxation allowed for the corresponding row or bound is not limited. The effect of a zero bound on a row or bound would be equivalent with not relaxing it, and can be achieved by setting its preference array value to zero instead, or not including it in the preference arrays.
11. If an irreducible infeasible set (IIS) has been identified, the generated IIS(s) are accessible through the IIS retrieval functions, see NUMIIS and problem.getIISData.
Related topics
problem.repairInfeas, problem.optimize, the Optimizer Reference Manual.
Parent Topic Xpress problem object methods

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