Linear Programming - satisfaction only at least one constraint

[wavingerwin]

Linear Programming - satisfaction of only at least one constraint

Hi

Is there a form of relaxation/modification of an LP of the form
\text{min }\;\;f^\mathsf{T}x\\\mathbf{A}x\leq b
such that if only anyone of the constraints is satisfied, then the solution $x$ is regarded as feasible?

Here $\mathbf{A}x\leq b$ represents a row of linear constraints.

Gratitudes in advance.

 

[Erland]

I don't really understand what you mean. If we have a constraint with only one inequality instead of a system of more than one, then we have a new LP-problem.

 

[wavingerwin]

If we have a constraint with only one inequality instead of a system of more than one, then we have a new LP-problem.

The LP I am concerned with has a number of inequalities. However, I need to only have at least one of them satisfied. This can be any combination of the inequalities, not a particular one.

Say the LP has 5 inequalities to satisfy. I want to have that satisfying at least one of the 5 means that the solution is feasible, as opposed to having all of the 5 satisfied, like in a standard LP. However, it can be any combination of the constraint, for example [1 0 0 0 0], [0 0 1 1 0], and many others, where 1 and 0 correspond to each of the inequality being satisfied or unsatisfied respectively.

 

[Erland]

I see... If there are n inequalities, you could solve n LP problems, one for each inequality, and choose the one with smallest value.

 

[Stephen Tashi]

There is the concept of whether a constraint is "tight" at an optimal solution. If you find an optimal solution subject to n inequality constraints then to improvie on that solution, look at dropping the constraits that are satisified as exact equalities.