# Inverse problem

### From Glossary

Given a point in the decision space, find parameter values that render it optimal. For example, suppose we have the LP: . Let be a basis from , and we ask for for which this basis is optimal. This has a simple solution. Let and partition and conformally, so and Then, the set of for which the associated basic solution is optimal is the following cone:

A more difficult inverse problem is when there is some target value for the parameters. We might, for example, fix and seek to minimize subject to , where is a target value for .

The problem can be combinatorial. We might want to minimize for some norm where 's are costs on the arcs or edges of a network. The solution at hand might be a spanning tree or a TSP tour. We might also impose constraints on directly, such as simple bounds.

The general inverse problem may thus be stated:

where is the target, is some norm, is a non-empty, closed set, which generally does not contain , and is the feasible region for , given .

There are variations, such as when only a partial solution is given. An example is a partial TSP tour or some partial scheduling of jobs, and we ask for cost values for which there is an optimal

solution that contains these subsets.