The Inverse Problems of some Mathematical Programming Problems

TL;DR

This paper investigates the computational complexity of inverse problems related to certain non-convex quadratic programming and linear complementarity problems, revealing polynomial solvability in some cases and NP-hardness in others, and addressing an open question in complexity theory.

Contribution

It demonstrates that the inverse problem of finding a KKT point in non-convex quadratic programming is polynomial, and characterizes the complexity of inverse linear complementarity problems, solving an open problem.

Findings

Inverse KKT problem for non-convex quadratic programming is polynomial.

Inverse linear complementarity problems are polynomial in some cases.

Inverse NP-hard problems relate to the equality of NP and CoNP.

Abstract

The non-convex quadratic orogramming problem and the non-monotone linear complementarity problem are NP-complete problems. In this paper we first show taht the inverse problem of determinning a KKT point of the non-convex quadratic programming problem is polynomial. We then show that the inverse problems of non-monotone linear complementarity problem are polynomial solvable in some cases, and in another case is NP-hard. Therefore we solve an open question raised by Heuberger on inverse NP-hard problems and prove that CoNP=NP.