Theoretical approach for the calculation of a feasible point of the special non-convex set described by a polyhedron and a quadratic function

Abstract

Let M be a set described as the intersection of a polyhedron and the set . In this work a method is proposed for detecting if , and in this case it will approach to a point . Roughly speaking, we define a parametric optimization problem P(t), adapted to the particular structure of M such that its solvability at t = 1 is equivalent to . Moreover x is a feasible point of P(1). The main part of the article is to prove that under generic hypothesis P(t) will be regular in the sense of Jongen–Jonker and Twilt and so the strategy of path following and jumps can be applied to P(t).

Keywords:

• parametric optimization
• generalized critical points
• Jongen-Jonker-Twilt regularity
• embedding methods

Acknowledgements

I would like to thank Prof. Dr. Jurgen Guddat and Prof. Dr. Sira Allende for their support and their valuable suggestions.