Some properties of solution sets to nonconvex quadratic programming problems

• Published in: Optimization Vol. 56; no. 3; pp. 369 - 383
• Main Author: Phu, H. X.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 01.06.2007; Taylor & Francis LLC
• Subjects: Global minimizer set; Kuhn-Tucker point; Local minimizer set; Mathematical problems; Mathematics Subject Classification 2000; Primary: 90C20; Quadratic programming; Quadratic programming problem; Solution set; Stationary point set; Studies
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331930600819597

This article deals with some properties of the global minimizer set G Q , the local minimizer set L Q , and the stationary point set S Q to the quadratic programming problem (Q) of minimizing the function f(x)=(1/2)x T Ax+b T x on the polyhedron , where , i∈I={1,2,..., m}. In particular, we investigate the intersection of these solution sets with faces and pseudofaces , where J⊂I. Some selected results are the following. If G Q ∩D J ≠ =∅ then G Q  ∩ D J and are relatively affine in the following sense: G Q ∩D J =aff(G Q ∩D J )∩D J and . If L Q ∩D J ≠ =∅ then L Q  ∩ D J is open relative to aff(L Q ∩D J )∩D J , is open relative to , and L Q  ∩ D J and are convex. If G Q ∩D J ≠ =∅ then each stationary point (in particular, each local minimizer) in is a global minimizer. If x 0 ∈L Q ∩D J , , and x 0 ≠ = x 1 , then [x 0 ,x 1 )⊂L Q ∩D J ⊂L Q . Let and denote the maximal number of nonempty faces and the maximal cardinality of an antichain of nonempty faces of a polyhedron defined as intersection of m closed halfspaces in . Then G Q (or L Q , or S Q , respectively) contains a segment connecting two distinct points if it possesses more than (or , or , respectively) different points.