Some properties of solution sets to nonconvex quadratic programming problems - Phu, H. X. (Optimization, 2007)
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.