Continuity and Directional Differentiability of the Value Function in Parametric Quadratically Constrained Nonconvex Qua
- PDF / 505,622 Bytes
- 26 Pages / 439.642 x 666.49 pts Page_size
- 77 Downloads / 190 Views
Continuity and Directional Differentiability of the Value Function in Parametric Quadratically Constrained Nonconvex Quadratic Programs Tran Van Nghi1 · Nguyen Nang Tam1
Received: 10 January 2016 / Revised: 11 March 2016 / Accepted: 31 March 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract The aim of this paper is to investigate the continuity and the directional differentiability of the value function in quadratically constrained nonconvex quadratic programming problem. Our result can be used in some cases where the existing results on differential stability in nonlinear programming (applied to quadratic programming) cannot be used. Keywords Nonconvex quadratic programming · Quadratically constrained · Value function · Continuity · Directional differentiability Mathematics Subject Classification (2010) 90C20 · 90C30 · 90C31
1 Introduction and Preliminaries Properties of the value function play an important role in parametric quadratic programming (QP for brevity) problems. Various aspects of the value function in QP problems under linear constraints have been studied in [9] and the references therein. Since QP is a class of nonlinear optimization problems, the results in nonlinear optimization can be applied to convex and nonconvex QP problems. In [1], A. Auslender and R. Cominetti considered first- and second-order sensitivity analysis of nonlinear programs under directional constraint qualification conditions. In [13], L. Minchenko and A. Tarakanov discussed directional derivatives of value functions under the calmness of solution mapping assumption. In [6, 10, 12, 14– 16] and the references given there, some similar topics were investigated. A survey of recent results on stability of nonlinear programming problems was given by J. F. Bonnans and
Tran Van Nghi
[email protected] Nguyen Nang Tam [email protected] 1
Hanoi Pedagogical University 2, Hanoi, Vietnam
T. V. Nghi and N. N. Tam
A. Shapiro [3], where many interesting results can be applied for QP. However, the special structure of QP problems allows one to have deeper and more comprehensive results on differential stability in QP. This paper deals with the continuity and directional differentiability of value function in nonconvex quadratic programming under quadratic constraints. Among our proposed assumptions, there are some weaker than those used in the cited works (applied for QP). The rest of the paper is organized as follows. Some preliminaries are given in Section 1. In Section 2, continuity of value function is studied. Sections 3 and 4 establish the sufficient condition such that there exist first- and second-order directional derivatives of value function. Finally, in Section 5, we present some examples to illustrate those results. The following notation will be adopted. Let Rn be the n-dimensional Euclidean space equipped with the standard scalar product and the Euclidean norm, Rn×n be the space of S real symmetric (n× n)–matrices equipped with
Data Loading...
