Canonical Dual Solutions to Quadratic Optimization over One Quadratic Constraint

Abstract

Dedication: This paper is dedicated to Professor Jiye Han on the occasion of his 80th birthday.A quadratic optimization problem with one nonconvex quadratic constraint is studied using the canonical dual approach. Under the dual Slater's condition, we show that the canonical dual has a smooth concave objective function over a convex feasible domain, and this dual has a finite supremum unless the original quadratic optimization problem is infeasible. This supremum, when it exists, always equals to the minimum value of the primal problem. Moreover, a global minimizer of the primal problem can be provided by a dual-to-primal conversion plus a "boundarification" technique. Application to solving * Corresponding author 1540007-1 Asia Pac. J. Oper. Res. 2015.32. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WESTERN ONTARIO on 04/13/15. For personal use only. W. Xing et al.a quadratic programming problem over a ball is included and an error bound estimation is provided.

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part I

“…Note that there exists a list of research work on solving (QP1QC). In the rest of this paper, we extend the results of [7,8] to study the problem (10) in an explicit manner.…”

“…; k: In the former case (27), wðs 0 Þ ¼ lim s!s þ 0 wðsÞ is not an optimal solution since it is located right at the center of the sphere. The boundarification technique developed in [7,8] may move wðs 0 Þ from the center to the boundary of the sphere along a null space direction of Diagða 1 ; a 2 ; . .…”

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

“…Email: rsheu@mail.ncku.edu.tw where A is an n × n real symmetric matrix, B 6 ¼ 0 is an m × n real matrix, c 2 R m , d 2 R and f 2 R n . By introducing a continuous variable transformation j = 1 2 k Bx À ck 2 À d, the double well potential problem (DWP) can be transformed into the following equivalent quadratic program over one nonhomogeneous quadratic constraint (QP1QC) [2,3]:…”

RETRACTED: Global solutions to spherically constrained quadratic minimization via canonical duality theory

“…In mathematical programming, the problem (P) is known as a trust region subproblem, which arises in trust region methods [4,5]. In literatures, two similar problems are also discussed: in [6][7][8] the convexity of the quadratic constraint is removed, while in [9,10] the constraint is replaced by a two-sided (lower and upper bounded) quadratic constraint. Although the function P(x) may be nonconvex, it is proved that the problem (P) possesses the hidden convexity, i.e.…”