TY - JOUR

T1 - On Extensions of the Frank-Wolfe Theorems

AU - Luo, Zhi Quan

AU - Zhang, Shuzhong

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1999

Y1 - 1999

N2 - In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.

AB - In this paper we consider optimization problems defined by a quadratic objective function and a finite number of quadratic inequality constraints. Given that the objective function is bounded over the feasible set, we present a comprehensive study of the conditions under which the optimal solution set is nonempty, thus extending the so-called Frank-Wolfe theorem. In particular, we first prove a general continuity result for the solution set defined by a system of convex quadratic inequalities. This result implies immediately that the optimal solution set of the aforementioned problem is nonempty when all the quadratic functions involved are convex. In the absence of the convexity of the objective function, we give examples showing that the optimal solution set may be empty either when there are two or more convex quadratic constraints, or when the Hessian of the objective function has two or more negative eigenvalues. In the case when there exists only one convex quadratic inequality constraint (together with other linear constraints), or when the constraint functions are all convex quadratic and the objective function is quasi-convex (thus allowing one negative eigenvalue in its Hessian matrix), we prove that the optimal solution set is nonempty.

KW - Convex quadratic system

KW - Existence of optimal solutions

KW - Quadratically constrained quadratic programming

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U2 - 10.1023/a:1008652705980

DO - 10.1023/a:1008652705980

M3 - Article

AN - SCOPUS:0008971685

VL - 13

SP - 87

EP - 110

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

SN - 0926-6003

IS - 1-3

ER -