TY - JOUR

T1 - S-lemma with equality and its applications

AU - Xia, Yong

AU - Wang, Shu

AU - Sheu, Ruey Lin

N1 - Funding Information:
This research was supported by Taiwan National Science Council under Grant 102-2115-M-006-010, by National Center for Theoretical Sciences (South), by National Natural Science Foundation of China under grant 11471325 and by Beijing Higher Education Young Elite Teacher Project 29201442.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/3/1

Y1 - 2016/3/1

N2 - Let (Formula presented.) and (Formula presented.) be two quadratic functions having symmetric matrices (Formula presented.) and (Formula presented.). The S-lemma with equality asks when the unsolvability of the system (Formula presented.) implies the existence of a real number (Formula presented.) such that (Formula presented.). The problem is much harder than the inequality version which asserts that, under Slater condition, (Formula presented.) is unsolvable if and only if (Formula presented.) for some (Formula presented.). In this paper, we show that the S-lemma with equality does not hold only when the matrix (Formula presented.) has exactly one negative eigenvalue and (Formula presented.) is a non-constant linear function ((Formula presented.)). As an application, we can globally solve (Formula presented.) as well as the two-sided generalized trust region subproblem (Formula presented.) without any condition. Moreover, the convexity of the joint numerical range (Formula presented.) where (Formula presented.) is a (possibly non-convex) quadratic function and (Formula presented.) are affine functions can be characterized using the newly developed S-lemma with equality.

AB - Let (Formula presented.) and (Formula presented.) be two quadratic functions having symmetric matrices (Formula presented.) and (Formula presented.). The S-lemma with equality asks when the unsolvability of the system (Formula presented.) implies the existence of a real number (Formula presented.) such that (Formula presented.). The problem is much harder than the inequality version which asserts that, under Slater condition, (Formula presented.) is unsolvable if and only if (Formula presented.) for some (Formula presented.). In this paper, we show that the S-lemma with equality does not hold only when the matrix (Formula presented.) has exactly one negative eigenvalue and (Formula presented.) is a non-constant linear function ((Formula presented.)). As an application, we can globally solve (Formula presented.) as well as the two-sided generalized trust region subproblem (Formula presented.) without any condition. Moreover, the convexity of the joint numerical range (Formula presented.) where (Formula presented.) is a (possibly non-convex) quadratic function and (Formula presented.) are affine functions can be characterized using the newly developed S-lemma with equality.

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U2 - 10.1007/s10107-015-0907-0

DO - 10.1007/s10107-015-0907-0

M3 - Article

AN - SCOPUS:84958116091

VL - 156

SP - 513

EP - 547

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -