S-lemma with equality and its applications

• Published in: Mathematical programming Vol. 156; no. 1-2; pp. 513 - 547
• Main Authors: Xia, Yong, Wang, Shu, Sheu, Ruey-Lin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016; Springer Nature B.V
• Subjects: Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computer programming; Control theory; Eigenvalues; Equality; Full Length Paper; Inequalities; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematics; Mathematics and Statistics; Mathematics of Computing; Matrices (mathematics); Numerical Analysis; Optimization; Quadratic equations; Studies; Symmetry; Texts; Theoretical; United States > US; Generalized trust region subproblem; 90C20; Joint numerical range; Quadratically constrained quadratic program; 90C22; Hidden convexity; 90C26; Slater condition; S-lemma
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-015-0907-0

Let f ( x ) = x T A x + 2 a T x + c and h ( x ) = x T B x + 2 b T x + d be two quadratic functions having symmetric matrices A and B . The S-lemma with equality asks when the unsolvability of the system f ( x ) < 0 , h ( x ) = 0 implies the existence of a real number μ such that f ( x ) + μ h ( x ) ≥ 0 , ∀ x ∈ R n . The problem is much harder than the inequality version which asserts that, under Slater condition, f ( x ) < 0 , h ( x ) ≤ 0 is unsolvable if and only if f ( x ) + μ h ( x ) ≥ 0 , ∀ x ∈ R n for some μ ≥ 0 . In this paper, we show that the S-lemma with equality does not hold only when the matrix A has exactly one negative eigenvalue and h ( x ) is a non-constant linear function ( B = 0 , b ≠ 0 ). As an application, we can globally solve inf { f ( x ) : h ( x ) = 0 } as well as the two-sided generalized trust region subproblem inf { f ( x ) : l ≤ h ( x ) ≤ u } without any condition. Moreover, the convexity of the joint numerical range { ( f ( x ) , h 1 ( x ) , … , h p ( x ) ) : x ∈ R n } where f is a (possibly non-convex) quadratic function and h 1 ( x ) , … , h p ( x ) are affine functions can be characterized using the newly developed S-lemma with equality.