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