S-lemma with equality and its applications

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.