Necessary and Sufficient Conditions for S-Lemma and Nonconvex Quadratic Optimization∗

The celebrated S-lemma establishes a powerful equivalent condition for the nonnegativity of a quadratic function over a single quadratic inequality. However, this lemma fails without the technical condition, known as the Slater condition. In this paper, we first show that the Slater condition is indeed necessary for the S-lemma and then establishes a regularized form of the S-lemma in the absence of the Slater condition. Consequently, we present characterizations of global optimality and the Lagrangian duality for quadratic optimization problems with a single quadratic constraint. Our method of proof makes use of Brickman’s theorem and conjugate analysis, exploiting the hidden link between the convexity and the S-lemma.