Abstract This paper introduces a generalization of the powerful S-lemma, extending it to minimax quadratic functions for the first time. Notably, the convexity of the involved functions is not assumed; instead, our approach leverages the convexifiability of the associated quadratic systems. It provides various verifiable conditions to identify this convexifiability by exploiting the system’s separability. Using the new minimax S-lemma and the simultaneous diagonalization property, the paper presents exact second-order cone program reformulations for classes of distributionally robust optimization problems involving minimax quadratic functions, ensuring that they share the same optimal value and can efficiently be solved. Finally, it also provides numerical validations of our results for concrete models of insurance risk assessment, maximum revenue estimation, and option pricing under distributional uncertainty.

We establish a geometric condition guaranteeing exact copositive relaxation for the nonconvex quadratic optimization problem under two quadratic and several linear constraints, and present sufficient conditions for global optimality in terms of generalized Karush–Kuhn–Tucker multipliers. The copositive relaxation is tighter than the usual Lagrangian relaxation. We illustrate this by providing a whole class of quadratic optimization problems that enjoys exactness of copositive relaxation while the usual Lagrangian duality gap is infinite. Finally, we also provide verifiable conditions under which both the usual Lagrangian relaxation and the copositive relaxation are exact for an extended CDT (two-ball trust-region) problem. Importantly, the sufficient conditions can be verified by solving linear optimization problems.