**ABSTRACT**

Many operational questions arising in science and engineering can be approximately addressed as systems of quadratic inequalities in continuous variables. Sometimes these are additionally stated as optimization problems, possibly by relying on a quadratic objective which measures an error, or deviation from a target operating point. The resulting problems can be, and often are, extremely challenging computational tasks, where even an order of magnitude estimate of the value of the problem can prove very difficult to attain. In this talk we will describe the broad background of such problems, together with two new results that we have obtained. First, we will describe an extension of the "lattice-free cuts" methodology originally developed for linear mixed-integer programming, to the continuous setting, obtaining a characterization of the epigraph for a nonconvex optimization problem relying on linear inequalities. Second, we will describe new results that provide efficient algorithms for extensions of the classical trust-region subproblem in nonlinear programming. Joint work with Alex Michalka.