Extending the Scope of Robust Quadratic Optimization

In this paper, we derive tractable reformulations of the robust counterparts of convex quadratic and conic quadratic constraints with concave uncertainties for a broad range of uncertainty sets. For quadratic constraints with convex uncertainty, it is well-known that the robust counterpart is, in general, intractable. Hence, we derive tractable inner and outer approximations of the robust counterparts of such constraints. The approximations are made by replacing the quadratic terms in the uncertain parameters with suitable linear upper and lower bounds. Furthermore, when the uncertain parameters consist of a mean vector and covariance matrix, we construct a natural uncertainty set using an asymptotic confidence level and show that its support function is semi-definite representable. Finally, we apply our results to a portfolio choice, a norm approximation, and a regression line problem.