The exact solution of multiparametric quadratically constrained quadratic programming problems

In this paper, we present a strategy for the exact solution of multiparametric quadratically constrained quadratic programs (mpQCQPs). Specifically, we focus on multiparametric optimization problems with a convex quadratic objective function, quadratic inequality and linear equality constraints, described by constant matrices. The proposed approach is founded on the expansion of the Basic Sensitivity Theorem to a second-order Taylor approximation, which enables the derivation of the exact parametric solution of mpQCQPs. We utilize an active set strategy to implicitly explore the parameter space, based on which (i) the complete map of parametric solutions for convex mpQCQPs is constructed, and (ii) the determination of the optimal parametric solution for every feasible parameter realization reduces to a nonlinear function evaluation. Based on the presented results, we utilize the second-order approximation to the Basic Sensitivity Theorem to expand to the case of nonconvex quadratic constraints, by employing the Fritz John necessary conditions. Example problems are provided to illustrate the algorithmic steps of the proposed approach.