Uncertainty Quantification for Multiobjective Stochastic Convex Quadratic Programs

A multiobjective stochastic convex quadratic program (MOSCQP) is a multiobjective optimization problem with convex quadratic objectives that are observed with stochastic error. MOSCQP is a useful problem formulation arising, for example, in model calibration and nonlinear system identification when a single regression model combines data from multiple distinct sources, resulting in a multiobjective least squares problem. We consider structured uncertainty quantification for MOSCQPs, which includes the questions of estimating the efficient and Pareto sets, inference through central limit theorems (CLTs), and constructing asymptotically exact confidence regions on the efficient and Pareto sets. We use parameterization to first write the efficient and Pareto set estimators in closed form, then expand the closed-form expression in a matrix geometric series resulting in a key lemma characterizing the Fréchet derivatives of the efficient and Pareto sets. The key lemma enables a delta theorem analogue for MOSCQPs, resulting in structured uniform CLTs on the estimated efficient and Pareto sets. Finally, we formulate a direct procedure for constructing asymptotically valid confidence regions that retain the efficient and Pareto set shapes endowed by the MOSCQP problem structure. We illustrate the confidence regions through a numerical example.