The Problem of Optimizing a Set of Quadratic Criteria Relative to a Linear Multi-Step Control System

Source abstract

A linear-quadratic problem of optimal control in a discrete-time formulation and multi-objective setting is considered: a linear system, a set of quadratic objective functions in state and control variables, multi-objective optimization, and Pareto-optimal solutions. Based on a linear scalarization of criteria, a reduction to a parametric scalar optimization problem is performed, requiring a global solution without any convexity assumptions. The dependence of the aggregate objective function on state variables is eliminated. Using extremal eigenvalues, conditions on the parameters that ensure the convexity property in the scalar problem are formulated. Furthermore, two optimization problems with respect to the parameters are announced; these problems improve the defining matrix of the resulting quadratic form in terms of conditioning and convergence. As a result of the proposed regularization procedure, a family of multi-extremal problems is reduced to convex programming problems with a quadratic objective function that admit a guaranteed solution. This solution constitutes a Pareto-optimal control for the original non-convex linear-quadratic problem.