Global optimality in model predictive control via hidden invariant convexity

TL;DR

This paper demonstrates that certain non-convex optimal control problems become invex, allowing local solutions to be near-global optima, supported by theoretical analysis and a numerical river control example.

Contribution

It introduces the concept of hidden invariant convexity in optimal control problems, providing a theoretical basis for near-global optimality of solutions.

Findings

Objective function is invex when state variables are eliminated.

Local search yields high-quality solutions in invex problems.

Theoretical framework explains near-global optimality in non-convex control.

Abstract

Non-convex optimal control problems occurring in, e.g., water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to find a globally optimal solution, and numerical experience indicates that algorithms aiming for Karush-Kuhn-Tucker points often find (near-)optimal solutions. In our paper, we provide a theoretical underpinning for this phenomenon, showing that on a broad class of problems the objective can be shown to be an invariantly convex function (invex function) of the control decision variables when state variables are eliminated using implicit function theory. In this way, near-global optimality can be demonstrated, where the exact nature of the global optimality guarantee depends on the position of the solution within the feasible set. In a numerical example, we show how high-quality solutions are obtained with local search for a river control problem where invexity holds.