Abstract
Non-convex discrete-time optimal control problems 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 solutions that are indistinguishable from global optima. 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 invariant convex function (invex function) of the control decision variables when state variables are eliminated using implicit function theory. In this way, optimality guarantees can be obtained, the exact nature of which 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.
Highlights
Discrete-time optimal control typically involves the solution of an optimization problem, which need not be convex
(c) From the nonlinear analysis angle, we show that the hidden invexity of a discrete-time optimal control problem yields so-called conic-intersection optimality guarantees for KKT points with active inequality constraints, and global optimality for KKT points in the interior of the domain of the control variables
We used the invariant convexity to provide a theoretical framework to understand the good performance of local solvers on a class of nonconvex problems that arise in discrete-time optimal control
Summary
We describe a numerical experiment involving a discrete-time optimization problem for a system driven by hyperbolic partial differential equations. The experiment illustrates how a search for a KKT point using a local solver always leads to the same conicintersection optimum This suggests that this solution is a conic-intersection or global optimum, as predicted by the theory. Downstream outflow, and initial conditions, the time evolution of the level and flow of water at each point of the river is governed by hyperbolic partial differential equations [39]. These are known as the Saint-Venant equations, and are given by the momentum equation. We will not consider wetting and drying of the channel, whence we assume that A > 0 and P > 0
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