Global dynamic optimization using edge-concave underestimator

Abstract

Optimization of problems with embedded system of ordinary differential equations (ODEs) is challenging and the difficulty is amplified due to the presence of nonconvexity. In this article, a deterministic global optimization method is presented for systems consisting of an objective function and constraints with integral terms and an embedded set of nonlinear parametric ODEs. The method is based on a branch-and-bound algorithm that uses a new class of underestimators recently proposed by Hasan (J Glob Optim 71:735–752, 2018). At each node of the branch-and-bound tree, instead of using a convex relaxation, an edge-concave underestimator or the linear facets of its convex envelope is used to compute a lower bound. The underestimator is constructed by finding valid upper bounds on the diagonal elements of the Hessian matrix of the nonconvex terms. Time dependent bounds on the state variables and diagonal elements of the Hessian are obtained by solving an auxiliary set of ODEs that is derived using the notion of differential inequalities. The performance of the edge-concave relaxation is compared to other approaches on several test problems.