An Eikonal Equation with Vanishing Lagrangian Arising in Global Optimization

Applied Mathematics & Optimization, vol. 87

Abstract

Abstract We show a connection between global unconstrained optimization of a continuous function f and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution v of the critical Hamilton–Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then v is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of v. Such trajectories are proved to converge to the set of minima of f, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.

Authors 2

1. Martino Bardi corresponding

   University of Padua

   Affiliation as printed

   Department of Mathematics “T. Levi-Civita”, University of Padova, via Trieste, 63, 35121, Padua, Italy

   Department of Mathematics "T. Levi-Civita", University of Padova, via Trieste, 63, 35121, Padua, Italy

2. Hicham Kouhkouh Aachen Institut für Mathematik, RTG Energy

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik, RTG Energy, Entropy, and Dissipative Dynamics, RWTH Aachen University, Templergraben 55 (111810), 52062, Aachen, Germany