Novel optimization algorithms for evaluating solution to certain high dimension (multi-time) Hamilton-Jacobi PDEs arising from optimal controldissertationsKim, TaeWoocreatorDarbon, JeromeAdvisorShu, Chi-WangReaderGuzman, JohnnyReaderBrown University. Department of Applied Mathematicssponsor2022, None p.born digitalThesis (Ph. D.)--Brown University, 2022thesesIn this thesis, we study optimization algorithms for evaluating solutions to certain high dimension (multi-time) Hamilton-Jacobi Partial Differential equations arising from optimal control. There are three main chapters to cover these topics. The first chapter describes known connections between optimal control problems and Hamilton-Jacobi PDEs and the generalized Hopf formula that corresponds to the viscosity solution of certain Hamilton-Jacobi PDEs. We then focus on an important class of optimal control problem where the control has to remain in a full ellipsoid and the terminal cost is a convex function. We show that these problems can be solved using standard convex optimization methods. The efficiency of these optimization methods rely on the performance of computing the projection on the Minkowsky sum of ellipsoids. We also describe connections between the generalized Hopf formula and multi-time Hamilton-Jacobi PDEs. The second chapter presents a novel and efficient Newton based algorithm to compute the projection onto the Minkowski sum of full ellipsoids. The third chapter extends the class of linear optimal control problems considered in this thesis by allowing the control to live in a degenerate ellipsoid. Several possible algorithms to cope with these degenerate cases are presented. Finally, the last chapter focuses on developing the algorithm for computing the proximal point of a convex polyhedral function whose domain is a polyhedral set. This algorithm finds applications in optimal control and multi-time Hamilton Jacobi PDEs.Convex Optimizationoptimal controlHamilton-Jacobi equationsprimal dualEnglishRPB20220706