Version 1
: Received: 30 June 2019 / Approved: 2 July 2019 / Online: 2 July 2019 (07:56:14 CEST)
How to cite:
Hamel, A.H.; Visetti, D. The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization. Preprints2019, 2019070032. https://doi.org/10.20944/preprints201907.0032.v1
Hamel, A.H.; Visetti, D. The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization. Preprints 2019, 2019070032. https://doi.org/10.20944/preprints201907.0032.v1
Hamel, A.H.; Visetti, D. The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization. Preprints2019, 2019070032. https://doi.org/10.20944/preprints201907.0032.v1
APA Style
Hamel, A.H., & Visetti, D. (2019). The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization. Preprints. https://doi.org/10.20944/preprints201907.0032.v1
Chicago/Turabian Style
Hamel, A.H. and Daniela Visetti. 2019 "The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization" Preprints. https://doi.org/10.20944/preprints201907.0032.v1
Abstract
The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. For example, the duality issue for multi-criteria and vector optimization problems could be solved using the complete-lattice approach, compare [11]. So far, it has been applied to set-valued dynamic risk measures (in the stochastic case), as discussed in Feinstein, Rudloff etc. (see [11], for example), but it has not been applied to deterministic calculus of variations and optimal control problems. In this paper, the following problem of set-valued optimization is considered: minimize the functional $$ \overline J_t[y]=\int_0^t \overline L(s,y(s),\dot y(s))\ ds + U_0(y(0)) $$ over all admissible arcs $y$, where $\overline L$ is the associated multifunction to a vector-valued Lagrangian $L$, the integral is in the Aumann sense and $U_0$ is the initial cost. A new concept of \emph{value function}, for which a Bellman's optimality principle holds, is introduced. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time $t$ and a directional derivative with respect to $x$ of the value function are defined, based on ideas close to the concepts in [12]. The value function is proved to be solution of a suitable Hamilton-Jacobi equation.
Keywords
multicriteria calculus of variations; value function; Hopf-Lax formula; Bellman's principle; Hamilton-Jacobi equation; set relations
Subject
Computer Science and Mathematics, Analysis
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
