Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# The Value Functions Approach and Hopf-Lax Formula for Multiobjective Costs via Set Optimization

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. Preprints 2019, 2019070032 (doi: 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 (doi: 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 . So far, it has been applied to set-valued dynamic risk measures (in the stochastic case), as discussed in Feinstein, Rudloff etc. (see , 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 . 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

MATHEMATICS & COMPUTER SCIENCE, Analysis

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