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. So far, it has only been applied to set-valued dynamic risk measures within a stochastic framework, but not to deterministic calculus of variations and optimal control problems. In this paper, a multi-objective calculus of variations problem is considered which is turned into a set-valued problem by a straightforward extension. A new set-valued value function is introduced, for which a Bellman's optimality principle holds. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time and a directional derivative with respect to the state variable of the value function are defined. The value function is proved to be a solution of a corresponding Hamilton-Jacobi equation.

Hamel, A., Visetti, D. (2020). The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 483(1) [10.1016/j.jmaa.2019.123605].

### The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization

#### 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. So far, it has only been applied to set-valued dynamic risk measures within a stochastic framework, but not to deterministic calculus of variations and optimal control problems. In this paper, a multi-objective calculus of variations problem is considered which is turned into a set-valued problem by a straightforward extension. A new set-valued value function is introduced, for which a Bellman's optimality principle holds. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time and a directional derivative with respect to the state variable of the value function are defined. The value function is proved to be a solution of a corresponding Hamilton-Jacobi equation.
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Bellman's principle; Hamilton-Jacobi equation; Hopf-Lax formula; Keywords multicriteria calculus of variations; Set relations; Value function;
English
14-ott-2019
2020
483
1
123605
reserved
Hamel, A., Visetti, D. (2020). The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 483(1) [10.1016/j.jmaa.2019.123605].
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