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

Characteristic curves for set-valued Hamilton-Jacobi equations

“…The value function was proved in [8] to satisfy a Hamilton-Jacobi equation and we report here the result:…”

“…See [7] and [8] for a motivation and many features including a discussion of the 'improper' function values R d and ∅.…”

“…The form of this Hamiltonian is due to the fact that sometimes it is the Fenchel conjugate (5) of a Lagrangian function (see [8]).…”

“…This problem was never studied before, to the knowledge of the author, and it is not only interesting per se, but also as a natural completion of a previous paper. In fact, in [8] the authors considered a calculus of variations' problem with a multicriteria Lagrangian function. The embedding of it into a set-valued framework was fundamental to prove a Hopf-Lax formula for the value function.…”

“…For set-valued optimization an infimizer is a set. It is proved that an infimizer for the Hopf-Lax formula proved in [8] can be constituted only by minimizers of the scalarized problem with respect to an element of the dual cone.…”