Mass optimization problem with convex cost

TL;DR

This paper studies a mass optimization problem with a convex cost functional, providing a PDE-based characterization of optimal solutions, extending previous linear models to nonlinear cases such as conductor optimization with variable conductivities.

Contribution

It introduces a generalized framework for mass optimization with convex costs, moving beyond linear models to nonlinear applications like conductor design.

Findings

Characterization of optimal solutions via PDEs.

Extension to nonlinear cost functions.

Application to conductor optimization.

Abstract

In this paper we consider a mass optimization problem in the case of scalar state function, where instead of imposing a constraint on the total mass of the competitors, we penalize the classical compliance by a convex functional defined on the space of measures. We obtain a characterization of optimal solutions to the problem through a suitable PDE. This generalizes the case considered in the literature of a linear cost and applies to the optimization of a conductor where very low and very high conductivities have both a high cost, and then the study of nonlinear models becomes relevant.