Numerical shape optimization among convex sets

TL;DR

This paper introduces a new discrete numerical framework for convex shape optimization that guarantees convexity, handles various constraints and objectives, and is compatible with standard optimization tools.

Contribution

It presents a novel, implementable discretization method based on support and gauge functions for convex shape optimization problems.

Findings

Framework guarantees convexity of shapes

Handles width and diameter constraints effectively

Compatible with standard optimization software

Abstract

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate discrete convex shapes and is easily implementable using standard optimization software. The framework can handle various objective functions ranging from geometric quantities to functionals depending on partial differential equations. Width or diameter constraints are handled using the support function. Functionals depending on a convex body and its polar body can be handled using a unified framework.