# Minimizing within Convex Bodies Using a Convex Hull Method

@article{LachandRobert2005MinimizingWC,
title={Minimizing within Convex Bodies Using a Convex Hull Method},
author={Thomas Lachand-Robert and {\'E}douard Oudet},
journal={SIAM J. Optim.},
year={2005},
volume={16},
pages={368-379}
}
• Published 1 June 2005
• Mathematics, Computer Science
• SIAM J. Optim.
We present numerical methods to solve optimization problems on the space of convex functions or among convex bodies. Hence convexity is a constraint on the admissible objects, whereas the functionals are not required to be convex. To deal with this, our method mixes geometrical and numerical algorithms. We give several applications arising from classical problems in geometry and analysis: Alexandrov's problem of finding a convex body of prescribed surface function; Cheeger's problem of a…

## Figures and Tables from this paper

### On Convex Functions and the Finite Element Method

• Computer Science
SIAM J. Numer. Anal.
• 2009
This paper proposes a finite element description of the Hessian, and proves convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes.

### Parametric shape optimization using the support function

• Computer Science
Comput. Optim. Appl.
• 2022
Meissner's conjecture, regarding three dimensional bodies of constant width with minimal volume, is confirmed by directly solving an optimization problem by constructing the numerical framework.

### Handling Convexity-Like Constraints in Variational Problems

• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2014
This work provides a general framework to construct finite-dimensional approximations of the space of convex functions and shows how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints.

### Numerical shape optimization among convex sets

A new discrete framework for approximating solutions to shape optimization problems under convexity constraints is proposed, based on the support function or the gauge function, which is guaranteed to generate discrete convex shapes and easily implementable using standard optimization software.

### A Numerical Method for Variational Problems with Convexity Constraints

This work considers the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex and proposes an approach to approximate the (nonpolyhedral) cone of convex functions by a polyhedral cone which can be represented by linear inequalities.

### Local properties of the surface measure of convex bodies

It is well known that any measure in S^2 satisfying certain simple conditions is the surface measure of a bounded convex body in R^3. It is also known that a local perturbation of the surface measure

### Approximating optimization problems over convex functions

• Computer Science, Mathematics
Numerische Mathematik
• 2008
A finite difference approximation using positive semidefinite programs and discrete Hessians is proposed, and convergence under very general conditions is proved, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes.

### Numerical Approximation of Optimal Convex Shapes

• Mathematics
SIAM J. Sci. Comput.
• 2020
Numerical experiments show that optimal convex shapes are generally non-smooth and that three-dimensional problems require an appropriate relaxation of the convexity condition and prove the convergence of discretizations in two-dimensional situations.

### Non-optimality of conical parts for Newton’s problem of minimal resistance in the class of convex bodies and the limiting case of infinite height

• Mathematics
Calculus of Variations and Partial Differential Equations
• 2022
We consider Newton’s problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial

### Conforming approximation of convex functions with the finite element method

The main motivation for this study is the investigation of a novel discretization of optimization problems with convexity constraints by the finite element method, which shows that the conforming approximation is convergent if the finite elements are at least piecewise quadratic.

## References

SHOWING 1-10 OF 30 REFERENCES

### Newton's problem of the body of minimal resistance in the class of convex developable functions

• Mathematics
• 2000
We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height ${M>0$ cite{butt in the class of convex developable functions defined in a

### Convex Bodies of Optimal Shape

• Mathematics
• 1997
L'article complet peuttrouvel'adresse : http://www.lama.univ-savoie.fr/~lachand/Publications.html Abstract Given a continuous function f : S n 1 ! R, we consider the min- imization of the functional

### H1-projection into the set of convex functions : a saddle-point formulation

• Mathematics
• 2001
We investigate numerical methods to approximate the projection-operator from H1 0 into the set of convex functions. We introduce a new formulation of the problem, based on gradient fi elds. It

### NON-CONVERGENCE RESULT FOR CONFORMAL APPROXIMATION OF VARIATIONAL PROBLEMS SUBJECT TO A CONVEXITY CONSTRAINT

• Mathematics
• 2001
In this article, we are interested in the minimization of functionals in the set of convex functions. We investigate the discretization of the convexity through various numerical methods and find a

### A numerical approach to variational problems subject to convexity constraint

• Mathematics
Numerische Mathematik
• 2001
An algorithm to approximate the minimizer of an elliptic functional in the form of $\int_\Omega j(x, u, \nabla u) on the set${\cal C}$of convex functions u in an appropriate functional space X is described. ### Generalized Cheeger sets related to landslides • Mathematics • 2005 Abstract.We study the maximization problem, among all subsets X of a given domain$\Omega$, of the quotient of the integral in X of a given function f by the integral on the boundary of X of another ### Regularity of solutions for some variational problems subject to a convexity constraint • Mathematics • 2001 We first study the minimizers, in the class of convex functions, of an elliptic functional with nonhomogeneous Dirichlet boundary conditions. We prove C1 regularity of the minimizers under the ### CHARACTERIZATION OF CHEEGER SETS FOR CONVEX SUBSETS OF THE PLANE • Mathematics • 2006 Given a planar convex domain Q, its Cheeger set CΩ is defined as the unique minimizer of |∂X|/|(X| among all nonempty open and simply connected subsets X of Ω. We prove an interesting geometric ### Isoperimetric estimates for the first eigenvalue of the$p\$-Laplace operator and the Cheeger constant

• Mathematics
• 2003
First we recall a Faber-Krahn type inequality and an estimate forp() in terms of the so-called Cheeger constant. Then we prove that the eigenvaluep() converges to the Cheeger constant h() as p → 1.

### A symmetry problem in the calculus of variations

We consider a class of integral functionals defined in a Sobolev space of functions vanishing at the boundary of a nonempty bounded connected open n-dimensional set. We prove that, if the functional