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}
}
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… 

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References

SHOWING 1-10 OF 30 REFERENCES

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

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

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

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

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

TLDR
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

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

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

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

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