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 subdomain minimizing the ratio surface area on volume; Newton's problem of the body of minimal resistance.
In particular for the latter application, the minimizers are still unknown, except in some particular classes. We give approximate solutions better than the theoretical known ones, hence demonstrating that the minimizers do not belong to these classes.

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Minimizing within Convex Bodies Using a Convex Hull Method

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 disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite{lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with~$n$ sides centered in the disc, and numerical experiments indicate that the natural number $ngeq2$ is a non-decreasing function of $M$. The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height~$M$.

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Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

We study the minima of the functional ∫ Ω f(∇u) . The function f is not convex, the set Ω is a domain in R 2 and the minimum is sought over all convex functions on Ω with values in a given bounded interval. We prove that a minimum u is almost everywhere ‘on the boundary of convexity’, in the sense that there exists no open set on which u is strictly convex. In particular, wherever the Gaussian curvature is finite, it is zero. An important application of this result is the problem of the body of least resistance as formulated by Newton (where f ( p )=1/(1+| p | 2 ) and Ω is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in [1].

/pdf/an-example-of-non-convex-minimization-and-an-application-to-3xo5f2d0cf.pdf

An example of non-convex minimization and an application to Newton's problem of the body of least resistance

A fully nonlinear partial differential equation for the convex envelope was recently introduced by the author. In this paper, the equation is discretized using a finite difference method. The resulting scheme yields an explicit local method to compute the convex envelope. The scheme is shown to converge. Computational results are presented for smooth and nonsmooth data. Extensions to higher dimensions and unstructured grids are discussed.

Computing the convex envelope using a nonlinear partial differential equation

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove 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. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.

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On Convex Functions and the Finite Element Method

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

We study the minima of the functional $int_Omega f(nabla u)$. The function $f$ is not convex, the set $Omega$ is a domain in $R^2$ and the minimum is sought over all convex functions on $Omega$ with values in a given bounded interval. We prove that a minimum $u$ is almost everywhere `on the boundary of convexity', in the sense that there exists no open set on which $u$ is strictly convex. In particular, wherever the Gaussian curvature is {finite, it is zero. An important application of this result is the problem of the body of least resistance as formulated by Newton (where $f(p) = 1/(1+abs{p^2)$ and $Omega$ is a ball), implying that the minimizer is not radially symmetric. This generalizes a result in~cite{bro.

Peletier

Papers

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

An example of non-convex minimization and an application to Newton's problem of the body of least resistance