Related papers: Polygons as optimal shapes with convexity constrai…
We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon $P$ with $n$ vertices. We give a randomized near-linear-time $(1-\varepsilon)$-approximation algorithm for…
We derive an upper bound on the size of a ball such that the image of the ball under quadratic map is strongly convex and smooth. Our result is the best possible improvement of the analogous result by Polyak in the case of quadratic map. We…
An application of dimensional reduction results for gradient constrained problems is provided for 3D-2D dimension reduction for supremal functionals, in the case when the domain is convex.
In this paper, we focus on the problem of stochastic optimization where the objective function can be written as an expectation function over a closed convex set. We also consider multiple expectation constraints which restrict the domain…
Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…
This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz…
An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…
Let us consider the autonomous obstacle problem \begin{equation*} \min_v \int_\Omega F(Dv(x)) \, dx \end{equation*} on a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and…
Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed.…
To every nearly convex optimization problem, that is a minimization problem with a nearly convex objective function and a nearly convex constraint set, we associate a uniquely defined convex optimization problem with a lower semicontinuous…
Let $K$ be a convex pentagon in the plane and let $K_1$ be the pentagon bounded by the diagonals of $K$. It has been conjectured that the maximum of the ratio between the areas of $K_1$ and $K$ is reached when $K$ is an affine regular…
We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of…
Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of…
We consider the problem of minimizing a sum of non-convex functions over a compact domain, subject to linear inequality and equality constraints. Approximate solutions can be found by solving a convexified version of the problem, in which…
This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one…
This paper presents a canonical dual approach to the problem of minimizing the sum of a quadratic function and the ratio of nonconvex function and quadratic functions, which is a type of non-convex optimization problem subject to an…
Estimation of convex functions finds broad applications in engineering and science, while convex shape constraint gives rise to numerous challenges in asymptotic performance analysis. This paper is devoted to minimax optimal estimation of…
We consider discretized two-dimensional PDE-constrained shape optimization problems, in which shapes are represented by triangular meshes. Given the connectivity, the space of admissible vertex positions was recently identified to be a…