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Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we…
Let $K$ be a $d$ dimensional convex body with a twice continuously differentiable boundary and everywhere positive Gauss-Kronecker curvature. Denote by $K_n$ the convex hull of $n$ points chosen randomly and independently from $K$ according…
In formal safety verification, many proposed algorithms use parametric set representations and convert the computation of the relevant sets into an optimization problem; consequently, the choice of parameterization and objective function…
We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon…
We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given $n$ rectangular items where the $i^{\textrm{th}}$ item has width $w(i)$, height $h(i)$, and…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on…
In this article, we survey the primary research on polyhedral computing methods for constrained linear control systems. Our focus is on the modeling power of convex optimization, featured to design set-based robust and optimal controllers.…
This paper presents a piecewise convexification method for solving non-convex multi-objective optimization problems with box constraints. Based on the ideas of the $\alpha$-based Branch and Bound (${\rm \alpha BB}$) method of global…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
This paper studies hidden convexity properties associated with constrained optimization problems over the set of rotation matrices $\text{SO}(n)$. Such problems are nonconvex due to the constraint $X \in \text{SO}(n)$. Nonetheless, we show…
A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes…
We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In…
We consider several basic questions pertaining to the geometry of image of a general quadratic map. In general the image of a quadratic map is non-convex, although there are several known classes of quadratic maps when the image is convex.…
A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low…
We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of…
The isoperimetric problem asks for the maximum area of a region of given perimeter. It is natural to consider other measurements of a region, such as the diameter and width, and ask for the extreme value of one when another is fixed. The…
We consider the problem of establishing dense correspondences within a set of related shapes of strongly varying geometry. For such input, traditional shape matching approaches often produce unsatisfactory results. We propose an ensemble…
We explore optimal circular nonconvex partitions of regular k-gons. The circularity of a polygon is measured by its aspect ratio: the ratio of the radii of the smallest circumscribing circle to the largest inscribed disk. An optimal…