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We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…

We show that for multivariate Freud-type weights $W_\alpha(x)=\exp(-|x|^\alpha)$, $\alpha>1$, any convex function $f$ on $R^d$ satisfying $fW_\alpha\in L_p(R^d)$ if $1\le p<\infty$, or $\lim_{|x|\to\infty}f(x)W_\alpha(x)=0$ if $p=\infty$,…

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…

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.…

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance…

There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the…

Bilevel optimization has been developed for many machine learning tasks with large-scale and high-dimensional data. This paper considers a constrained bilevel optimization problem, where the lower-level optimization problem is convex with…

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

In this paper, we consider the problem of covering a plane region with unit discs. We present an improved upper bound and the first nontrivial lower bound on the number of discs needed for such a covering, depending on the area and…

A number of results related to statistical classification on convex sets are presented. In particular, the focus is on the case where some of the covariates in the data and observation being classified can be missing. The form of the…

We consider solving nonconvex composite optimization problems in which the sum of a smooth function and a nonsmooth function is minimized. Many of convergence analyses of proximal gradient-type methods rely on global descent property…

We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of…

Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…

In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$…

We identity the optimal non-infinitesimal direction of descent for a convex function. An algorithm is developed that can theoretically minimize a subset of (non-convex) functions.

A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed…

In this paper, we introduce a new class of structured polynomials, called separable plus lower degree (SPLD) polynomials. The formal definition of an SPLD polynomial, which extends the concept of SPQ polynomials (Ahmadi et al. in Math Oper…

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows…

In this paper, we develop a direct formula for determining the coefficients in the canonical basis of the best polynomial of degree $M$ that approximates a polynomial of degree $N>M$ on a symmetric interval for the $\mathcal{L}^2$-norm. We…