arXiv subjects

Vera Roshchina

Publications and source records attributed to Vera Roshchina.

At least 19 recordsLinked to original sources

arXiv2026

Exposed extreme rays of the SONC cone

We provide a complete and explicit characterization of the exposed extreme rays of the cone of sums of nonnegative circuit (SONC) polynomials. The criterion we derive is purely combinatorial and depends only on the existence of certain circuits within the ground set and on the nature of the corresponding extreme ray. Our constructive proofs also yield explicit exposing functionals, offering a basis for algorithmic detection of exposed rays in SONC-based optimization.

math.AG↗

arXiv2025

Inner approximations of convex sets and intersections of projectionally exposed cones

A convex cone is said to be projectionally exposed (p-exposed) if every face arises as a projection of the original cone. It is known that, in dimension at most four, the intersection of two p-exposed cones is again p-exposed. In this paper we construct two p-exposed cones in dimension $5$ whose intersection is not p-exposed. This construction also leads to the first example of an amenable cone that is not projectionally exposed, showing that these properties, which coincide in dimension at most $4$, are distinct in dimension $5$. In order to achieve these goals, we develop a new technique for constructing arbitrarily tight inner convex approximations of compact convex sets with desired facial structure. These inner approximations have the property that all proper faces are extreme points, with the exception of a specific exposed face of the original set.

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arXiv2024

The problem of optimal camouflaging

We consider the problem of camouflaging for bodies with specular surface in the framework of geometric optics. The index of visibility introduced in [Plakhov 2017] measures the mean deviation of light rays incident on the body's surface. We study the problem of minimal visibility index for bodies with fixed volume contained in the unit sphere. This problem is reduced to a special vector-valued problem of optimal mass transfer, which is solved partly analytically and partly numerically. This paper is a continuation of the study started in [Plakhov 2009], [Plakhov 2017], and [Plakhov and Roshchina 2011].

math.DS↗

arXiv2024

Everything is possible: constructing spectrahedra with prescribed facial dimensions

Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. In this work, we show that such sets can be realised as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral. It also follows that these sets are facially exposed, in contrast to earlier constructions. We obtain a lower bound on the minimum number of convex quadratic inequalities needed to represent a closed convex set with prescribed facial dimension signature, and show that our bound is tight for some special cases. Finally, we relate the question of finding efficient representations with indecomposability of integer sequences and other topics, and discuss a substantial number of open questions.

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arXiv2024

Face relative interior of convex sets in topological vector spaces

A new notion of face relative interior for convex sets in topological real vector spaces is introduced in this work. Face relative interior is grounded in the facial structure, and may capture the geometry of convex sets in topological vector spaces better than other generalisations of relative interior. We show that the face relative interior partitions convex sets into face relative interiors of their closure-equivalent faces (different to the partition generated by intrinsic cores), establish the conditions for nonemptiness of this new notion, compare the face relative interior with other concepts of convex interior and prove basic calculus rules.

math.OC↗

arXiv2023

Douglas--Rachford is the best projection method

We prove that the Douglas--Rachford method applied to two closed convex cones in the Euclidean plane converges in finitely many steps if and only if the set of fixed points of the Douglas--Rachford operator is nontrivial. We analyze this special case using circle dynamics. We also construct explicit examples for a broad family of projection methods for which the set of fixed points of the relevant projection method operator is nontrivial, but the convergence is not finite. This three-parametric family is well known in the projection method literature and includes both the Douglas--Rachford method and the classic method of alternating projections. Even though our setting is fairly elementary, this work contributes in a new way to the body of theoretical research justifying the superior performance of the Douglas--Rachford method compared to other techniques. Moreover, our result leads to a neat sufficient condition for finite convergence of the Douglas--Rachford method in the locally polyhedral case on the plane, unifying and expanding several special cases available in the literature.

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arXiv2023

Hyperbolicity cones are amenable

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or "nice") which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.

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arXiv2022

Amenable cones are particularly nice

Amenability is a geometric property of convex cones that is stronger than facial exposedness and assists in the study of error bounds for conic feasibility problems. In this paper we establish numerous properties of amenable cones, and investigate the relationships between amenability and other properties of convex cones, such as niceness and projectional exposure. We show that the amenability of a compact slice of a closed convex cone is equivalent to the amenability of the cone, and prove several results on the preservation of amenability under intersections and other convex operations. It then follows that homogeneous, doubly nonnegative and other cones that can be represented as slices of the cone of positive semidefinite matrices, are amenable. It is known that projectionally exposed cones are amenable and that amenable cones are nice, however the converse statements have been open questions. We construct an example of a four-dimensional cone that is nice but not amenable. We also show that amenable cones are projectionally exposed in dimensions up to and including four. We conclude with a discussion on open problems related to facial structure of convex sets that we came across in the course of this work, but were not able to fully resolve.

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arXiv2022

The intrinsic core and minimal faces of convex sets in general vector spaces

Intrinsic core generalises the finite-dimensional notion of the relative interior to arbitrary (real) vector spaces. Our main goal is to provide a self-contained overview of the key results pertaining to the intrinsic core and to elucidate the relations between intrinsic core and facial structure of convex sets in this general context. We gather several equivalent definitions of the intrinsic core, cover much of the folklore, review relevant recent results and present examples illustrating some of the phenomena specific to the facial structure of infinite-dimensional sets.

math.OC↗

arXiv2021

Deep Learning with Nonsmooth Objectives

We explore the potential for using a nonsmooth loss function based on the max-norm in the training of an artificial neural network. We hypothesise that this may lead to superior classification results in some special cases where the training data is either very small or unbalanced. Our numerical experiments performed on a simple artificial neural network with no hidden layers (a setting immediately amenable to standard nonsmooth optimisation techniques) appear to confirm our hypothesis that uniform approximation based approaches may be more suitable for the datasets with reliable training data that either is limited size or biased in terms of relative cluster sizes.

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arXiv2020

Refining the partition for multifold conic optimization problems

In this paper we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in [J. F. Bonnans and H. Ramírez C., Math. Program. 104 (2005), no. 2-3, Ser. B, 205--227] for conic optimization problems, and in [J. Peña and V. Roshchina, Math. Program. 142 (2013), no 1-2, Ser. A, 579--589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework, and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions.

math.OC↗

arXiv2020

A data-independent distance to infeasibility for linear conic systems

We offer a unified treatment of distinct measures of well-posedness for homogeneous conic systems. To that end, we introduce a distance to infeasibility based entirely on geometric considerations of the elements defining the conic system. Our approach sheds new light on and connects several well-known condition measures for conic systems, including {\em Renegar's} distance to infeasibility, the {\em Grassmannian} condition measure, a measure of the {\em most interior} solution, and other geometric measures of {\em symmetry} and of {\em depth} of the conic system.

math.OC↗

arXiv2019

Dimension and structure of higher-order Voronoi cells on discrete sites

We study the structure of higher-order Voronoi cells on a discrete set of sites in $\mathbb{R}^n$, focussing on the relations between cells of different order, and paying special attention to the ill-posed case when a large number of points lie on a sphere. In particular, we prove that higher order cells of dimension $n-1$ do not exist, even though high-order Voronoi cells may have empty interior. We also present a number of open questions.

math.MG↗

arXiv2019

Uniqueness of solutions in multivariate Chebyshev approximation problems

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for the ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions.

math.OC↗

arXiv2019

On the Structure of Higher Order Voronoi Cells

The classic Voronoi cells can be generalized to a higher-order version by considering the cells of points for which a given $k$-element subset of the set of sites consists of the $k$ closest sites. We study the structure of the $k$-order Voronoi cells and illustrate our theoretical findings with a case study of two-dimensional higher-order Voronoi cells for four points.

math.MG↗

arXiv2019

Facially Dual Complete (Nice) cones and lexicographic tangents

We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the well-known and commonly used concept of tangent cones in nonlinear optimization, we introduce some new notions for exposure of faces of convex sets. Based on these new notions, we obtain a necessary condition and a sufficient condition for a cone to be facially dual complete. In our sufficient condition, we utilize a new notion called lexicographic tangent cones (these are a family of cones obtained from a recursive application of the tangent cone concept). Lexicographic tangent cones are related to Nesterov's lexicographic derivatives and to the notion of subtransversality in the context of variational analysis.

math.OC↗

arXiv2019

Projection Algorithms for Finite Sum Constrained Optimization

Parallel and cyclic projection algorithms are proposed for minimizing the sum of a finite family of convex functions over the intersection of a finite family of closed convex subsets of a Hilbert space. These algorithms are of predictor-corrector type, with each main iteration consisting of an inner cycle of subgradient descent process followed by a projection step. We prove the convergence of these methods to an optimal solution of the composite minimization problem under investigation upon assuming boundedness of the gradients at the iterates of the local functions and the stepsizes being chosen appropriately, in the finite-dimensional setting. We also discuss generalizations and limitations of the proposed algorithms and our techniques.

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arXiv2018

Comparing Averaged Relaxed Cutters and Projection Methods: Theory and Examples

We focus on the convergence analysis of averaged relaxations of cutters, specifically for variants that---depending upon how parameters are chosen---resemble \emph{alternating projections}, the \emph{Douglas--Rachford method}, \emph{relaxed reflect-reflect}, or the \emph{Peaceman--Rachford} method. Such methods are frequently used to solve convex feasibility problems. The standard convergence analyses of projection algorithms are based on the \emph{firm nonexpansivity} property of the relevant operators. However if the projections onto the constraint sets are replaced by cutters (projections onto separating hyperplanes), the firm nonexpansivity is lost. We provide a proof of convergence for a family of related averaged relaxed cutter methods under reasonable assumptions, relying on a simple geometric argument. This allows us to clarify fine details related to the allowable choice of the relaxation parameters, highlighting the distinction between the exact (firmly nonexpansive) and approximate (strongly quasi-nonexpansive) settings. We provide illustrative examples and discuss practical implementations of the method.

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