On the Approximation of Unbounded Convex Sets by Polyhedra

• Why. Convex sets have many useful special topological properties. The challenging concept of recession directions of a convex set has to be mastered: this is needed for work with unbounded convex sets. Here is an example of the use of recession directions: they can turn ‘non-existence’ (of a bound for a convex set or of an optimal solution for a convex optimization problem) into existence (of a recession direction). This gives a certificate for non-existence.

Learning convex set boundaries via primal-dual neural approximation with application to reachable set computation.

It is known that the curvature of the feasible set in convex optimization allows for algorithms with better convergence rates, and there is renewed interest in this topic for both off-line and online problems. In this paper, leveraging results on geometry and convex analysis, we further our understanding of the role of curvature in optimization: We first show the equivalence of two notions of curvature, namely, strong convexity and gauge bodies, proving a conjecture of Abernethy et al. As a consequence, this shows that the Frank–Wolfe–type method of Wang and Abernethy has accelerated convergence rate [Formula: see text] over strongly convex feasible sets without additional assumptions on the (convex) objective function. In online linear optimization, we identify two main properties that help explaining why/when follow the leader (FTL) has only logarithmic regret over strongly convex sets. This allows one to directly recover and slightly extend a recent result of Huang et al., and to show that FTL has logarithmic regret over strongly convex sets whenever the gain vectors are nonnegative. We provide an efficient procedure for approximating convex bodies by strongly convex ones while smoothly trading off approximation error and curvature. This allows one to extend the improved algorithms over strongly convex sets to general convex sets. As a concrete application, we extend results on online linear optimization with hints to general convex sets. Funding: This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grant Bolsa de Produtividade em Pesquisa #4310516/2017-0] Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Finance Code 001) Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (Jovem Cientista do Nosso Estado), Bolsa de Produtividade em Pesquisa #312751/2021-4 from CNPq, FAPERJ [Grant “Jovem Cientista do Nosso Estado”].

In this paper, we study properties of general closed convex sets that determine the closedness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the convex hull of integer points in general convex sets to be a polyhedron. This result generalizes the well-known result due to Meyer (Math Program 7:223–225, 1974). Under a simple technical assumption, we show that these sufficient conditions are also necessary for the convex hull of integer points contained in general convex sets to be a polyhedron.

Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.

In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

In the work we study the solvability of the convolution equation (in particular, of differential - difference equation) in the Schwartz space on an unbounded convex set in $\mathbb R^n$.

The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes, and the question of how to relax Bregman projections onto convex sets that are not linear (i.e., not hyperplanes or half-spaces) has remained open. A definition of the underrelaxation of Bregman projections onto general convex sets is given here, which includes as special cases the underrelaxed orthogonal projections and the underrelaxed Bregman projections onto linear sets as given by De Pierro and Iusem [J. Optim. Theory Appl., 51 (1986), pp. 421--440]. With this new definition, we construct a block-iterative projection algorithmic scheme and prove its convergence to a solution of the convex feasibility problem. The practical importance of relaxation parameters in the application of such projection algorithms to real-world problems is demonstrated on a problem of image reconstruction from projections.

A potential of fuzzy relations with a linear structure: The unbounded case

To efficiently solve online problems with complicated constraints, projection-free algorithms including online frank-wolfe (OFW) and its variants have received significant interest recently. However, in the general case, existing efficient projection-free algorithms only achieved the regret bound of O(T^{3/4}), which is worse than the regret of projection-based algorithms, where T is the number of decision rounds. In this paper, we study the special case of online learning over strongly convex sets, for which we first prove that OFW can enjoy a better regret bound of O(T^{2/3}) for general convex losses. The key idea is to refine the decaying step-size in the original OFW by a simple line search rule. Furthermore, for strongly convex losses, we propose a strongly convex variant of OFW by redefining the surrogate loss function in OFW. We show that it achieves a regret bound of O(T^{2/3}) over general convex sets and a better regret bound of O(T^{1/2}) over strongly convex sets.

Min-max and max-min identities are found for inner products on the boundaries of compact, convex sets whose interiors contain the origin. The identities resemble the minimax theorem but they are different from it. Specifically, the value of each min-max (or max-min) equals the value of a dual problem of the same type. Their solution sets can be characterized geometrically in terms of the enclosed convex sets and their polar sets. However, the solution sets need not be convex nor even connected.

Article Free Access Share on Voronoi diagrams and containment of families of convex sets on the plane Authors: M. Abellanas Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , G. Hernandez Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , R. Klein Fern Universität Hagen, Germany Fern Universität Hagen, GermanyView Profile , V. Neumann-Lara Universidad Nacional Autonoma de México, Mexico Universidad Nacional Autonoma de México, MexicoView Profile , J. Urrutia University of Ottawa, Canada University of Ottawa, CanadaView Profile Authors Info & Claims SCG '95: Proceedings of the eleventh annual symposium on Computational geometrySeptember 1995 Pages 71–78https://doi.org/10.1145/220279.220287Published:01 September 1995Publication History 3citation259DownloadsMetricsTotal Citations3Total Downloads259Last 12 Months12Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Minkowski’s classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result.

We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm's behavior in the inconsistent case. We present numerical results of computational experiments that illustrate the computational advantage of the new method.

This book describes and analyzes all available alternating projection methods for solving the general problem of finding a point in the intersection of several given sets belonging to a Hilbert space. For each method the authors describe and analyze convergence, speed of convergence, acceleration techniques, stopping criteria, and applications. Different types of algorithms and applications are studied for subspaces, linear varieties, and general convex sets. The authors also unify these algorithms into a common theoretical framework. Alternating Projection Methods is a comprehensive and accessible source of information, providing readers with the theoretical and practical aspects of the most relevant alternating projection methods. It features several acceleration techniques for every method it presents and analyzes, including schemes that cannot be found in other books. It also provides full descriptions of several important mathematical problems and specific applications for which the alternating projection methods represent an efficient option. Examples and problems that illustrate this material are also included. Audience: This book can be used as a textbook for advanced undergraduate or first-year graduate students. Because it is comprehensive, it can also be used as a tutorial or a reference by mathematicians and nonmathematicians from many fields of application who need to solve alternating projection problems in their work. Contents: Preface; Chapter 1: Introduction; Chapter 2: Overview on Spaces; Chapter 3: The MAP on Subspaces; Chapter 4: Row-Action Methods; Chapter 5: Projecting on Convex Sets; Chapter 6: Applications of MAP for Matrix Problems; Bibliography; Author Index; Subject Index.