Generalized Alternating Projections on Manifolds and Convex Sets

We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface. As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} . For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex.

Some properties of the spaces of paths are studied in order to define and characterize the local convexity of sets belonging to smooth manifolds and the local convexity of functions defined on local convex sets of smooth manifolds.

While convex sets in Euclidean space can easily be approximated by convex sets with C∞ -boundary, the C∞ -approximation of convex sets in Riemannian manifolds is a non-trivial problem. Here we prove that C∞-approximation is possible for a compact, locally convex set C in a Riemannian manifold if (i) C has strictly convex boundary or if (ii) the sectional curvature is positive or negative on C.

Uniform tail-decay of Lipschitz functions implies Cheeger's isoperimetric inequality under convexity assumptions

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”].

A non-empty set X of vertices of an acyclic digraph is called connected if the underlying undirected graph induced by X is connected and it is called convex if no two vertices of X are connected by a directed path in which some vertices are not in X. The set of convex sets (connected convex sets) of an acyclic digraph D is denoted by \(\mathcal{CO}(D) (\mathcal{CC}(D))\) and its size by co(D) (cc(D)). Gutin et al. (2008) conjectured that the sum of the sizes of all convex sets (connected convex sets) in D equals Θ(n · co(D)) (Θ(n · cc(D))) where n is the order of D. In this paper we exhibit a family of connected acyclic digraphs with \(\sum_{C\in \mathcal{CO}(D)}|C| = o(n\cdot \mathrm{co}(D))\) and \(\sum_{C\in \mathcal{CC}(D)}|C| = o(n\cdot \mathrm{cc}(D))\). We also show that the number of connected convex sets of order k in any connected acyclic digraph of order n is at least n − k + 1. This is a strengthening of a theorem of Gutin and Yeo.

The problem of finding a point in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , from which the sum-of-distances to a finite number of nonempty, closed and convex sets is minimum is called generalized Fermat-Torricelli Problem (FTP). In applications, along with the point that minimizes sum-of-distances, it is important to know the points in the convex sets at which the minimum sum-of-distances is achieved. Various formulations existing in literature do not involve finding the optimal points in the convex sets. In this letter, we formulate a non-smooth convex optimization problem, with both the point/set of points which yields the minimum sum-of-distances as well as the corresponding points in the convex sets as primal variables. We term this problem as extended FTP (eFTP). We adopt non-smooth projected primal-dual dynamical approach to solve this problem. The proposed dynamical system can exhibit a continuum of equilibria. Hence we show semistability of the set of optimal points, which is the pertinent notion of stability for such systems. A distributed implementation of the primal-dual dynamical system is also presented in this letter. Four illustrative examples are considered for the simulation based validation of the solution proposed for eFTP.

The split feasibility problem (SFP), which refers to the task of finding a point that belongs to a given nonempty, closed and convex set, and whose image under a bounded linear operator belongs to another given nonempty, closed and convex set, has promising applicability in modeling a wide range of inverse problems. Motivated by the increasingly data-driven regularization in the areas of signal/image processing and statistical learning, in this paper, we study the regularized split feasibility problem (RSFP), which provides a unified model for treating many real-world problems. By exploiting the split nature of the RSFP, we shall gainfully employ several efficient splitting methods to solve the model under consideration. A remarkable advantage of our methods lies in their easier subproblems in the sense that the resulting subproblems have closed-form representations or can be efficiently solved up to a high precision. As an interesting application, we apply the proposed algorithms for finding Dantzig selectors, in addition to demonstrating the effectiveness of the splitting methods through some computational results on synthetic and real medical data sets.

Previous article Next article Uniform Distributions on Convex Sets: Inequality for Characteristic FunctionsA. A. Kulikova and Yu. V. ProkhorovA. A. Kulikova and Yu. V. Prokhorovhttps://doi.org/10.1137/S0040585X97980099PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis paper proves an inequality for a characteristic function of the uniform distribution on a compact convex body $D\subset{\bf R}^s$.[1] N. G. Ushakov, Some inequalities for characteristic functions of unimodal distributions, Theory Probab. Appl., 26 (1983), pp. 595–599. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkGoogle Scholar[2] Google Scholar[3] A. A. Kulikova, Estimate of the rate of convergence of probability distributions to a uniform distribution, Theory Probab. Appl., 47 (2002), pp. 693–699. tba TPRBAU 0040-585X Theor. Probab. Appl. LinkGoogle ScholarKeywordsuniform distributioncharacteristic function Previous article Next article FiguresRelatedReferencesCited ByDetails Four Areas of Yu. V. Prokhorov's Studies and Their PerspectivesO. V. Viskov and V. I. KhokhlovTheory of Probability & Its Applications, Vol. 60, No. 2 | 7 June 2016AbstractPDF (155 KB)Limit theorems for additive functionals of stationary fields, under integrability assumptions on the higher order spectral densitiesStochastic Processes and their Applications, Vol. 125, No. 4 | 1 Apr 2015 Cross Ref Convex and star-shaped sets associated with multivariate stable distributions, I: Moments and densitiesJournal of Multivariate Analysis, Vol. 100, No. 10 | 1 Nov 2009 Cross Ref On an inequality by Kulikova and ProkhorovStatistics & Probability Letters, Vol. 79, No. 14 | 1 Jul 2009 Cross Ref Estimate of the Rate of Convergence of Probability Distributions to a Uniform DistributionA. A. KulikovaTheory of Probability & Its Applications, Vol. 47, No. 4 | 25 July 2006AbstractPDF (145 KB) Volume 47, Issue 4| 2003Theory of Probability & Its Applications567-744 History Published online:25 July 2006 InformationCopyright © 2003 Society for Industrial and Applied MathematicsKeywordsuniform distributioncharacteristic functionPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97980099Article page range:pp. 700-701ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here, projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present, then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavioural patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use and like. If you know of books or review papers that should be added to this list please contact us.

We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids are special cases of local pre-Leibniz algebroids, which are still general enough to include many physically motivated algebroids such as Lie, Courant, metric and higher-Courant algebroids. They create a natural framework for generalizations of differential geometric structures on a smooth manifold. The symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid satisfies a certain property depending on a choice of an equivalence class of connections which are called admissible. These admissible connections are shown to be necessary to generalize aforementioned structures on pre-Leibniz algebroids. Consequently, we prove that, provided certain conditions are met, statistical and conjugate connection structures are equivalent when defined for admissible connections. Moreover, we also show that for ‘projected-torsion-free’ connections, one can generalize Hessian metrics and Hessian structures. We prove that any Hessian structure yields a statistical structure, where these results are completely parallel to the ones in the manifold setting. We also prove a mild generalization of the fundamental theorem of statistical geometry. Moreover, we generalize a-connections, strongly conjugate connections and relative torsion operator, and prove some analogous results.

A number of inverse problems, in image reconstruction and elsewhere, can be formulated in terms of finding a vector in the intersection of certain convex sets that serve to constrain the solution. Finding such vectors is called the `convex feasibility problem' (CFP). Algorithms to solve the CFP are usually iterative `projection onto convex sets' (POCS) methods that employ orthogonal or more general projections onto the individual convex sets; Bregman's `successive generalized projection' (SGP) is one such method. When the intersection of the convex sets is heterogeneous one may wish to optimize a certain function over that intersection; then we have a constrained optimization problem. Generalized projections come from generalized distances, typically Bregman distances, chosen to incorporate prior information, such as non-negativity, about the image being reconstructed. Calculating a generalized projection onto a convex set can be simplified if the generalized distance can vary with the convex set. Censor and Elfving have discovered a simultaneous multiprojection algorithm that permits just such variation. Because simultaneous methods, which use all the convex sets at each step, can be slow to converge, there is considerable interest in faster, block-iterative methods that employ only some of the convex sets at each step. In this paper we present the first such method that permits multiple generalized projections. We introduce a new notion of convex combination and apply it to obtain an extension of the SGP, called the `multidistance SGP' (MSGP) method, that allows for projecton with respect to multiple generalized distances. We conclude with an extension of the MSGP to block-iterative algorithms involving relaxed generalized projections and paracontractive operators.

Introduction: Notation, Elementary Results.- Convex Sets: Generalities Convex Sets Attached to a Convex Set Projection onto Closed Convex Sets Separation and Applications Conical Approximations of Convex Sets.- Convex Functions: Basic Definitions and Examples Functional Operations Preserving Convexity Local and Global Behaviour of a Convex Function First- and Second-Order Differentiation.- Sublinearity and Support Functions: Sublinear Functions The Support Function of a Nonempty Set Correspondence Between Convex Sets and Sublinear Functions.- Subdifferentials of Finite Convex Functions: The Subdifferential: Definitions and Interpretations Local Properties of the Subdifferential First Examples Calculus Rules with Subdifferentials Further Examples The Subdifferential as a Multifunction.- Conjugacy in Convex Analysis: The Convex Conjugate of a Function Calculus Rules on the Conjugacy Operation Various Examples Differentiability of a Conjugate Function.

Let H = H ( ∗ , [ + ] ) H = H{(^ \ast },[ + ]) denote the real linear space of locally schlicht normalized functions in | z | &gt; 1 |z| &gt; 1 as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If M ⊂ H M \subset H there is a closed nonempty convex set in the α β \alpha \beta -plane such that for ( α , β ) (\alpha ,\beta ) in this set α ∗ f [ + ] β ∗ g ∈ C {\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C (in K) whenever f, g ∈ M g \in M . This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).