Search

In Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter $$\Omega $$ is expressed by the integration with respect to a measure $$P(\Omega ,\cdot )$$ which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of $$\Omega $$ . The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff–Gauss measure $$\mathscr {S}^{\infty -1}$$ restricted to the measure-theoretic boundary of $$\Omega $$ . In this paper, we consider an open convex set $$\Omega $$ and we provide an explicit formula for the density of $$P(\Omega ,\cdot )$$ with respect to $$\mathscr {S}^{\infty -1}$$ . In particular, the density can be written in terms of the Minkowski functional $$\mathfrak {p}$$ of $$\Omega $$ with respect to an inner point of $$\Omega $$ . As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces.

For $d\in\{5,6\}$, we classify arrangements of $d + 2$ points in $\mathbf{RP}^{d-1}$ for which the minimum distance is as large as possible. To do so, we leverage ideas from matrix and convex analysis to determine the best possible codes that contain equiangular lines, and we introduce a notion of approximate Positivstellensatz certificates that promotes numerical approximations of Stengle's Positivstellensatz certificates to honest certificates., Comment: code included in ancillary files

M ♮ -concavity is a key concept in discrete convex analysis. For set functions, the class of M ♮ -concavity is a proper subclass of submodularity. It is a well-known fact that the set of minimizers of a submodular function forms a distributive lattice, where every finite distributive lattice is possible to appear. It is a natural question whether every finite distributive lattice appears as the minimizer set of an M ♮ -concave set function. This paper affirmatively answers the question.

The paper is devoted to the abstract $${\mathcal{H}}$$ -convexity of functions (where $${\mathcal{H}}$$ is a given set of elementary functions) and its realization in the cases when $${\mathcal{H}}$$ is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular $${\mathcal{H}}$$ -convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) $${\mathcal{H}}$$ -minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support $${\mathcal{H}}$$ -minorants at a point and the set of lower $${\mathcal{H}}$$ -support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of $${\mathcal{H}}$$ -convexity are realized in the specific cases when functions are defined on a metric or normed space $$X$$ and the set of elementary functions is the space $${\mathcal{L}}(X,{\mathbb{R}})$$ of Lipschitz functions or the set $${\mathcal{L}}\widehat{C}(X,{\mathbb{R}})$$ of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower $${\mathcal{L}}$$ -support points and the set of lower $${\mathcal{L}}\widehat{C}$$ -support points coincide and are dense in the effective domain of the function. These results extend the known Brondsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (1998), and the tractability of the problem is ascribed to total dual integrality in a linear programming formulation by Schrijver (1982). Another view on the tractability of this problem is afforded by a valuated matroid intersection formulation by Takazawa (2012). In the present paper, we discuss the relationship between these two formulations for the shortest bibranching problem. We first demonstrate that the valuated matroid intersection formulation can be derived from the linear programming formulation through the Benders decomposition, where integrality is preserved in the decomposition process and the resulting convex programming is endowed with discrete convexity. We then show how a pair of primal and dual optimal solutions of one formulation is constructed from that of the other formulation, thereby providing a connection between polyhedral combinatorics and discrete convex analysis., Comment: 18 pages

We consider the null controllability problem fo linear systems of the form \begin{document}$ y'(t) = Ay(t)+Bu(t) $\end{document} on a Hilbert space \begin{document}$ Y $\end{document} . We suppose that the control operator \begin{document}$ B $\end{document} is bounded from the control space \begin{document}$ U $\end{document} to a larger extrapolation space containing \begin{document}$ Y $\end{document} . The control \begin{document}$ u $\end{document} is constrained to lie in a time-varying bounded subset \begin{document}$ \Gamma(t) \subset U $\end{document} . From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set \begin{document}$ \Gamma (t) $\end{document} contains the origin in its interior at each \begin{document}$ t>0 $\end{document} , the local constrained property turns out to be equivalent to a weighted dual observability inequality of \begin{document}$ L^{1} $\end{document} type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets \begin{document}$ \Gamma (t) $\end{document} in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that \begin{document}$ \Gamma (t) $\end{document} is a closed ball centered at the origin and its radius is time-varying.

In this paper, we compare the definitions of convex sets and convex functions in finite dimensional integer spaces introduced by Adivar and Fang, Borwein, and Giladi, respectively. We show that the...

Given two convex d-polytopes P and Q in R d for d ⩾ 3 , we study the problem of bundling P and Q in a smallest convex container. More precisely, our problem asks to find a minimum convex set containing P and a translate of Q that do not properly overlap each other. We present the first exact algorithm for the problem for any fixed dimension d ⩾ 3 . The running time is O ( n ( d − 1 ) ⌊ d + 1 2 ⌋ ) , where n denotes the number of vertices of P and Q. In particular, in dimension d = 3 , our algorithm runs in O ( n 4 ) time. We also give an example of polytopes P and Q such that in the smallest container the translates of P and Q do not touch.

Altres ajuts: Australian Research Council DP140103213 In this work, we achieve a complete characterization of the existence of a saddle value, for bifunctions which are convex, proper, and lower semi continuous in their first argument, by considering new suitably defined notions of special directions of recession. As special cases, we obtain some recent results of Lagrangian duality theory on zero duality gap for convex programs.

In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

We prove general nonlinear large deviation estimates similar to Chatterjee–Dembo’s original bounds, except that we do not require any second order smoothness. Our approach relies on convex analysis arguments and is valid for a broad class of distributions. Our results are then applied in three different setups. Our first application consists in the mean-field approximation of the partition function of the Ising model under an optimal assumption on the spectra of the adjacency matrices of the sequence of graphs. Next, we apply our general large deviation bound to investigate the large deviation of the traces of powers of Wigner matrices with sub-Gaussian entries and the upper tail of cycles counts in sparse Erdős–Renyi graphs down to the sparsity threshold $n^{-1/2}$.

In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis. Mathematics Subject Classification (2010) 49J52 · 49J53

• Why. The central object of convex analysis is a convex set. Whenever weighted averages play a role, such as in the analysis by Nash of the question ‘what is a fair bargain?’, one is led to consider convex sets.

For two matroids $M_1$ and $M_2$ with the same ground set $V$ and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl, Peis, and Timmermans (2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where the constraint is $|X_1 \cap X_2|=k$. The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $\mathrm{M}$-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games., Comment: This is a post-peer-review, pre-copyedit version of an article published in Mathematical Programming. The final authenticated version is available online at: https://doi.org/10.1007/s10107-021-01625-2

• Why. A second main object of convex analysis is a convex function. Convex functions can help to describe convex sets: these are infinite sets, but they can often be described by a formula for a convex function, so in finite terms. Moreover, in many optimization applications, the function that has to be minimized is convex, and then the convexity is used to solve the problem.

For two matroids \(M_1\) and \(M_2\) with the same ground set V and two cost functions \(w_1\) and \(w_2\) on \(2^V\), we consider the problem of finding bases \(X_1\) of \(M_1\) and \(X_2\) of \(M_2\) minimizing \(w_1(X_1)+w_2(X_2)\) subject to a certain cardinality constraint on their intersection \(X_1 \cap X_2\). Lendl, Peis, and Timmermans (2019) discussed modular cost functions: They reduced the problem to weighted matroid intersection for the case where the cardinality constraint is \(|X_1 \cap X_2|\le k\) or \(|X_1 \cap X_2|\ge k\); and designed a new primal-dual algorithm for the case where \(|X_1 \cap X_2|=k\). The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or \(\mathrm {M}\)-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in matroid congestion games and combinatorial optimization problems with interaction costs.

The brain encodes spatial structure through a combinatorial code of neural activity. Experiments suggest such codes correspond to convex areas of the subject’s environment. We present an intrinsic condition that implies a neural code may correspond to a collection of convex sets and give a bound on the minimal dimension underlying such a realization.

The multiple exchange property for matroid bases is generalized for valuated matroids and M-natural concave set functions. The proof is based on the Fenchel-type duality theorem in discrete convex analysis. The present result has an implication in economics: The strong no complementarities condition of Gul and Stacchetti is, in fact, equivalent to the gross substitutes condition of Kelso and Crawford.

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so...

M-convex functions, which are a generalization of valuated matroids, play a central role in discrete convex analysis. Quadratic M-convex functions constitute a basic and important subclass of M-convex functions, which has a close relationship with phylogenetics as well as valued constraint satisfaction problems. In this paper, we consider the quadratic M-convexity testing problem (QMCTP), which is the problem of deciding whether a given quadratic function on $\{0,1\}^n$ is M-convex. We show that QMCTP is co-NP-complete in general, but is polynomial-time solvable under a natural assumption. Furthermore, we propose an $O(n^2)$-time algorithm for solving QMCTP in the polynomial-time solvable case., 13 pages

For a symmetric positive semidefinite linear system of equations $$\mathcal{Q}{{\varvec{x}}}= {{\varvec{b}}}$$ , where $${{\varvec{x}}}= (x_1,\ldots ,x_s)$$ is partitioned into s blocks, with $$s \ge 2$$ , we show that each cycle of the classical block symmetric Gauss–Seidel (sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $$\frac{1}{2}\Vert {{\varvec{x}}}-{{\varvec{x}}}^k\Vert _\mathcal{T}^2$$ , where $$\mathcal{T}$$ is a symmetric positive semidefinite matrix related to the sGS decomposition of $$\mathcal{Q}$$ and $${{\varvec{x}}}^k$$ is the previous iterate. By leveraging on such a connection to optimization, we are able to extend the result (which we name as the block sGS decomposition theorem) for solving convex composite QP (CCQP) with an additional possibly nonsmooth term in $$x_1$$ , i.e., $$\min \{ p(x_1) + \frac{1}{2}\langle {{\varvec{x}}},\,\mathcal{Q}{{\varvec{x}}}\rangle -\langle {{\varvec{b}}},\,{{\varvec{x}}}\rangle \}$$ , where $$p(\cdot )$$ is a proper closed convex function. Based on the block sGS decomposition theorem, we extend the classical block sGS method to solve CCQP. In addition, our extended block sGS method has the flexibility of allowing for inexact computation in each step of the block sGS cycle. At the same time, we can also accelerate the inexact block sGS method to achieve an iteration complexity of $$O(1/k^2)$$ after performing k cycles. As a fundamental building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the inexact semiproximal ALM/ADMM for linearly constrained multi-block convex composite conic programming (CCCP), and the accelerated block coordinate descent method for multi-block CCCP.

In this paper, we study properties of a special class of ordered median functions, called block-functions. These are ordered median functions which belong to a generating binary (row)-vector of the form called a block vector. The aim of this paper is to explicitly determine the simplicial complexes and all steepest descent and ascent directions of descent and ascent cones of ordered median block-function.

$${\mathrm{M}}^\natural $$ -concave functions form a class of discrete concave functions in discrete convex analysis, and are defined by a certain exchange axiom. We show in this paper that $${\mathrm{M}}^\natural $$ -concave functions can be characterized by a combination of two simpler exchange properties. It is also shown that for a function defined on an integral polymatroid, a much simpler exchange axiom characterizes $${\mathrm{M}}^\natural $$ -concavity. These results have some significant implications in discrete convex analysis.

An infinitely smooth convex body in R n is called polynomially integrable of degree N if its parallel section functions are polynomials of degree N. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if N ≥ n − 1 . This is in contrast with the case of even dimensions and the case of odd dimensions with N n − 1 , where such bodies do not exist, as it was recently shown by Agranovsky.

In this paper we consider the rank and zero norm regularized least squares loss minimization problem with a spectral norm ball constraint. For this class of NP-hard optimization problems, we propose a two-stage convex relaxation approach by majorizing some suitable locally Lipschitz continuous surrogates. Furthermore, the Frobenius norm error bound for the optimal solution of each stage is characterized and the theoretical guarantee is established for the two-stage convex relaxation approach by showing that the error bound of the first stage convex relaxation (i.e., the nuclear norm and $$\ell _1$$ -norm regularized minimization problem), can be reduced much by the second stage convex relaxation under a suitable restricted eigenvalue condition. Also, we verify the efficiency of the proposed approach by applying it to some random test problems and some real problems.

One of the central questions of polyhedral combinatorics is the question of the algorithmic relationship between the vertex and facet descriptions of convex polytopes. From the standpoint of combinatorial optimization, the main reason for the actuality of this question is the possibility of applying the methods of convex analysis to solving the extremal combinatorial problems. In this paper, we consider the combinatorial polytopes of a sufficiently general form. We obtain a few of necessary conditions and a sufficient condition for a supporting inequality of a polytope to be a facet inequality and give an illustration of the use of the developed technology to the polytope of some graph approximation problem.

In this paper we give higher order sufficient optimality conditions for a fractional vector optimization problem over cones, using higher order cone-convex functions. A higher order Schaible type dual program is formulated over cones.Weak, strong and converse duality results are established by using the higher order cone convex and other related functions.

Do we have enough examples of convex bodies that we truly understand? Is out standard set of examples diverse enough to understand convexity? In this note, we will dramatically increase our set of examples. More specifically, we will present several new constructions of convex bodies: the geometric mean of two convex bodies, the power function K α (which in general exists only for | α | ≤ 1 ), and even the logarithm log ⁡ K .

The general notion of convexity also applies to a graph G = ( V , E ) . A subset V c of V is called a convex set if all shortest paths with end vertices in V c remain within V c . A convex cut of G is a partition ( V 1 , V 2 ) of V such that V 1 and V 2 are convex sets (halfspaces). Finding convex cuts is NP -hard for general graphs. In this paper we first characterize the convex cuts of a connected bipartite graph G ′ in terms of the Djokovic relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges' end vertices. Specifically, we show that a cut of G ′ is convex if and only if the Djokovic relation holds for any pair of edges in its cut-set. As a consequence, all convex cuts of G ′ = { V ′ , E ′ } can be found in O ( | V ′ | | E ′ | ) . We then characterize the convex cuts of a general connected graph G using the Djokovic–Winkler relation θ and another relation τ on the edges of G. Based on this general characterization, we show how one can find all convex cuts of G in polynomial time. The key parts here are (i) describing the Djokovic–Winkler relation on the edges of G in terms of the Djokovic relation θ ′ on the bipartite graph obtained from G by subdividing each edge of G into two edges, (ii) studying the interplay of θ ′ and τ on plane curves representing convex cuts, and (iii) a running time analysis of our algorithm for finding the convex cuts. Our method for characterizing and finding convex cuts of a connected plane graph G is motivated by the concept of alternating cuts and conditions on the latter to be convex. In the last part of this paper we represent alternating cuts as plane curves and focus on their intersection pattern. If the plane curves form an arrangement of pseudolines, G is scale embedded in its dual, and any edge of G is contained in the cut-set of a convex cut of G.

Choose uniform random points $$X_1, \dots , X_n$$ in a given convex set and let $${\text { conv}}[X_1, \dots , X_n]$$ be their convex hull. It is shown that in dimension three the expected volume of this convex hull is in general not monotone with respect to set inclusion. This answers a question by Meckes in the negative. The given counterexample is formed by uniformly distributed points in the three-dimensional tetrahedron together with a small perturbation of it. As side result we obtain an explicit formula for all even moments of the volume of a random simplex which is the convex hull of three uniform random points in the tetrahedron and the center of one facet.

We introduce and study the family of sets in a finite-dimensional Euclidean space which can be written as the Minkowski sum of an open, bounded, and convex set and a closed convex cone. We establish several properties of the class of such sets, called OM-decomposable, some of which extend related properties holding for the class of Motzkin decomposable sets (i.e., those for which the convex and bounded set in the decomposition is requested to be closed, hence compact).

In this paper, we introduce the notion of strictly convex n-normed space as a generalization strictly convex normed space and give few characterizations for such space. We also prove that every map...

We establish unique existence of $p$-barycenter of any probability measure for $p \ge 2$ on CAT(1)-spaces of small radii. In our proof, we employ Kendall's convex function on a ball of CAT(1)-spaces instead of the convexity of distance function. Various properties of $p$-barycenter on those spaces are also presented. They extend the author's previous work [Yo].

Estimation of convex functions finds broad applications in science and engineering; however, the convex shape constraint complicates the asymptotic performance analysis of such estimators. This technical note is devoted to the minimax optimal estimation of univariate convex functions in a given Holder class. Particularly, a minimax lower bound in the supremum norm (or simply sup-norm) is established by constructing a novel family of piecewise quadratic convex functions in the Holder class. This result, along with a recent result on the minimax upper bound, gives rise to the optimal rate of convergence for the minimax sup-norm risk of convex functions with the Holder order between one and two. The present technical note provides the first rigorous justification of the optimal minimax risk for convex estimation on the entire interval of interest in the sup-norm.

In this paper we establish a generalization of the left Fejer inequality for general Lebesgue integral on measurable spaces as well as various upper bounds for the di¤erence 1 R b a g (x) dx Z b a h (x) g (x) dx h a+ b 2 where h : [a; b] ! R is a convex function and g : [a; b] ! [0;1) is an integrable weight. Applications for discrete means and Hermite-Hadamard type inequalities are also provided.

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively.

The aim of this paper is to obtain some new bounds having Riemann type quantum integrals within the class of strongly convex functions. The results obtained are sharp on limit q → 1. These new results reduce to Tariboon-Ntouyas, Merentes-Nikodem and other previously known results when q → 1, where 0 < q < 1. The sharpness of the results of Tariboon-Ntouyas and Merentes-Nikodem is proved as a consequence.

Central configurations of the n-body problem have been studied for more than 200 years since the pioneer works of Euler and Lagrange. In this article we study convex central configurations which are not strictly convex. We give explicit examples of such configurations in both planar and spatial n-body problems. Particularly, in the spatial case, we consider regular polyhedra with bodies of same mass m at the vertices and bodies of same mass M at the middle points of each edge. In this setting we prove that the cube is the unique regular polyhedron such that this construction leads to a convex central configuration which is not strictly convex.

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e. being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.

We investigate the bend-minimization problem with respect to a new drawing style called an orthogonally convex drawing, which is an orthogonal drawing with an additional requirement that each inner face is drawn as an orthogonally convex polygon. For the class of biconnected plane graphs of maximum degree 3, we present a necessary and sufficient condition for the existence of a no-bend orthogonally convex drawing, which in turn, enables a linear time algorithm to check and construct such a drawing if one exists. We also develop a flow network formulation for bend-minimization in orthogonally convex drawings, yielding a polynomial time solution for the problem. An interesting application of our orthogonally convex drawing technique is to characterize internally triangulated plane graphs that admit floorplans using only orthogonally convex modules subject to given orthogonally convex boundary constraints.

In this paper we show that the celebrated Davis-Choi-Jensen’s inequality for normalised positive linear maps can be extended in a weakened form for convex functions. A reverse inequality and applications for important instances of convex (concave) functions are also given.