Midpoint Convexity Research Articles

Ameso optimization: A relaxation of discrete midpoint convexity

In this paper we introduce the Ameso optimization problem, a special class of discrete optimization problems. We establish its basic properties and investigate the relation between Ameso optimization and the convex optimization. Further, we design an algorithm to solve a multi-dimensional Ameso problem by solving a sequence of one-dimensional Ameso problems. Finally, we demonstrate how the knapsack problem can be solved using the Ameso optimization framework.

On basic operations related to network induction of discrete convex functions

Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to investigate basic operations such as direct sum, splitting, and aggregation that are related to network induction of discrete convex functions as well as discrete convex sets. Various kinds of discrete convex functions in discrete convex analysis are considered such as integrally convex functions, L-convex functions, M-convex functions, multimodular functions, and discrete midpoint convex functions.

Convexity of sets in metric Abelian groups

Abstract In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

Directed discrete midpoint convexity

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L^natural-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L^natural-convexity and integral convexity and is independent of global/local discrete midpoint convexity. The new convexity, named DDM-convexity, has nice properties satisfied by L^natural-convexity and global/local discrete midpoint convexity. DDM-convex functions are stable under scaling, satisfy the so-called parallelogram inequality and a proximity theorem with the same small proximity bound as that for L^{natural }-convex functions. Several characterizations of DDM-convexity are given and algorithms for DDM-convex function minimization are developed. We also propose DDM-convexity in continuous variables and give proximity theorems on these functions.

Discrete Midpoint Convexity

For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L♮-convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ℓ∞-distance equal to 2 or not smaller than 2, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L♮-convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

Null-finite sets in topological groups and their applications

In the paper we introduce a new family of "small" sets which is tightly connected with two well known $\sigma$-ideals: of Haar-null sets and of Haar-meager sets. We define a subset $A$ of a topological group $X$ to be $\mathit{null}$-$\mathit{finite}$ if there exists an infinite compact subset $K\subset X$ such that for every $x\in X$ the intersection $K\cap (x+A)$ is finite. We prove that each null-finite Borel set in a complete metric Abelian group is Haar-null and Haar-meager. The Borel restriction in the above result is essential as each non-discrete metric Abelian group is the union of two null-finite sets. Applying null-finite sets to the theory of functional equations and inequalities, we prove that a mid-point convex function $f:G\to\mathbb R$ defined on an open convex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a subset $B$ which is not null-finite and whose closure is contained in $G$. This gives an alternative short proof of a known generalization of Bernstein-Doetsch theorem (saying that a mid-point convex function $f:G\to\mathbb R$ defined on an open covex subset $G$ of a metric linear space $X$ is continuous if it is upper bounded on a non-empty open subset $B$ of $G$). Since Borel null-finite sets are Haar-meager and Haar-null, we conclude that a mid-point convex function $f:G\to\mathbb{R}$ defined on an open convex subset $G$ of a complete linear metric space $X$ is continuous if it is upper bounded on a Borel subset $B\subset G$ which is not Haar-null or not Haar-meager in $X$. The last result resolves an old problem in the theory of functional equations and inequalities posed by Baron and Ger in 1983.

On the Steinhaus Property and Ergodicity via the Measure-Theoretic Density of Sets

It is shown how the Steinhaus property and ergodicity of a translation invariant extension \(\mu\) of the Lebesgue measure depend on the measure-theoretic density of \(\mu\)-measurable sets. Some connection of the Steinhaus property with almost convex sets is considered and a translation invariant extension of the Lebesgue measure is presented, for which the generalized Steinhaus property together with the mid-point convexity do not imply the almost convexity.

Projection and convolution operations for integrally convex functions

This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under projection, and this is also the case with the subclasses of integrally convex functions satisfying local or global discrete midpoint convexity. As is known in the literature, the convolution of two integrally convex functions may possibly fail to be integrally convex. We show that the convolution of an integrally convex function with a separable convex function remains integrally convex. We also point out in terms of examples that the similar statement is false for integrally convex functions with local or global discrete midpoint convexity.

Quantitative measure of nonconvexity for black-box continuous functions

Metaheuristic algorithms usually aim to solve nonconvex optimization problems in black-box and high-dimensional scenarios. Characterizing and understanding the properties of nonconvex problems is therefore important for effectively analyzing metaheuristic algorithms and their development, improvement and selection for problem solving. This paper establishes a novel analysis framework called nonconvex ratio analysis, which can characterize nonconvex continuous functions by measuring the degree of nonconvexity of a problem. This analysis uses two quantitative measures: the nonconvex ratio for global characterization and the local nonconvex ratio for detailed characterization. Midpoint convexity and Monte Carlo integral are important methods for constructing the measures. Furthermore, as a practical feature, we suggest a rapid characterization measure that uses the local nonconvex ratio and can characterize certain black-box high-dimensional functions using a much smaller sample. Throughout this paper, the effectiveness of the proposed measures is confirmed by numerical experiments using the COCO function set.

Category-measure duality: convexity, midpoint convexity and Berz sublinearity

Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over {mathbb {Q}} to prove that the graph of a measurable sublinear function that is {mathbb {Q}}_{+}-homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set.

A further generalization of midpoint convexity of multimaps towards common fixed point theorems and applications

We furtherly generalize midpoint convexity for multivalued maps and derive Fixed Point Theorems and Common Fixed Point Theorems without requiring strong compactness. As an application we obtain some Best Approximation results, and minimax and variational inequalities.

Semigroups of midpoint convex set-valued functions

Semigroups of midpoint convex set-valued functions

On the continuity of midpoint hull convex set-valued functions

The concept of “hull convexity” (“midpoint hull convexity”) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of “convexity” (“midpoint convexity”).

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R n is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature

Midpoint convex functions majorized by midpoint concave functions

Theoreme de representation pour les fonctions convexes medianes et les fonctions concaves medianes satisfaisant a une inegalite

On midpoint convex set-valued functions

Etude de quelques proprietes des fonctions d'ensembles convexes medianes. Conditions suffisantes de continuite

Midpoint convex functions majorized by midpoint concave functions

Midpoint convex functions majorized by midpoint concave functions

On generalized mid-point convexity

On generalized mid-point convexity

On the Continuity of Mid-Point Convex Functions

On the Continuity of Mid-Point Convex Functions

Discrete Approximation of Continuous Convex Blobs

Given a pattern of squares on a grid for approximating arbitrary figures, one would like to have a criterion for determining whether that pattern could have been produced by a convex figure. The main result shows that an intuitively sparse criterion, the midpoint convexity property,is actually stronger than an intuitively neat global criterion, that of having a convex smallest bounding polygon. We also show that any pattern produced by a convex figure can be produced by a convex polygonal figure with its vertices at the corners of squares.