M-convex Function Research Articles

Legendre-Fenchel duality in m-convexity

We show that m-convex function in n-dimensional can be viewed as the Legendre-Fenchel dual problem. Particularly, we have prove this for certain convex functions.

Hermite-Hadamard and Simpson’s type of inequalities for first and second ordered derivatives using product of (h1, h2, s)-convex and m-convex function

This article is dedicated to find the extensions for Hermite-Hadamard (H-H) and Simpson’s type of inequalities. By combining multiple existing convex functions by placing specific restrictions on them is the most effective in many approaches to find a new convex function. Here, to find the new function (h1, h2, s)-Convex and m-Convex Function are used. Because of the product of two or even more convex functions does not necessarily have to be convex, we decided to investigate merging distinct convex functions. Combining more than two convex functions in a novel adaptive way advances to new applications in a range of disciplines, including mathematics and other fields. In this paper, some extensions for Hermite-Hadamard and Simpson’s inequalities is explored. The newly constructed extensions of these inequalities will be considered as the improvements and refinements of previously obtained results.

M-Convex Function Minimization Under L1-Distance Constraint and Its Application to Dock Reallocation in Bike-Sharing System

In this paper, we consider a problem of minimizing an M-convex function under an L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given “center.” This is motivated by a nonlinear integer programming problem for reallocation of dock capacity in a bike-sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock reallocation problem through the connection with M-convexity and show its polynomial-time solvability using this connection. For this, we first show that the dock reallocation problem and its generalizations can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on the steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock reallocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. For this purpose, we develop a polynomial-time algorithm for a relaxation of the dock reallocation problem by using a proximity-scaling approach, which is of interest in its own right.

Fractional Hadamard and Fejer-Hadamard inequalities for exponentially m-convex function

Fractional integral operators play a vital role in the advancement of mathematical inequalities. The aim of this paper is to present the Hadamard and the Fej er-Hadamard inequalities for generalized fractional integral operators con- taining Mittag-Le er function. Exponentially m-convexity is utilized to establish these inequalities. By xing parameters involved in the Mittag-Le er function Hadamard and the Fej er-Hadamard inequalities for various well known fractional integral operators can be obtained.

On Generalized Strongly Convex Functions and Unified Integral Operators

In this paper, we define a strongly exponentially α , h − m -convex function that generates several kinds of strongly convex and convex functions. The left and right unified integral operators of these functions satisfy some integral inequalities which are directly related to many unified and fractional integral inequalities. From the results of this paper, one can obtain various fractional integral operator inequalities that already exist in the literature.

Compression of M♮-convex functions — Flag matroids and valuated permutohedra

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M♮-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M♮-convex function that transforms the given M♮-convex function to an M-convex function, which we call a compression of an M♮-convex function. For the class of valuated generalized matroids, which are special M♮-convex functions, the compression induces a valuated permutohedron together with a decomposition of the valuated generalized matroid into flag-matroid strips, each corresponding to a maximal linearity domain of the induced valuated permutohedron. We examine the details of the structure of flag-matroid strips and the induced valuated permutohedron by means of discrete convex analysis of Murota.

Generalized Fractional Hadamard and Fejér–Hadamard Inequalities for Generalized Harmonically Convex Functions

In this paper, we define a new function, namely, harmonically α , h − m -convex function, which unifies various kinds of harmonically convex functions. Generalized versions of the Hadamard and the Fejér–Hadamard fractional integral inequalities for harmonically α , h − m -convex functions via generalized fractional integral operators are proved. From presented results, a series of fractional integral inequalities can be obtained for harmonically convex, harmonically h − m -convex, harmonically α , m -convex, and related functions and for already known fractional integral operators.

A note on M-convex functions on jump systems

A jump system is defined as a set of integer points (vectors) with a certain exchange property, generalizing the concepts of matroids, delta-matroids, and base polyhedra of integral polymatroids (or submodular systems). A discrete convexity concept is defined for functions on constant-parity jump systems and it has been used in graph theory and algebra. In this paper we call it “jump M-convexity” and extend it to “jump M♮-convexity” for functions defined on a larger class of jump systems. By definition, every jump M-convex function is a jump M♮-convex function, and we show the equivalence of these concepts by establishing an (injective) embedding of jump M♮-convex functions in n variables into the set of jump M-convex functions in n+1 variables. Using this equivalence we show further that jump M♮-convex functions admit a number of natural operations such as aggregation, projection (partial minimization), convolution, composition, and transformation by a network.

Estimation of different entropies via Hermite interpolating polynomial using Jensen type functionals

We estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Hermite interpolating polynomial is used to generalize the new inequalities for m-convex function.

Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

M-Convex Functions of Higher Order

Abstract In this research we introduce the concept of m-convex function of higher order by means of the so called m-divided difference; elementary properties of this type of functions are exhibited and some examples are provided.

New inequalities for generalized m-convex functions via generalized fractional integral operators and their applications

In the present work, we prove a parametrized identity for a differentiable function via generalized integral operators. By applying the established identity and the new so-called generalized m-convex function, some generalized trapezium, Ostrowski and Simpson type integral inequalities have been discovered. Various special cases have been studied as well. Some applications of the present results to special means and new error estimates for the trapezium and midpoint quadrature formula have been investigated. It is hoped that the methods and techniques of this paper could further stimulate the research conducted in the field of integral inequalities.

A Tractable Class of Binary VCSPs via M-Convex Intersection

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions. An important line of VCSP research is to investigate what functions can be solved in polynomial time. Cooper and Živný classified the tractability of binary VCSP instances according to the concept of “triangle,” and showed that the only interesting tractable case is the one induced by the joint winner property (JWP). Recently, Iwamasa, Murota, and Živný made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two quadratic M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given. In this article, we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be represented as the sum of two quadratic M-convex functions and can be solved in polynomial time via an M-convex intersection algorithm? We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case. Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

In this work, we estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Taylor one point and Taylor two points interpolations are used to generalize the new inequalities for m-convex function.

Fractional Exponentially m-Convex Functions and Inequalities

In this article, we introduce a new class of convex functions involving m ∈ [0, 1], which is called exponentially m-convex function. Some new Hermite-Hadamard inequalities for exponentially m-convex functions via Reimann-Liouville fractional integral are deduced. Several special cases are discussed. Results proved in this paper may stimulate further research in different areas of pure and applied sciences.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

Refinement of Jensen\u2019s inequality and estimation of f- and R\xe9nyi divergence via Montgomery identity

Jensen’s inequality is important for obtaining inequalities for divergence between probability distribution. By applying a refinement of Jensen’s inequality (Horváth et al. in Math. Inequal. Appl. 14:777–791, 2011) and introducing a new functional based on an f-divergence functional, we obtain some estimates for the new functionals, the f-divergence, and Rényi divergence. Some inequalities for Rényi and Shannon estimates are constructed. The Zipf–Mandelbrot law is used to illustrate the result. In addition, we generalize the refinement of Jensen’s inequality and new inequalities of Rényi Shannon entropies for an m-convex function using the Montgomery identity. It is also given that the maximization of Shannon entropy is a transition from the Zipf–Mandelbrot law to a hybrid Zipf–Mandelbrot law.

On Hadamard-type inequalities for <i>m</i>-convex functions via Riemann-Liouville fractional integrals

In this paper we prove Hadamard-type inequalities for m-convex function via Riemann-Liouville fractional integrals and Hadamard-type in- equalities for convex function via Riemann-Liouville fractional integral are deduced. Also we find connection with some well known results related to Hadamard inequality.

ON POLYHEDRAL APPROXIMATION OF L-CONVEX AND M-CONVEX FUNCTIONS

In discrete convex analysis, L-convexity and M-convexity are defined for functions in both discrete and continuous variables. Polyhedral L-/M-convex functions connect discrete and continuous versions. Specifically, polyhedral L-/M-convex functions with certain integrality can be identified with discrete versions. Here we show another role of polyhedral L-/M-convex functions: every closed L-/M-convex function in continuous variables can be approximated by polyhedral L-/M-convex functions, uniformly on every compact set. The proof relies on L-M conjugacy under the Legendre-Fenchel transformation.

A note on M-convexity in polyhedral split decomposition of distances

Buneman’s decomposition of a metric can be obtained in a geometric way as the polyhedral split decomposition of a polyhedral convex function associated with a metric. In discrete convex analysis, such a polyhedral convex function is considered as an important example of M-convex functions. Motivated by this, we shed light on the decomposition from the viewpoint of discrete convex analysis. We firstly show that the decomposition is a sum of M-convex functions, where a sum of M-convex functions is not necessarily M-convex in general. We explain why the M-convexity is preserved in the decomposition. We next show that a quadratic M-convex function is split-decomposable at every point. This indicates simplicity of the geometric structure of a quadratic M-convex function. We finally give a characterization of M-convex polyhedra in terms of their facets. This can be used to describe the M-convexity of a polyhedron associated with a function on a cross-free family.