Fejér and Hermite–Hadamard type results for H-invex functions with applications

Positivity - We investigate the class of H-invex functions including, e.g., the subclasses of convex, c-strongly convex, $$ \varphi $$ -uniformly convex and superquadratic functions. For H-invex...     

Sup-Sums Principles for F-Divergence and a New Definition for t-Entropy

Journal of Theoretical Probability - The article presents new $${\sup }$$ -sums principles for integral F-divergence for arbitrary convex functions F on the whole real axis and arbitrary (not...     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

Mathematical Programming - We consider the problem of minimizing composite functions of the form $$f(g(x))+h(x)$$ , where f and h are convex functions (which can be nonsmooth) and...     

On boundedness of (quasi-)convex integer optimization problems

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained integer optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or quasi-convex polynomial function over the feasible set contained in , and defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities. The conditions for boundedness are provided in the form of an implementable algorithm, terminating after a finite number of iterations, showing that for the considered class of functions, the integer programming problem with nonempty feasible region is unbounded if and only if the associated continuous optimization problem is unbounded. We also prove that for a broad class of objective functions (which in particular includes polynomials with integer coefficients), an optimal solution set of the constrained integer problem is nonempty over any subset of .     

Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex

Positivity - For a compact convex subset K of a locally convex Hausdorff space, a measurement on A(K) is a finite family of positive elements in A(K) normalized to the unit constant $$1_K , $$...     

On the symmetry function of a convex set

We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}We attempt a broad exploration of properties and connections between the symmetry function of a convex set S and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point , let sym(x,S) denote the symmetry value of x in S: , which essentially measures how symmetric S is about the point x, and define x ^* is called a symmetry point of S if x ^* achieves the above maximum. The set S is a symmetric set if sym (S)=1. There are many important properties of symmetric convex sets; herein we explore how these properties extend as a function of sym (S) and/or sym (x,S). By accounting for the role of the symmetry function, we reduce the dependence of many mathematical results on the strong assumption that S is symmetric, and we are able to capture and otherwise quantify many of the ways that the symmetry function influences properties of convex sets and functions. The results in this paper include functional properties of sym (x,S), relations with several convex geometry quantities such as volume, distance, and cross-ratio distance, as well as set approximation results, including a refinement of the L?wner-John rounding theorems, and applications of symmetry to probability theory on convex sets. We provide a characterization of symmetry points x ^* for general convex sets. Finally, in the polyhedral case, we show how to efficiently compute sym(S) and a symmetry point x ^* using linear programming. The paper also contains discussions of open questions as well as unproved conjectures regarding the symmetry function and its connection to other areas of convexity theory. Dedicated to Clovis Gonzaga on the occasion of his 60th birthday.     

Sufficient conditions for the convexity of the level sets of ground-state solutions

Let Ω be a bounded convex domain in . We consider constrained minimization problems related to the Euler-Lagrange equation

On compact anisotropic Weingarten hypersurfaces in Euclidean space

Let $$\Omega \subset \mathbb {R}^n$$ be a bounded mean convex domain. If $$\alpha <0$$ , we prove the existence and uniqueness of classical solutions of the Dirichlet problem in $$\Omega $$ for the $$\alpha $$ -singular minimal surface equation with arbitrary continuous boundary data.     

The problem concerning the epimorphicity of a convolution operator in convex domains

We deduce an epimorphicity criterion for the convolution operator $$(a * x) (z) = \frac{1}{{2\pi i}}\oint {x (t) \tilde a (t - z) dt} ,$$ acting from a space of functions analytic in a convex domain into another such space;a(z) is the Borel transformation of the exponential functiona(z).     

On lattices of convex sets in {\mathbb{R}}^{n}

Properties of several sorts of lattices of convex subsets of are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of and the lattice of all convex subsets of The lattices of arbitrary, of open bounded, and of compact convex sets in all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date.     

Tight convex underestimators for $${\mathcal{C}^2}$$ -continuous problems: II. multivariate functions

In Part I (Gounaris, C.E., Floudas, C.A.: Tight convex understimators for -continuous functions: I: Univariate functions. J. Global Optim. (2008). doi: ), we introduced a novel approach for the underestimation of univariate functions which was based on a piecewise application of the well-known αBB underestimator. The resulting underestimators were shown to be very tight and, in fact, can be driven to coincide with the convex envelopes themselves. An approximation by valid linear supports, resulting in piecewise linear underestimators was also presented. In this paper, we demonstrate how one can make use of the high quality results of the approach in the univariate case so as to extend its applicability to functions with a higher number of variables. This is achieved by proper projections of the multivariate αBB underestimators into select two-dimensional planes. Furthermore, since our method utilizes projections into lower-dimensional spaces, we explore ways to recover some of the information lost in this process. In particular, we apply our method after having transformed the original problem in an orthonormal fashion. This leads to the construction of even tighter underestimators, through the accumulation of additional valid linear cuts in the relaxation.     

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Acta Mathematica Hungarica - We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences...     

When is a spherical body of constant diameter of constant width?

Aequationes mathematicae - We prove that a smooth convex body of diameter $$\delta &lt; \frac{\pi }{2}$$ on the d-dimensional unit sphere $$S^d$$ is of constant diameter $$\delta $$ if and only...     

Solving the Sum-of-Ratios Problem by an Interior-Point Method

We consider the problem of minimizing the sum of a convex function and of p1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasi-convex. We give a brief discussion of the problem and prove that in spite of its special structure, the problem is -complete even when only p=1 fraction is involved. We then show how the problem can be reduced to the minimization of a function of p variables where the function values are given by the solution of certain convex subproblems. Based on this reduction, we propose an algorithm for computing the global minimum of the problem by means of an interior-point method for convex programs.     

Trajectory of an Observer Tracking the Motion of an Object around a Convex Set in $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An object t moving in $${{\mathbb{R}}^{3}}$$ goes around a solid convex set along the shortest path $$\mathcal{T}$$ under observation. The task of an observer f (moving at the...     

Complexity bounds for primal-dual methods minimizing the model of objective function

Mathematical Programming - We provide Frank–Wolfe ( $$\equiv $$ Conditional Gradients) method with a convergence analysis allowing to approach a primal-dual solution of convex optimization...     

Explicit Solutions for a Series of Optimization Problems with 2-Dimensional Control via Convex Trigonometry

Doklady Mathematics - We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $$\Omega $$ . Solutions to these problems are obtained...     

On Hermite-Hadamard-type characterizations of higher-order differential inequalities

Let $I$ be an open interval of $\mathbb{R}$ and $f: I\to \mathbb{R}$. It is well-known that $f$ is convex in $I$ if and only if, for all $x,y\in I$ with $x相似文献