Integral inequalities for generalized concave or convex functions

Let ψ be convex with respect to ?, B a convex body in Rⁿ and f a positive concave function on B. A well-known result by Berwald states that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{ψ(f(x)) dx ? n ∝}{}_0{}^1\text{ψ(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$ (1) if ξ is chosen such that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{?(f(x)) dx = n ∝}{}_0{}^1\text{?(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$.The main purpose in this paper is to characterize those functions f : B → R₊ such that (1) holds.     

Convex envelopes of products of convex and component-wise concave functions

In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and g(y), y ? \mathbb Rⁿ{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.     

A method for globally minimizing concave functions over convex sets

A method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear. The algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region. The algorithm does not involve cuts of the feasible region, requires only simplex pivot operations and univariate search computations to be performed, allows the objective function to be lower semicontinuous and nonseparable, and is guaranteed to converge to the global solution. Computational aspects of the algorithm are discussed.     

Submajorisation inequalities for convex and concave functions of sums of measurable operators

It is shown that non-negative, increasing, convex (respectively, concave) functions are superadditive (respectively, subadditive) with respect to submajorisation on the positive cone of the space of all τ-measurable operators affiliated with a semifinite von Neumann algebra. This extends recent results for n × n-matrices by Ando-Zhan, Kosem and Bourin-Uchiyama. This work was partially supported by the Australian Research Council.     

Conditions for boundedness in concave programming under reverse convex and convex constraints

In this paper, we are concerned with the problem of boundedness in the constrained global maximization of a convex function. In particular, we present necessary and sufficient conditions for boundedness of a feasible region defined by reverse convex constraints and we establish sufficient and necessary conditions for existence of an upper bound for a convex objective function defined over the system of concave inequality constraints. We also address the problem of boundedness in the global maximization problem when a feasible region is convex and unbounded.     

Explicit convex and concave envelopes through polyhedral subdivisions

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.     

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

Integrals of logarithmically concave functions

In this note we consider integrals of the form

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing conve