Functional inequalities for the Wright functions

In this paper, our aim is to show some mean value inequalities for the Wright function, such as Turán-type inequalities, Lazarevi?-type inequalities, Wilker-type inequalities and Redheffer-type inequalities. Moreover, we prove monotonicity of ratios for sections of series of Wright functions, the result is also closely connected with Turán-type inequalities. In the end of the paper, we present some other inequalities for the Wright function.     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

Operator-norm inequalities and interpolated trace inequalities

In this article, we investigate some operator-norm inequalities related to some conjectures posed by Hayajneh and Kittaneh that are related to questions of Bourin regarding a special type of inequalities referred to as subadditivity inequalities. While some inequalities are meant to answer these conjectures, other inequalities present reverse-type inequalities for these conjectures. Then, we present some new trace inequalities related to Heinz means inequality and use these inequalities to prove some variants of the aforementioned conjectures.     

Benders,metric and cutset inequalities for multicommodity capacitated network design

Solving multicommodity capacitated network design problems is a hard task that requires the use of several strategies like relaxing some constraints and strengthening the model with valid inequalities. In this paper, we compare three sets of inequalities that have been widely used in this context: Benders, metric and cutset inequalities. We show that Benders inequalities associated to extreme rays are metric inequalities. We also show how to strengthen Benders inequalities associated to non-extreme rays to obtain metric inequalities. We show that cutset inequalities are Benders inequalities, but not necessarily metric inequalities. We give a necessary and sufficient condition for a cutset inequality to be a metric inequality. Computational experiments show the effectiveness of strengthening Benders and cutset inequalities to obtain metric inequalities.     

Gap inequalities for non-convex mixed-integer quadratic programs

Laurent and Poljak introduced a very general class of valid linear inequalities, called gap inequalities, for the max-cut problem. We show that an analogous class of inequalities can be defined for general non-convex mixed-integer quadratic programs. These inequalities dominate some inequalities arising from a natural semidefinite relaxation.     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

Relationships Among Flag <Emphasis Type="Italic">f</Emphasis>-Vector Inequalities for Polytopes

We examine linear inequalities satisfied by the flag $f$-vectors of polytopes. One source of these inequalities is the toric $g$-vector; convolutions of its entries are non-negative for rational polytopes. We prove a conjecture of Meisinger about a redundancy in these inequalities. Another source of inequalities is the {\bf cd}-index; among all $d$-polytopes, each {\bf cd}-index coefficient is minimized on the $d$-simplex. We show that not all of the {\bf cd}-index inequalities are implied by the toric $g$-vector inequalities, and that not all of the toric $g$-vector inequalities are implied by the {\bf cd}-index inequalities. Finally, we show that some inequalities from convolutions of {\bf cd}-index coefficients are implied by other {\bf cd}-index inequalities.     

Hemivariational-like inequalities

In this paper, we introduce and consider a new class of variational inequalities, known as the hemivariational-like inequalities. It is shown that the hemivariational-like inequalities include hemivariational inequalities, variational-like inequalities and the classical variational inequalities as special cases. The auxiliary principle is used to suggest and analyze some iterative methods for solving hemivariational-like inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique.     

A Class of Differential Vector Variational Inequalities in Finite Dimensional Spaces

In this paper, we introduce and study a class of differential vector variational inequalities in finite dimensional Euclidean spaces. We establish a relationship between differential vector variational inequalities and differential scalar variational inequalities. Under various conditions, we obtain the existence and linear growth of solutions to the scalar variational inequalities. In particular we prove existence theorems for Carathéodory weak solutions of the differential vector variational inequalities. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the initial-value differential vector variational inequalities.     

Projection Method Approach for General Regularized Non-convex Variational Inequalities

In this paper, we investigate or analyze non-convex variational inequalities and general non-convex variational inequalities. Two new classes of non-convex variational inequalities, named regularized non-convex variational inequalities and general regularized non-convex variational inequalities, are introduced, and the equivalence between these two classes of non-convex variational inequalities and the fixed point problems are established. A projection iterative method to approximate the solutions of general regularized non-convex variational inequalities is suggested. Meanwhile, the existence and uniqueness of solution for general regularized non-convex variational inequalities is proved, and the convergence analysis of the proposed iterative algorithm under certain conditions is studied.     

Weighted Variational Inequalities

In this paper, we introduce weighted variational inequalities over product of sets and system of weighted variational inequalities. It is noted that the weighted variational inequality problem over product of sets and the problem of system of weighted variational inequalities are equivalent. We give a relationship between system of weighted variational inequalities and systems of vector variational inequalities. We define several kinds of weighted monotonicities and establish several existence results for the solution of the above-mentioned problems under these weighted monotonicities. We introduce also the weighted generalized variational inequalities over product of sets, that is, weighted variational inequalities for multivalued maps and systems of weighted generalized variational inequalities. Extensions of weighted monotonicities for multivalued maps are also considered. The existence of a solution of weighted generalized variational inequalities over product of sets is also studied. The existence results for a solution of weighted generalized variational inequality problem give also the existence of solutions of systems of generalized vector variational inequalities. The first and third author express their thanks to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities. The authors are also grateful to the referees for comments and suggestions improving the final draft of this paper.     

Capacitated facility location: Separation algorithms and computational experience

We consider the polyhedral approach to solving the capacitated facility location problem. The valid inequalities considered are the knapsack cover, flow cover, effective capacity, single depot, and combinatorial inequalities. The flow cover, effective capacity and single depot inequalities form subfamilies of the general family of submodular inequalities. The separation problem based on the family of submodular inequalities is NP-hard in general. For the well known subclass of flow cover inequalities, however, we show that if the client set is fixed, and if all capacities are equal, then the separation problem can be solved in polynomial time. For the flow cover inequalities based on an arbitrary client set and general capacities, and for the effective capacity and single depot inequalities we develop separation heuristics. An important part of these heuristics is based on the result that two specific conditions are necessary for the effective cover inequalities to be facet defining. The way these results are stated indicates precisely how structures that violate the two conditions can be modified to produce stronger inequalities. The family of combinatorial inequalities was originally developed for the uncapacitated facility location problem, but is also valid for the capacitated problem. No computational experience using the combinatorial inequalities has been reported so far. Here we suggest how partial output from the heuristic identifying violated submodular inequalities can be used as input to a heuristic identifying violated combinatorial inequalities. We report on computational results from solving 60 medium size problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

A new affine invariant for polytopes and Schneider's projection problem

New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.

     

Ramsey theory and integrality gap for the independent set problem

We discuss the effectiveness of integer programming for solving large instances of the independent set problem. Typical LP formulations, even strengthened by clique inequalities, yield poor bounds for this problem. We show that a strong bound can be obtained by the use of the so-called rank inequalities, which generalize the clique inequalities. For some problems the clique inequalities imply the rank inequalities, and then a strong bound is guaranteed already by the simpler formulation.     

Wasserstein-Divergence transportation inequalities and polynomial concentration inequalities

In this paper, we study Wasserstein-Divergence transportation inequalities which are the generalization of classical transportation inequalities. We present sufficient and necessary conditions for them separately, which coincide in the limit case. Using this kind of inequalities, we establish polynomial concentration inequalities for probability measures with no exponential moments.     

A recursive procedure to generate all cuts for 0–1 mixed integer programs

We study several ways of obtaining valid inequalities for mixed integer programs. We show how inequalities obtained from a disjunctive argument can be represented by superadditive functions and we show how the superadditive inequalities relate to Gomory's mixed integer cuts. We also show how all valid inequalities for mixed 0–1 programs can be generated recursively from a simple subclass of the disjunctive inequalities.The research of this author was supported by NSF Contract No. ECS-8540898.     

Inequalities between intrinsic volumes

Removing the dependence on dimension of the inequalities between quermassintegrals resulting from the Aleksandrov-Fenchel inequalities leads to universal quadratic inequalities between intrinsic volumes, and to an inequality for the Wills functional. The inequalities correspond to equations which hold in the polytope algebra.     

New valid inequalities for the fixed-charge and single-node flow polytopes

The most effective software packages for solving mixed 0–1linear programs use strong valid linear inequalities derived from polyhedral theory. We introduce a new procedure which enables one to take known valid inequalities for the knapsack polytope, and convert them into valid inequalities for the fixed-charge and single-node flow polytopes. The resulting inequalities are very different from the previously known inequalities (such as flow cover and flow pack inequalities), and define facets under certain conditions.     

Sharp Inequalities for Ratios of Partition Functions of Schrödinger Operators

This paper derives inequalities for multiple integrals from which inequalities for ratios of integrals of heat kernels of certain Schrödinger operators follows. Such ratio inequalities imply inequalities for the partition functions of these operators which extend the spectral gap results proved by R. Bañuelos and P. Méndez-Hernández and B. Davis.     

Merit functions for general mixed quasi-variational inequalities

In this paper, we present some merit functions for general mixed quasi-variational inequalities, and we obtain the equivalent optimization problems to general mixed quasi-variational inequalities. Since the general mixed quasi-variational inequalities include general variational inequalities, quasi-variational inequalities and nonlinear (implicit) complementarity problems as special cases, our results continue to hold for these problems. In this respect, results obtained in this paper represent an extension of previously known results.