Vector optimization problems with nonconvex preferences

In this paper, some vector optimization problems are considered where pseudo-ordering relations are determined by nonconvex cones in Banach spaces. We give some characterizations of solution sets for vector complementarity problems and vector variational inequalities. When the nonconvex cone is the union of some convex cones, it is shown that the solution set of these problems is either an intersection or an union of the solution sets of all subproblems corresponding to each of these convex cones depending on whether these problems are defined by the nonconvex cone itself or its complement. Moreover, some relations of vector complementarity problems, vector variational inequalities, and minimal element problems are also given. While this paper was being revised in September 2006, Professor Alex Rubinov (the second author of the paper) left us due to the illness. This is a very sad news to us. We dedicate this paper to the memory of Professor Rubinov as a mathematician and truly friend.     

On nonconvex equilibrium problems

In this paper, we introduce a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We suggest some iterative schemes for solving nonconvex equilibrium problems by using the auxiliary principle technique. The convergence of the proposed methods either requires partially relaxed strongly monotonicity or pseudomonotonicity. As special cases, we obtain a number of known and new results for solving various classes of equilibrium and variational inequality problems.     

Mixed quasi nonconvex equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving nonconvex equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier known results for solving equilibrium problems and variational inequalities involving the convex sets.     

Discrete ordering relations

Discreteness is usually defined in terms of topology. In discrete mathematics, however, it was sometimes felt that other notions of discreteness should be at hand. We propose the notion of discreteness as a fixed point property of a relation. Using relational algebra, we prove formally that a strict discrete ordering possesses a (generalized) Hasse-diagram.     

Vector complementarity problems with a variable ordering relation

In this paper, we investigate vector complementarity problems with a variable ordering relation. We establish existence results of a solution of a vector complementarity problem under an inclusive type condition. We obtain equivalence results among a vector complementarity problem, a vector variational inequality problem and other related problems.     

Simple chaotic flows with a line equilibrium

Using a systematic computer search, nine simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a line equilibrium. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.     

Vector network equilibrium problems with elastic demands

We consider vector models for complex systems with spatially distributed elements which arise in communication and transportation networks. In order to describe the flow distribution within such a network, we utilize the equilibrium approach, which extends the shortest path one. Being based on this approach, we investigate several networking control problems, with taking into account many factors. As a result, general vector equilibrium problems models with complex behavior of elements are suggested. In particular, they involve elastic demand functions. Due to the presence of many factors, we utilize multicriteria models with respect to different preference relations. The corresponding problems admit efficient solution methods within optimization and equilibrium frameworks.     

On nonconvex optimization problems with separated nonconvex variables

A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x ₁,...,x _n ) and y=( y ₁,...,y _n ), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.     

Vector equilibrium problems with elastic demands and capacity constraints

In this paper, a (weak) vector equilibrium principle for vector network problems with capacity constraints and elastic demands is introduced. A sufficient condition for a (weak) vector equilibrium flow to be a solution for a system of (weak) vector quasi-variational inequalities is obtained. By virtue of Gerstewitz’s nonconvex separation functional ξ, a (weak) ξ-equilibrium flow is introduced. Relations between a weak vector equilibrium flow and a (weak) ξ-equilibrium flow is investigated. Relations between weak vector equilibrium flows and two classes of variational inequalities are also studied.     

Postoptimizing equilibrium flows on large scale networks

In a large variety of applications, equilibrium traffic flows corresponding to a set of slightly modified input data must be computed sequentially. Until now, it is believed that only a disaggregate decomposition approach, that works explicitly on the path flow space, offers the postoptimization capability. This note proposes a new postoptimization method to deal with perturbations of the traffic demand input that does not require path information. Numerical experiments on practical size networks show a drastic reduction in the number of iterations with respect to the naive restart approach.     

Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry

In this paper, we investigate the system of equations modelling multicomponent reactive flows with detailed transport and complex chemistry in the limit of partial equilibrium. The reduced system is obtained using a projection step compatible with the chemical entropy production. The reduced multicomponent transport and convection fluxes are shown to be compatible with the mathematical entropy thus providing a symmetric form as well as normal forms for the reduced system. This yields global existence and asymptotic stability around constant equilibrium states for the Cauchy problem on the partial equilibrium manifold in all space dimensions. Copyright © 2004 John Wiley & Sons, Ltd.     

On existence of vector equilibrium flows with capacity constraints of arcs

In this paper, we propose a (weak) vector equilibrium principle with capacity constraints of arcs. By proving the existence of solutions for the weighted variational inequality, we establish the existence results of (weak) vector traffic equilibrium flows with capacity constraints of arcs.     

Isohedra with nonconvex faces

An isohedron is a 3-dimensional polyhedron all faces of which are equivalent under symmetries of the polyhedron. Many well known polyhedra are isohedra; among them are the Platonic solids, the polars of Archimedean polyhedra, and a variety of polyhedra important in crystallography. Less well known are isohedra with nonconvex faces. We establish that such polyhedra must be starshaped and hence of genus 0, that their faces must be star-shaped pentagons with one concave vertex, and that they are combinatorially equivalent to either the pentagonal dodecahedron, or to the polar of the snub cube or snub dodecahedron.Supported in part by grants from the USA National Science Foundation.     

On the equilibrium of circular flows under gravity

This paper investigates the stability of an incompressible inviscid rotatory couette flow confined between two corotating and coaxial vertical cylinders under the force field of gravity. The stationary distributions of density and pressure are functions of the radial and axial coordinates with the azimuthal component of velocity being an arbitrary function of radial coordinate. The perturbations concerned are axisymmetric and infinitesimaly small in nature. Sufficient criteria of instability are derived for both vertically stable and unstable density stratifications¹ and this clearly shows that the necessary and sufficient character of Rayleigh's criterion does not hold good in the present circumstances.     

Dominance relations and ranking of units by using interval number ordering with cross-efficiency intervals

This paper proposes an approach to the cross-efficiency evaluation that considers all the optimal data envelopment analysis (DEA) weights of all the decision-making units (DMUs), thus avoiding the need to make a choice among them according to some alternative secondary goal. To be specific, we develop a couple of models that allow for all the possible weights of all the DMUs simultaneously and yield individual lower and upper bounds for the cross-efficiency scores of the different units. As a result, we have a cross-efficiency interval for the evaluation of each unit. Existing order relations for interval numbers are used to identify dominance relations among DMUs and derive a ranking of units based on the cross-efficiency intervals provided. The approach proposed may also be useful for assessing the stability of the cross-efficiency scores with respect to DEA weights that can be used for their calculation.     

Vector network equilibrium problems and nonlinear scalarization methods

The conventional equilibrium problem found in many economics and network models is based on a scalar cost, or a single objective. Recently, equilibrium problems based on a vector cost, or multicriteria, have received considerable attention. In this paper, we study a scalarization method for analyzing network equilibrium problems with vector-valued cost function. The method is based on a strictly monotone function originally proposed by Gerstewitz. Conditions that are both necessary and sufficient for weak vector equilibrium are derived, with the prominent feature that no convexity assumptions are needed, in contrast to other existing scalarization methods.