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.     

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.     

On non-smooth α-invex functions and vector variational-like inequality

In this paper, we establish some relationships between vector variational-like inequality and non-smooth vector optimization problems under the assumptions of α-invex non-smooth functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality, under non-smooth pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends an earlier work of Ruiz-Garzon et al. (J Oper Res 157:113–119, 2004) to a wider class of functions, namely the non-smooth pseudo-α-invex functions. Moreover, this work extends an earlier work of Mishra and Noor (J Math Anal Appl 311:78–84, 2005) to non-differentiable case.     

Dynamic Walrasian price equilibrium problem: evolutionary variational approach with sensitivity analysis

Many authors have been devoted to the study of the static general economic equilibrium problem regulated to Walras’ law (see e.g. Arrow and Debreu in Econometrica 22:265–290, 1954; Arrow and Hahn in General competitive analysis, 1991; Arrow et al. in Econometrica 27:82–109, 1959; Border in Fixed point theorems with application to economics and game theory, Cambridge University Press, Cambridge, 1985; Dafermos in Math Programm 46:391–402, 1990; Dafermos and Zhao in Oper Res Lett 10:396–376, 1991; Donato et al. in J Glob Optim, 2007; Hahn in Stability, North Holland, Amsterdam, 1982; Jofré et al. in Math Oper Res, 2007; Nagurney in Network economics—a variational inequality approach, Kluwer, Dordrecht, 1999; Nagurney and Zhao in Network formalism for pure exchange economic equilibria, World Scientific Press, Singapore, 1993; Walker in J Polit Econ 94(4), 1987; Walras in Elements d’Economique Politique Pure, Corbaz, Lausanne, Switzerland, 1874; Zhao in Variational inequalities in general equilibrium: analysis and computation, PhD thesis, Brown University, 1988; and their bibliography). The aim of this paper is to provide a first approach to a particular dynamic general economic equilibrium problem: a Walrasian price equilibrium problem when the data are time-dependent. The equilibrium conditions that describe this pure exchange economic model are expressed in terms of an evolutionary variational inequality, for which existence and sensitivity results are given. Moreover, our problem can be expressed in a common way to many other equilibrium problems.     

Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azéma [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process. Received: 30 April 1997 / Revised version: 17 September 1999 /?Published online: 11 April 2000     

Lagrange Multipliers for <Emphasis Type="Italic">ε</Emphasis>-Pareto Solutions in Vector Optimization with Nonsolid Cones in Banach Spaces

This paper presents some results concerning the existence of the Lagrange multipliers for vector optimization problems in the case where the ordering cone in the codomain has an empty interior. The main tool for deriving our assertions is a scalarization by means of a functional introduced by Hiriart-Urruty (Math. Oper. Res. 4:79–97, 1979) (the so-called oriented distance function). Moreover, we explain some applications of our results to a vector equilibrium problem, to a vector control-approximation problem and to an unconstrainted vector fractional programming problem.     

Generalized Nash equilibrium

For accep/reject games and coalitionless games, the classical Roos-Nash equilibrium is generalized to a so-called strongly dependent equilibrium, which exists for a wider class of games than the classical equilibrium. The following hierarchical chain of progressively stronger equilibria is established: symmetrical activeA-equilibrium, strongly dependent equilibrium, symmetricalB-equilibrium, Roos-Nash classical dependent equilibrium. The first three of these have been proposed by the author as weaker versions of the classical coalitionless equilibrium. Translated from Nelineinaya Dinamika i Upravlenie, pp. 217–227, 1999.     

Revisiting Generalized Nash Games and Variational Inequalities

Generalized Nash games with shared constraints represent an extension of Nash games in which strategy sets are coupled across players through a shared or common constraint. The equilibrium conditions of such a game can be compactly stated as a quasi-variational inequality (QVI), an extension of the variational inequality (VI). In (Eur. J. Oper. Res. 54(1):81–94, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately defined VI solves the QVI. This is a particularly important result, given that VIs are generally far more tractable than QVIs. However Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007) suggested that the hypotheses of this result are difficult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harker’s result for these games with the aim of formally establishing its reach. Specifically, we show that if Harker’s result is applied in a natural manner, its hypotheses are impossible to satisfy in most settings, thereby supporting the observations of Facchinei et al. But we also show that an indirect application of the result extends the realm of applicability of Harker’s result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harker’s result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007), in which it is shown that a suitably defined VI provides a solution to the QVI of a shared-constraint game.     

Torsion and conformally Anosov flows in contact Riemannian geometry

We prove that on a compact (non Sasakian) contact metric 3-manifold with critical metric for the Chern-Hamilton functional, the characteristic vector field ξ is conformally Anosov and there exists a smooth curve in the contact distribution of conformally Anosov flows. As a consequence, we show that negativity of the ξ-sectional curvature is not a necessary condition for conformal Anosovicity of ξ (this completes a result of [4]). Moreover, we study contact metric 3-manifolds with constant ξ-sectional curvature and, in particular, correct a result of [13].     

On Implicit Multifunction Theorems

We give implicit multifunction results generalizing to multifunctions the Robinson’s implicit function theorem (Robinson, Math Oper Res 16(2):292–309, 1991). To this end, we use parametric error bounds estimates for a suitable function refining the one given in Azé and Corvellec (ESAIM Control Optim Calc Var 10:409–425, 2004). Sharp approximations of the implicit multifunctions are given extending the results of Nachi and Penot (Control Cybernet 35:871–901, 2005). Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

On Economic Equilibrium Type Problems with Applications

The existence solution for a class of economic equilibrium type problems in reflexive Banach space is considered. New results concerning the variational inequality approach to Arrow–Debreu model of economic equilibrium introduced in Naniewicz (Math Oper Res 32:436–466, 2007) are found and applied to ensure the existence of Pareto optimal solutions for a class of multiobjective optimization problems with so-called budget-like constraints. To achieve this goal, the theory of pseudo-monotone multivalued mappings combined with some fixed point technique for multivalued mappings with nonconvex values is used.     

A remark on a standard and linear vector network equilibrium problem with capacity constraints

In this paper, (weak) vector equilibrium principle with capacity constraints is introduced. A necessary condition that a vector minimum cost flow is a vector equilibrium flow with capacity constraints is obtained. When the number of paths connecting with each pair of source and sink is less than or equal to 2, a sufficient condition for a vector minimum cost flow to be a vector equilibrium flow is also obtained. A generalized (weak) vector equilibrium principle is also introduced. Without any additional assumption, a necessary and sufficient condition for a (weak) vector minimum cost flow to be a generalized (weak) vector equilibrium flow is obtained.     

On price competition in location-price models with spatially separated markets

In this paper we study price competition for two types of location-price models in which facility locations are set up and price decisions have to be made in order to maximise profit. We discuss the existence and determination of equilibrium prices in a general location space when facilities have different production costs. It is assumed that each price is bounded from below and demand for a single homogeneous product is price-inelastic. When facilities set mill prices, a price equilibrium rarely exists and necessary conditions for existence are obtained. In particular, when the location space is a tree network, we give a characterisation of the locations for which a unique equilibrium exists for two competitors. With spatial price discrimination, though equilibrium prices might not exist, it is shown that ε-equilibrium prices always exist for any locations of the facilities. A characterisation of ε-equilibrium is also given. Then the location-price problem is reduced to a location problem. A comparison of results with the two types of price determination is also presented. This work has been supported by the Ministry of Science and Technology of Spain under the research project BEC2002-01026, in part financed by the European Regional Development Fund (ERDF).     

A viscosity approximation method for equilibrium problems,fixed point problems of nonexpansive mappings and a general system of variational inequalities

In this paper, we introduce and study a new iterative scheme for finding the common element of the set of common fixed points of a sequence of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the general system of variational inequality for α and μ-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Ceng et al. (Math Meth Oper Res 67:375–390, 2008) and many others.     

A simple GAP-canceling algorithm for the generalized maximum flow problem

We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O(mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O(m log(mB/ε)) iterations of O(mn) time to compute an ε-optimal flow, or O(m ² log (mB)) iterations to compute an optimal flow, for an overall running time of O(m ³ nlog(mB)). The fastest known running time for this problem is , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997). David P. Williamson is supported in part by an IBM Faculty Partnership Award and NSF grant CCF-0514628.     

Algebraic Coding of Expansive Group Automorphisms and Two-sided Beta-Shifts

 Let α be an expansive automorphisms of compact connected abelian group X whose dual group is cyclic w.r.t. α (i.e. is generated by for some ). Then there exists a canonical group homomorphism ξ from the space of all bounded two-sided sequences of integers onto X such that , where σ is the shift on . We prove that there exists a sofic subshift such that the restriction of ξ to V is surjective and almost one-to-one. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue and all other eigenvalues of absolute value we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the two-sided beta-shift is surjective and almost one-to-one. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]). (Received 27 October 1998; in revised form 17 May 1999)     

State-space collapse in stationarity and its application to a multiclass single-server queue in heavy traffic

Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56–90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45–56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs).     

Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.     

Equivalence of Equilibrium Problems and Least Element Problems

In this paper, we introduce the concept of feasible set for an equilibrium problem with a convex cone and generalize the notion of a Z-function for bifunctions. Under suitable assumptions, we derive some equivalence results of equilibrium problems, least element problems, and nonlinear programming problems. The results presented extend some results of [Riddell, R.C.: Equivalence of nonlinear complementarity problems and least element problems in Banach lattices. Math. Oper. Res. 6, 462–474 (1981)] to equilibrium problems. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Educational Science Foundation of Chongqing (KJ051307).     

On vector variational-like inequality problems

In this paper, we establish some relationships between vector variational-like inequality and vector optimization problems under the assumptions of α-invex functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problems, under pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119] to a wider class of functions, namely the pseudo-α-invex functions studied in a recent work of Noor [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. 5 (2004) 1-9].