Criteria for efficiency in vector optimization

We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect, whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs 1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions, such as linear independence of the gradients, convexity or directional monotonicity. This work was partially supported by grant 55681 of the CONACyT.     

Characterizing robust set containments and solutions of uncertain linear programs without qualifications

Qualification-free dual characterizations are given for robust polyhedral set containments where a robust counterpart of an uncertain polyhedral set is contained in another polyhedral set or a polyhedral set is contained in a robust counterpart of an uncertain polyhedral set. These results are used to characterize robust solutions of uncertain linear programs, where the uncertainty is defined in terms of intervals or l₁-balls. The hidden separable sub-linearity of the robust counterparts allows qualification-free dual characterizations.     

A UNIFIED APPROACH TO A CLASS OF STERLING-TYPE PAIRS

Here presented is a unified approach to a wide class of symmetric Sfirling number pairs,which is determined by four complex parameters and includes as particular cases various previousextensions of Stirling numbers due to Carlicz, Howard, Koutras, Gould-Hopper, respectively.Certain Schlomilch-type formulas and congruence properties will be also exhibited.     

Evaluating computational efficiency: A stochastic approach

This paper introduces the use of stochastic models for the evaluation of relative computational efficiency of algorithms. Such an approach is used for the comparison of computational efficiency of three algorithms for quadratic programming.     

Convergence Analysis of a Class of Penalty Methods for Vector Optimization Problems with Cone Constraints

In this paper, we consider convergence properties of a class of penalization methods for a general vector optimization problem with cone constraints in infinite dimensional spaces. Under certain assumptions, we show that any efficient point of the cone constrained vector optimization problem can be approached by a sequence of efficient points of the penalty problems. We also show, on the other hand, that any limit point of a sequence of approximate efficient solutions to the penalty problems is a weekly efficient solution of the original cone constrained vector optimization problem. Finally, when the constrained space is of finite dimension, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original cone constrained vector optimization problem if Mangasarian–Fromovitz constraint qualification holds at the limit point.This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

A Brézis-Browder principle on partially ordered spaces and related ordering theorems

Through a simple extension of Brézis-Browder principle to partially ordered spaces, a very general strong minimal point existence theorem on quasi ordered spaces, is proved. This theorem together with a generic quasi order and a new notion of strong approximate solution allow us to obtain two strong solution existence theorems, and three general Ekeland variational principles in optimization problems where the objective space is quasi ordered. Then, they are applied to prove strong minimal point existence results, generalizations of Bishop-Phelps lemma in linear spaces, and Ekeland variational principles in set-valued optimization problems through a set solution criterion.     

The impact of tourist destination on hotel efficiency: A data envelopment analysis approach

This paper evaluates the impact of location on hotel efficiency using a sample of 400 Spanish hotels, the novel aspect being that location is considered at the tourist destination level. Moreover, for the first time, the location variables are based on the main theoretical models concerning location in the hotel sector, namely geographical positioning models, agglomeration and urbanization economic models and competitive environment models. The methodology consists of a four-stage data envelopment analysis (DEA) model that decomposes super-efficiency in the portion attributable to the tourist destination and the portion attributable to hotel management. Then, managerial efficiency is regressed against hotel characteristics, while tourist destination efficiency is explained by the characteristic of each location. The findings highlight the importance of tourist destinations, providing novel empirical support for the propositions of the main location models. Indeed, the tourist destination is the main cause of differences in the level of efficiency among hotels. The occupancy level, degree of seasonality and market concentration are the variables with the greater impact on efficiency.     

A potential of fuzzy relations with a linear structure: The contractive case

In this paper we develop a potential theory of fuzzy relations on the positive orthant in a Euclidean space. By introducing a linear structure for fuzzy relations, the existence of a potential and its characterization by fuzzy relational equation are derived under the assumption of contraction and compactness. In the one-dimensional unimodal case, a potential is given explicity. Also, a numerical example is shown to illustrate our approaches.     

Towards a unified territorial design approach — Applications, algorithms and GIS integration

Territory design may be viewed as the problem of grouping small geographic areas into larger geographic clusters called territories in such a way that the latter are acceptable according to relevant planning criteria. In this paper we review the existing literature for applications of territory design problems and solution approaches for solving these types of problems. After identifying features common to all applications we introduce a basic territory design model and present in detail two approaches for solving this model: a classical location-allocation approach combined with optimal split resolution techniques and a newly developed computational geometry based method. We present computational results indicating the efficiency and suitability of the latter method for solving large-scale practical problems in an interactive environment. Furthermore, we discuss extensions to the basic model and its integration into Geographic Information Systems.     

A special ordered set approach for optimizing a discontinuous separable piecewise linear function

We give a special ordered set (SOS) approach that optimizes a discontinuous separable piecewise linear function, even when a mixed-integer programming (MIP) model is not available for it. When a MIP model is available, our SOS model gives a linear programming relaxation bound that is as good as the MIPs.     

Continuous-time linear programming problems revisited: A perturbation approach

The class of continuous-time linear programming problems under the assumption that the constraints are satisfied almost everywhere in the time interval [0,?T]?is taken into account in this article. Under this assumption, its corresponding discretized problems cannot be formulated by equally dividing the time interval [0,?T]?as subintervals of [0,?T]?. In this article, we also introduce the perturbed continuous-time linear programming problems to prove the strong duality theorem when the constraints are assumed to be satisfied a.e. in [0,?T]?.     

Inverse problem on a tree-shaped network: unified approach for uniqueness

In this article, we prove uniqueness results for coefficient inverse problems regarding wave, heat or Schrödinger equation on a tree-shaped network, as well as the corresponding stability result of the inverse problem for the wave equation. The objective is the determination of the potential on each edge of the network from the additional measurement of the solution at all but one external end points. Several results have already been obtained in this precise setting or in similar cases, and our main goal is to propose a unified and simpler method of proof of some of these results. The idea which we will develop for proving the uniqueness is to use a more traditional approach in coefficient inverse problems by Carleman estimates. Afterwards, using an observability estimate on the whole network, we apply a compactness–uniqueness argument and prove the stability for the wave inverse problem.     

On the structure and properties of a linear multilevel programming problem

Many decision-making situations involve multiple planners with different, and sometimes conflicting, objective functions. One type of model that has been suggested to represent such situations is the linear multilevel programming problem. However, it appears that theoretical and algorithmic results for linear multilevel programming have been limited, to date, to the bounded case or the case of when only two levels exist. In this paper, we investigate the structure and properties of a linear multilevel programming problem that may be unbounded. We study the geometry of the problem and its feasible region. We also give necessary and sufficient conditions for the problem to be unbounded, and we show how the problem is related to a certain parametric concave minimization problem. The algorithmic implications of the results are also discussed.This research was supported by National Science Foundation Grant No. ECS-85-15231.     

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

A problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering.     

Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

     

Efficiency and effectiveness of Spanish football teams: a three-stage-DEA approach

In this paper, we apply a three-stage-DEA model to the Spanish Professional Football League, which means separating the teams’ economic behaviour into three components: operating efficiency—of the offence and defence—athletic or operating effectiveness, and social effectiveness. The results showed that the technical inefficiency of the defence is greater than that of the offence, both being caused by aspects linked to the poor management of players’ abilities and by the football team’s size. Teams showed a favourable evolution of their offensive and defensive efficiency during the 2004/2005 season and to a lesser extent in the season before. The point system assigned by the professional football league regulations evaluates the teams’ athletic effectiveness, but we detected that the teams with the most experience perform athletically in a more effective manner. Their social effectiveness is strongly related to the level of play in itself and to factors linked to their PFL ranking: participation in international competitions for important football teams; or the struggle of minor football teams to stay in the first division.     

Treatment of set order relations by means of a nonlinear scalarization functional: a full characterization

In this paper, we show how a nonlinear scalarization functional can be used in order to characterize several well-known set order relations and which thus plays a key role in set optimization. By means of this functional, we derive characterizations for minimal elements of set-valued optimization problems using a set approach. Our methods do not rely on any convexity assumptions on the considered sets. Furthermore, we develop a derivative-free descent method for set optimization problems without convexity assumptions to verify the usefulness of our results.     

Neighborhood assignments and cardinal functions: a unified approach to metrization and uniformity

We use neighborhood assignments and cardinal functions to give a unified approach to metrizability and uniformity. This leads to a number of characterizations of m(X), the metrizability degree of X, u(X), the uniform weight of X, and w(X), the weight of X. For X normal (and regular), m(X) = u(X); it is unknown whether this result extends to completely regular spaces.     

Harnack inequality on configuration spaces: The coupling approach and a unified treatment

In this paper, we establish the dimension-free Harnack inequality on configuration spaces by using the coupling argument. Furthermore, a unified treatment is also used to prove the equivalence between the Harnack inequality on configuration space and that on the corresponding base space under a very mild condition.