Many mathematical models are based on the coupling of two or more optimization problems. This paper surveys possibilities to couple two optimization problems and discusses how solutions of the different models are interrelated with each other. The considered pairs stem from the fields of standard and generalized Nash equilibrium problems, optimistic and pessimistic bilevel problems, saddle problems, standard and generalized semi-infinite problems, robust optimization, Lagrange duality, bicriteria optimization problems, minimax problems, decomposition, and two-stage stochastic optimization. Some connections to vector optimization and variational inequalities are discussed as well.
Compromise programming is based on the concept of the ideal point and relies on distance functions for evaluation purposes. Analysts usually rely on graphical methods to construct the non-dominated frontier to select the best alternatives when dealing with two criteria. However, this graphical analysis becomes more complex when dealing with three or more criteria. To solve this limitation, this work proposes a graphical extension of compromise programming in which the ideal point is replaced with an ideal polygon, and the distance functions to the ideal point are replaced with area functions within multiaxis graphs. We evaluate alternatives by relying on two novel area functions of the polygons formed by the degree of achievement of each criterion on a graph. In addition to reporting novel theoretical results, we illustrate our method with graphical and quantitative analyses of the degree of development of cooperative banking in Europe using five indicators.
It is well-known that duality theory is a fundamental tool in various areas of mathematics. There are great advantages to including or using the dual problem and duality statements. Especially, solving the dual problem can be done using other methods of analysis or numerical mathematics. We consider a primal vector optimization problem with an objective function acting between a linear topological space X and a linear topological space Y equipped with a pointed closed convex cone D⊂ Y with nonempty interior. The feasible set is supposed to be a closed convex cone. The aim of this paper is to construct a simple and easy-to-handle dual problem by exploiting the special structure of the primal problem and using a suitable nonlinear scalarization. The computation of the dual image set involves the minimization of a nonlinear scalarization of the primal vector-valued objective function subject to only one linear inequality constraint. We introduce a new concept of upper semicontinuity for a vector-valued function and prove (weak and strong) duality statements under the assumption that the vector-valued objective function is D-quasiconvex and D-upper semicontinuous. Furthermore, we study special cases.
This article investigates the steady-state behavior of a single-server retrial queue in which arriving units select one of two heterogeneous service types, each followed by an optional immediate re-service of the same type. Units encountering a busy server join a retrial orbit and attempt to access the server after a random delay governed by classical retrial policy. Using the supplementary variable technique, we have derived the Laplace–Stieltjes transforms of the probability generating functions for the number of units in the orbit and in the system. The stochastic decomposition property is established, and pivotal system characteristics, such as the mean and variance of the orbit size, system size, and waiting times both in orbit and in the system are obtained. To validate the analytical results and study the effect of key system parameters, numerical experiments are conducted across three service time distributions namely, exponential, 2-stage Erlang and Hyper-exponential distribution. A cost optimization is carried out employing the parabolic method to identify the optimal arrival rate that minimizes the total expected operating cost per unit time.
In this paper, we give a different vision of a cooperative game inspired by the filtration theory for simplicial complexes. Two important elements are introduced for a filtration associated with a game: bases and vercets. Each of them determines the definition of one new value for cooperative games. We show the kind of situations where both can be applied, with examples. We also introduce operations and algorithms of filtration manipulation to give an axiomatization of the defined values.
Lexicographical extensions of well-known separation theorems for convex sets in ℝ^n are provided in the literature. Particularly, recent theorems regarding open and closed separation of a convex set from any outside point by linear operators from ℝ^n to ℝ^m , in the sense of the lexicographical order of ℝ^m , for each m∈{1,… ,n} , allow to define two new families of properties for convex sets. Based on these results, that we review and extend in this paper, we provide dual characterizations for the consistency of two kinds of systems defined by weak and/or strict lexicographical linear inequalities, and for those inequalities which are satisfied for every solution of a given system. Such results are formulated in terms of appropriate convex hulls of certain sets depending on the coefficients of the system.
In this paper, we examine parametric vector equilibrium problems in which the objective mappings take values in a linear space. We introduce generalized concepts of semistrict and explicit quasiconvexity for functions. Utilizing these concepts, along with a recent algebraic semicontinuity notion for vector-valued maps and an algebraic version of a non-linear scalarization function, we establish sufficient conditions for the non-emptiness of solution sets and the continuity of solution maps for the reference problems. Our results are novel and different from those previously presented in the literature. An application is provided at the end of the paper.
In this paper, we present a new class of “Incompatibility Graphs”(IG) for Data Mining. They are used in Supervised Learning for Box Clustering problems, where an instance is given by a training set of observations, classified as positive and negative, and the objective is to predict the class of any new observation. The Box Clustering algorithm outputs a set of clusters corresponding to labeled hyper-rectangles (homogeneous boxes). In an IG, the vertices correspond to positive observations (points in R^d ), and an edge exists between two vertices if they cannot be clustered together, because all boxes, including both of them contain also some negative points. In this paper, we formalize the notion of IG, and explain how a Box Clustering problem can be modeled using incompatibility graphs. We also show that IGs have an intrinsic interest from a theoretical viewpoint, since we can prove relationships with other well-known graph classes, as, for example, comparability graphs. In particular, for IGs in the plane, we prove strong structural properties, and we provide a list of forbidden induced subgraphs. Furthermore, we show that Incompatibility Graphs can be exploited to solve some key-problems related to Box Clustering, such as the “Maximum Box”and the “Minimum Covering by Boxes”. In fact, we show that these two problems can be formulated as a vertex packing and a vertex coloring on an Incompatibility Graph, respectively, and that one can solve in polynomial time the former and, for two important subclasses of instances, also the latter.
We take into consideration the classical saddle point conditions for programming problems involving convex functions, linear affine functions and a convex set constraint. We prove a nonlinear theorem of the alternative, and we discuss the relationships between the Slater constraint qualification and a constraint qualification proposed by Neustadt (Neustadt LW. Academic Press, New York, pp 323–348,1970) .
We discuss the strategic situation in the Barcelona Town Council after the elections held on May 28, 2023. This institution is described as a simple game, and the Shapley–Shubik index is applied to measure the initial power distribution among the parties. We introduce some ideological incompatibilities in the game and, finally, we discuss the existence of stable coalition structures in the different stated scenarios when the power payoffs to the players are computed using the proportional partitional Shapley value. The key point is the formation of a stable coalition structure. Broadly speaking, a coalition structure is stable if the payoff obtained by each player cannot be increased by a unilateral change in their choice of coalition. In a previous paper, we showed that any simple game admits stable coalition structures when the payoffs are computed using the proportional partitional Shapley value. We find the stable coalition structures for each of the situations analyzed. The results show a fairly complicated problem that the involved politicians and parties should try to solve for the benefit of Barcelona citizens.
This article provides a brief review of the fundamentals of semi-infinite optimization and presents a selection of applications in various fields, as well as a short tutorial on the mathematical approach to optimality theory for linear/convex and general models. Alternative formulations of the KKT optimality conditions are established, emphasizing the relevance of the (local) Farkas–Minkowski property as a constraint qualification. Some personal reminiscences are included at the end of the article.
Existing data envelopment analysis (DEA) methods mainly solve the fixed cost allocation problem in an environment with a powerful central decision-maker. However, there is no central decision-maker in some decision-making environments, which brings new challenges to the fixed cost allocation problem. To this end, this paper proposes a consensus DEA method for allocating fixed costs in the absence of a central decision-maker. First, we propose bounded altruism principle, based on which the cost preference is closer to the actual situation. Second, this paper designs a new DEA allocation model that considers the cost preferences of all decision-making units (DMUs), rendering the individual cost schemes reasonable and acceptable. In addition, we propose a consensus reaching process in that all DMUs can reach a consensus in terms of the cost scheme through egoistic and altruistic adjustment. Finally, the proposed method is applied to two empirical studies of truck fleets and insurance alliance to illustrate its feasibility and practicality.
This paper was initially motivated by the computation of the Lipschitz modulus of the metric projection on polyhedral convex sets in the Euclidean space when both the reference point and the polyhedron where it is projected are subject to perturbations. The paper tackles the more general problem of computing the Lipschitz modulus of the argmin mapping in the framework of canonically perturbed convex quadratic problems. We point out the fact that a point-based formula (depending only on the nominal data) for such a modulus is provided. In this way, the paper extends to the current quadratic setting some results previously developed in linear programming. As an application, we provide a point-based formula for the Lipschitz modulus of the metric projection on a polyhedral convex set.
This paper addresses the challenge of congestion in time-expanded networks, focusing on a case study related to maritime evacuations. The problem is made complex by an endogenous relationship between inputs and outputs, where the assignment of flow to an edge leads to increased congestion, which reflects in later arrivals and changes on the overall network topology. This dynamic interaction between flow and congestion is central to the problem, as it results in a feedback loop that complicates the identification of optimal evacuation paths. The study presents an iterative algorithm inspired by the network-simplex method, designed to handle the evolving nature of congestion while minimizing evacuation time. While the primary case study involves cruise ship evacuations, the approach is generalizable to other cases where congestion and nonlinear flow dynamics are significant factors. By considering lifeboat capacity, passenger mobility restrictions, and the impact of congestion on network structure, this work provides a practical initial plan for an evacuation off-shore, considering a congested, time-expanded network setting.
The current paper analyzes minimum cost spanning tree problems having irreducible costs while incorporating revenues. We prove that, in this context, the core of the associated cost-revenues game (the r-core) is non-empty. In particular, we find an stable allocation in the r-core, based on the CEL bankruptcy rule, that ensures fairness by distributing revenues only among agents that belong to every effective coalition. Therefore, our findings contribute to the literature by identifying structural conditions that guarantee stability in minimum cost spanning tree problems with revenues, overcoming previous results which showed that the r-core could be empty.
Triggered by earlier work on random walks in the quarter-plane, we study the issue of two-queue systems whereby, at least for states (m, n) in some interior part of the state space, the stationary joint system-content distribution u(m, n) can be expressed as a finite linear combination of bivariate geometric terms of type γ ^m δ ^n . Using a transform-based approach, we prove that this is certainly the case if the steady-state joint probability generating function U(z_1,z_2) of the two system contents can be expressed as a bivariate rational function of its two arguments, with mutually prime numerator and denominator, whereby the denominator is the product of two univariate polynomials in z_1 and z_2 , respectively, whose zeroes ẑ_̂1̂ and ẑ_̂2̂ all have multiplicity one. We show that the decay rates γ and δ appearing in u(m, n) are the inverse values of (some of) the zeroes ẑ_̂1̂ and ẑ_̂2̂ , but, in general, there may be zero-pairs (ẑ_̂1̂, ẑ_̂2̂) that do not contribute a bivariate geometric term in u(m, n). For two specific classes of discrete-time two-queue systems, we prove that, when U(z_1,z_2) has the prescribed form, only the zero-pairs that are zero-tuples of the kernel of the system contribute a term in u(m, n). In an extended series of examples, we then demonstrate that, within the two classes, specific instances (corresponding with specific arrival processes) that comply with the condition on U(z_1,z_2) indeed exist. In some examples, we can use existing solutions, but in other cases, we also construct entirely new solutions, thereby identifying several new solvable two-queue models. We observe that, in most cases, the zero-pairs that do contribute terms in u(m, n) are mutually connected, in the sense that each of them has its first or second component in common with at least one other pair that contributes, but we also construct a remarkable example where this is not the case.
In this paper, we generalize some of the inventory models with nonlinear costs presented in Rosling (Oper Res 50:1007–1017, 2002). In particular, we consider random lead times and study models with continuous and discrete demand. In the unified model, we study conditions on the lead time and the demand distribution that ensure the quasiconvexity of the cost function, thereby guaranteeing the existence of optimal (s, S) policies. These conditions are applied to specific well-known models. Particularly, we study the case where backlog costs per time equal to zero, thus providing more general conditions than those available in the literature for specific models. Moreover, we introduce a random lead time for the compound renewal model with periodic review. Our results are mainly based on the reliability properties of the random variables under consideration.
In this paper, the inconsistency index provided by Barzilai [J. Barzilai, J, Consistency Measures for Pairwise Comparison Matrices, Journal of MultiCriteria Decisision Analysis, 7: 123–132, 1998] is revisited. Simpler proofs and expressions giving more insight of the inconsistency index are provided. A new procedure to decrease the inconsistency with the feedback for experts are also provided.
In this paper, we consider a (k,ℓ ) -out-of-n system with three states: up, partial performance, and down. The system has n binary components and is in up state if at least (n-k+1) out of its components work. The state of partial performance is defined when the number of working components is at least (n-ℓ +1) and less than (n-k+1) ; k<ℓ . It is assumed that the system is subject to Marshall–Olkin type of shocks where there are n shocks, each of them affects one component and destroys it, and there is one shock that affects all components and destroys simultaneously all of them. Under this scenario, the joint reliability function of state lifetimes and the corresponding singular and absolutely continuous parts are obtained. For the system with the age of t, the mean residual lifetimes of the system states are explored. Some other aging, stochastic, and dependence properties of the system states are investigated, too. We also extend the model for the case where the system is subject to Marshall–Olkin type of shocks in which the arrived shocks may affect one, two,..., or all components and destroy them. Some illustrative examples are also provided to show the applications of the proposed model.
