The liberal paradox and the pareto set

We say that the liberal paradox occurs where the set of alternatives is a disjoint union of sets $\text{X}{}_{\mathrm{i}}$, one for each voter, if and only if the Pareto set contains no element $\text{u}$ of any $\text{X}{}_{\mathrm{i}}$ such that voter $\text{i}$ prefers $\text{u}$ to any other element of $\text{X}{}_{\mathrm{i}}$.For random profiles, we estimate the size of the Pareto set, and show that (1) if the number of voters is held fixed but the number of alternatives tends to infinity, the probability of the liberal paradox tends to 1; (2) if the number of alternatives is a fixed ratio to the number of voters and the number of voters tends to infinity, the probability of the liberal paradox tends to zero; (3) the rights can always be redistributed to the voters so that the liberal paradox does not occur; and (4) for single-peaked preferences if the sets of rights form intervals, the liberal paradox does not occur.     

Inexact Multi-Objective Local Search Proximal Algorithms: Application to Group Dynamic and Distributive Justice Problems

We introduce and examine an inexact multi-objective proximal method with a proximal distance as the perturbation term. Our algorithm utilizes a local search descent process that eventually reaches a weak Pareto optimum of a multi-objective function, whose components are the maxima of continuously differentiable functions. Our algorithm gives a new formulation and resolution of the following important distributive justice problem in the context of group dynamics: In each period, if a group creates a cake, the problem is, for each member, to get a high enough share of this cake; if this is not possible, then it is better to quit, breaking the stability of the group.     

The Borda rule and Pareto stability: a further comment

In their paper “The Borda rule and Pareto stability: a comment” published in 1979 by Econometrica, Farkas and Nitzan revealed the “intimate relationship” between the Borda rule and the Pareto criterion. The idea was the following: in a profile of total orders, when there is a candidate who obviously wins under unanimous agreement of the voters, that candidate should be in the choice set. In a profile where there is no obvious winner, the candidates that are the closest to unanimity should be chosen. According to this principle, they defined a choice rule called “closeness to unanimity” and they showed that it is equivalent to the Borda rule. In our paper, we give an equivalent result for a ranking rule. Then we try to obtain similar results when aggregating profiles of tournaments, weak orders, semiorders, fuzzy relations, … and we show that the definition of an obvious winner is no more obvious.     

Some Paradoxes in Special Relativity and the Resolutions

The special theory of relativity is the foundation of modern physics, but its unusual postulate of invariant vacuum speed of light results in a number of plausible paradoxes. This situation leads to radical criticisms and suspicions against the theory of relativity. In this paper, from the perspective that the relativity is nothing but a geometry, we give a uniform resolution to some famous and typical paradoxes such as the ladder paradox, the Ehrenfest’s rotational disc paradox. The discussion shows that all the paradoxes are caused by misinterpretation of concepts. We misused the global simultaneity and the principle of relativity. As a geometry of Minkowski space-time, special relativity can never result in a logical contradiction.     

混合双参数广义Pareto分布的参数估计

Pareto分布族因其厚尾特点,在金融分析、寿命分析中都是非常重要的统计模型.但是对于混合双参广义Pareto分布,在模型参数估计时,传统的矩法估计和极大似然估计在理论上可以实现,实践时比较困难.本文应用EM算法之ECM算法,研究了混合广义Pareto分布在完全数据场合下的参数估计问题,并模拟说明EM算法来估计混合广义Pareto分布是一种容易实现又非常有效的方法.     

Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.     

Multiobjective evolutionary algorithms for complex portfolio optimization problems

This paper investigates the ability of Multiobjective Evolutionary Algorithms (MOEAs), namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving complex portfolio optimization problems. The portfolio optimization problem is a typical bi-objective optimization problem with objectives the reward that should be maximized and the risk that should be minimized. While reward is commonly measured by the portfolio’s expected return, various risk measures have been proposed that try to better reflect a portfolio’s riskiness or to simplify the problem to be solved with exact optimization techniques efficiently. However, some risk measures generate additional complexities, since they are non-convex, non-differentiable functions. In addition, constraints imposed by the practitioners introduce further difficulties since they transform the search space into a non-convex region. The results show that MOEAs, in general, are efficient and reliable strategies for this kind of problems, and their performance is independent of the risk function used.     

Mathematical and computer modelling of the Pareto principle

The well-known Pareto principle states that approximately 80% of wealth was concentrated in about 20% of a population. Two problems have been reported on the abuse of the 80/20 phenomenon: (1) the measure deviates from 80/20 in many applications of the principle; under what condition, then, will the 80/20 rule be true? And, (2) if the 80/20 measure is correct at the time of analysis, will it be true over time? In this paper, we address these two problems by conducting mathematical modelling and computer simulations of the Pareto principle. Several significant factors influencing the phenomenon are identified.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

Tax evasion and the prisoner's dilemma

This analysis expands the model of tax evasion suggested by Allingham and Sandmo (1972) to include public goods, financed by revenues from taxation and penalties. We argue that this leads to a Pareto inferior equilibrium outcome of individual declarations both in models of competitive and interdependent behaviour, thus linking the paradox to the Prisoner's Dilemma, well known from game theory. It is further claimed that a government led by utilitarian welfare standards will perpetuate tax evasion in the case of positive variable costs of detection.     

A New Pareto Optimal Solution in a Lagrange Decomposable Multi-Objective Optimization Problem

In this paper a committee decision-making process of a convex Lagrange decomposable multi-objective optimization problem, which has been decomposed into various subproblems, is studied. Each member of the committee controls only one subproblem and attempts to select the optimal solution of this subproblem most desirable to him, under the assumption that all the constraints of the total problem are satisfied. This procedure leads to a new solution concept of a Lagrange decomposable multi-objective optimization problem, called a preferred equilibrium set. A preferred equilibrium point of a problem, for a committee, may or may not be a Pareto optimal point of this problem. In some cases, a non-Pareto optimal preferred equilibrium point of a problem, for a committee, can be considered as a special type of Pareto optimal point of this problem. This fact leads to a generalization of the Pareto optimality concept in a problem.     

On goal programming formulations of the reference point method

This paper discusses connections between the multi-criteria techniques of goal programming (GP) and the reference point method (RPM). Both approaches use a certain target point in the criterion (outcome) space as a key element to model decision maker preferences. Therefore, RPM can be expressed, similarly to GP, in the modelling framework of deviational variables. The paper gives a systematic survey and analysis of the lexicographic GP models of RPM. The corresponding preference models are formalised and analysed with respect to target values interpretations as well as the Pareto efficiency of their solutions. The properties of equity among the individual achievements of solutions are also analysed with respect to the Rawlsian principle of justice.     

Pareto Analysis vis-à-vis Balance Space Approach in Multiobjective Global Optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.     

Local stability of the Cournot solution with increasing heterogeneous competitors

In this paper we try to solve a paradox related to the results of Theocharis (1960). When the number of competitors increases the Cournot–Nash equilibrium loses stability. We relax the assumption about homogeneity in the decision mechanism and show that if we admit heterogeneity than by increasing the number of competitors the stability region on the parameters’ space may enlarge instead of shrinking.     

New optimality principles for economic efficiency and equilibrium

This paper develops several optimization principles relating the fundamental concepts of Pareto efficiency and competitive equilibria. The beginning point for this development is the introduction of a new function describing individual preferences, closely related to willingness-to-pay, termed the benefit function. An important property of the benefit function is that it can be summed across individuals to obtain a meaningful measure of total benefit relative to a given set of utility levels; and the optimization principles presented in the paper are based on maximization of this total benefit.Specifically, it is shown that, under appropriate technical assumptions, a Pareto-efficient allocationX maximizes the total benefit relative to the utility levels it yields. Conversely, if an allocationX yields zero benefit and maximizes the total benefit function, then that allocation is Pareto efficient. The Lagrange multipliersp of the benefit maximization problem serve as prices; and the (X,p) pair satisfies a generalized saddle-point property termed a Lagrange equilibrium. This in turn is equivalent, under appropriate assumptions, to a competitive equilibrium.There are natural duals to all of the results stated above. The dual optimization principle is based on a surplus function which is a function of prices. The surplus is the total income generated at pricesp, minus the total income required to obtain given utility levels. The dual optimization principle states that prices that are dual (or indirect) Pareto efficient minimize total surplus and render it zero. Conversely, a set of prices that minimizes total surplus and renders it zero is a dual Pareto efficient set of prices.The results of the paper can be viewed as augmenting the first and second theorems of welfare economics (and their duals) to provide a family of results that relate the important economic concepts of Pareto efficiency, equilibrium, dual (or indirect) Pareto efficiency, total benefit, Lagrange equilibrium, and total surplus.The author wishes to thank Charles R. Bowman and Andrew J. Yates for several valuable suggestions and corrections.     

Using the idea of expanded core for the exact solution of bi-objective multi-dimensional knapsack problems

We propose a methodology for obtaining the exact Pareto set of Bi-Objective Multi-Dimensional Knapsack Problems, exploiting the concept of core expansion. The core concept is effectively used in single objective multi-dimensional knapsack problems and it is based on the “divide and conquer” principle. Namely, instead of solving one problem with n variables we solve several sub-problems with a fraction of n variables (core variables). In the multi-objective case, the general idea is that we start from an approximation of the Pareto set (produced with the Multi-Criteria Branch and Bound algorithm, using also the core concept) and we enrich this approximation iteratively. Every time an approximation is generated, we solve a series of appropriate single objective Integer Programming (IP) problems exploring the criterion space for possibly undiscovered, new Pareto Optimal Solutions (POS). If one or more new POS are found, we appropriately expand the already found cores and solve the new core problems. This process is repeated until no new POS are found from the IP problems. The paper includes an educational example and some experiments.     

Enumeration of Pareto optimal multi-criteria spanning trees – a proof of the incorrectness of Zhou and Gen's proposed algorithm

The minimum spanning tree (MST) problem is a well-known optimization problem of major significance in operational research. In the multi-criteria MST (mc-MST) problem, the scalar edge weights of the MST problem are replaced by vectors, and the aim is to find the complete set of Pareto optimal minimum-weight spanning trees. This problem is NP-hard and so approximate methods must be used if one is to tackle it efficiently. In an article previously published in this journal, a genetic algorithm (GA) was put forward for the mc-MST. To evaluate the GA, the solution sets generated by it were compared with solution sets from a proposed (exponential time) algorithm for enumerating all Pareto optimal spanning trees. However, the proposed enumeration algorithm that was used is not correct for two reasons: (1) It does not guarantee that all Pareto optimal minimum-weight spanning trees are returned. (2) It does not guarantee that those trees that are returned are Pareto optimal. In this short paper we prove these two theorems.     

Reduced Jacobian Method

In this paper, we present the Wolfe’s reduced gradient method for multiobjective (multicriteria) optimization. We precisely deal with the problem of minimizing nonlinear objectives under linear constraints and propose a reduced Jacobian method, namely a reduced gradient-like method that does not scalarize those programs. As long as there are nondominated solutions, the principle is to determine a direction that decreases all goals at the same time to achieve one of them. Following the reduction strategy, only a reduced search direction is to be found. We show that this latter can be obtained by solving a simple differentiable and convex program at each iteration. Moreover, this method is conceived to recover both the discontinuous and continuous schemes of Wolfe for the single-objective programs. The resulting algorithm is proved to be (globally) convergent to a Pareto KKT-stationary (Pareto critical) point under classical hypotheses and a multiobjective Armijo line search condition. Finally, experiment results over test problems show a net performance of the proposed algorithm and its superiority against a classical scalarization approach, both in the quality of the approximated Pareto front and in the computational effort.     

基于网络治理的公交服务总效益和各主体决策分析

为解决我国公交服务效率与公平难以兼容问题,本文引入网络治理理论假定政府、企业和社会群体三方合作的基础上,从最大化公交服务效益构建公交服务网络治理模型,剖析政府、企业和社会群体努力水平、控制权配置和产出效率对公交服务总效益的影响,并对模型分析结果进行数据模拟。结果表明:公交服务总效益与三方主体的努力水平、控制权和产出效率紧密相关,只有在合理范围内参与方的努力水平、控制权的提高有利于公交服务效益增加,当超出合理范围时会导致总效益的降低;而三方主体产出效率越高,公交服务总效益越大;控制权配置与自身的产出系数、努力水平相关,产出系数、努力水平越高,则其对应的控制权比例就越大。     

Stochastic convergence of random search methods to fixed size Pareto front approximations

In this paper we investigate to what extent random search methods, equipped with an archive of bounded size to store a limited amount of solutions and other data, are able to obtain good Pareto front approximations. We propose and analyze two archiving schemes that allow for maintaining a sequence of solution sets of given cardinality that converge with probability one to an ?-Pareto set of a certain quality, under very mild assumptions on the process used to sample new solutions. The first algorithm uses a hierarchical grid to define a family of approximate dominance relations to compare solutions and solution sets. Acceptance of a new solution is based on a potential function that counts the number of occupied boxes (on various levels) and thus maintains a strictly monotonous progress to a limit set that covers the Pareto front with non-overlapping boxes at finest resolution possible. The second algorithm uses an adaptation scheme to modify the current value of ? based on the information gathered during the run. This way it will be possible to achieve convergence to the best (smallest) ? value, and to a corresponding solution set of k solutions that ?-dominate all other solutions, which is probably the best possible result regarding the limit behavior of random search methods or metaheuristics for obtaining Pareto front approximations.