Weak Pareto Research Articles

Timeslice Prioritarianism, Prudence, and Weak Pareto

ABSTRACT Andrić and Herlitz (2022) object to Timeslice Prioritarianism on the basis that it violates two purportedly uncontroversial properties: prudence and Weak Pareto. We will claim that their objection does not undermine Timeslice Prioritarianism because the basis of their objection is just a straightforward implication of utilitarianism. To establish this argument, we will show that a timeslice view satisfies prudence and Weak Pareto if and only if it is utilitarianism.

Robust Optimality and Duality for Nonsmooth Multiobjective Programming Problems with Vanishing Constraints Under Data Uncertainty

This article investigates robust optimality conditions and duality results for a class of nonsmooth multiobjective programming problems with vanishing constraints under data uncertainty (UNMPVC). Mathematical programming problems with vanishing constraints constitute a distinctive class of constrained optimization problems because of the presence of complementarity constraints. Moreover, uncertainties are inherent in various real-life problems. The aim of this article is to identify an optimal solution to an uncertain optimization problem with vanishing constraints that remains feasible in every possible future scenario. Stationary conditions are necessary conditions for optimality in mathematical programming problems with vanishing constraints. These conditions can be derived under various constraint qualifications. Employing the properties of convexificators, we introduce generalized standard Abadie constraint qualification (GS-ACQ) for the considered problem, UNMPVC. We introduce a generalized robust version of nonsmooth stationary conditions, namely a weakly stationary point, a Mordukhovich stationary point, and a strong stationary point (RS-stationary) for UNMPVC. By employing GS-ACQ, we establish the necessary conditions for a local weak Pareto solution of UNMPVC. Moreover, under generalized convexity assumptions, we derive sufficient optimality criteria for UNMPVC. Furthermore, we formulate the Wolfe-type and Mond–Weir-type robust dual models corresponding to the primal problem, UNMPVC.

Unboundedness of the images of set-valued mappings having closed graphs: application to vector optimization

AbstractIn this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded.

Collective or individual rationality in the Nash bargaining solution: efficiency-free characterizations

AbstractIn the classical bargaining problem, we propose a very mild axiom of individual rationality, which we call possibility of utility gain. This requires that for at least one bargaining problem, there exists at least one player who reaches a higher utility level than their disagreement utility. This paper shows that the Nash solution (Nash in Econometrica 18(2):155–162, 1950) is characterized by possibility of utility gain and continuity with respect to feasible sets together with Nash’s axioms except weak Pareto optimality. We also show that in Nash’s theorem, weak Pareto optimality can be replaced by conflict-freeness (introduced by Rachmilevitch in Math Soc Sci 76(C):107–109, 2015). This demands that when the agreement most preferred by all players is feasible, this should be chosen. Furthermore, we provide alternative and unified proofs for other efficiency-free characterizations of the Nash solution. This clarifies the role of each axiom in the related results.

BARGAIN-MATCH: A Game Theoretical Approach for Resource Allocation and Task Offloading in Vehicular Edge Computing Networks

Vehicular edge computing (VEC) is emerging as a promising architecture of vehicular networks (VNs) by deploying the cloud computing resources at the edge of the VNs. However, efficient resource management and task offloading in the VEC network is challenging. In this work, we first present a hierarchical framework that coordinates the heterogeneity among tasks and servers to improve the resource utilization for servers and service satisfaction for vehicles. Moreover, we formulate a joint resource allocation and task offloading problem (JRATOP), aiming to jointly optimize the intra-VEC server resource allocation and inter-VEC server load-balanced offloading by stimulating the horizontal and vertical collaboration among vehicles, VEC servers, and cloud server. Since the formulated JRATOP is NP-hard, we propose a cooperative resource allocation and task offloading algorithm named BARGAIN-MATCH, which consists of a bargaining-based incentive approach for intra-server resource allocation and a matching method-based horizontal-vertical collaboration approach for inter-server task offloading. Besides, BARGAIN-MATCH is proved to be stable, weak Pareto optimal, and polynomial complex. Simulation results demonstrate that the proposed approach achieves superior system utility and efficiency compared to the other methods, especially when the system workload is heavy.

Multi-objective dynamic optimization of path-constrained switched systems

A multi-objective dynamic optimization algorithm is proposed for path-constrained switched systems to locate approximated local Pareto solutions with specified tolerances. The algorithm is based on iteratively solving single-objective modified ϵ-constraint dynamic optimization problems to avoid generating weak Pareto solutions. These problems are first reduced to semi-infinite programs embedded with ordinary differential equations by performing the control vector parameterization technique. Then the guaranteed feasibility is attained within a finite number of iterations by enforcing the path constraints at discretized time points with a restriction parameter on the right-hand side. Furthermore, the approximation error of the obtained local Pareto solutions is quantitatively analyzed. The result shows that the Pareto approximation error can be directly controlled via user-specified parameters. Finally, the effectiveness of the algorithm is illustrated via a numerical example.

Generalized Polarity and Weakest Constraint Qualifications in Multiobjective Optimization

In Haeser and Ramos (J Optim Theory Appl, 187:469–487, 2020), a generalization of the normal cone from single objective to multiobjective optimization is introduced, along with a weakest constraint qualification such that any local weak Pareto optimal point is a weak Kuhn–Tucker point. We extend this approach to other generalizations of the normal cone and corresponding weakest constraint qualifications, such that local Pareto optimal points are weak Kuhn–Tucker points, local proper Pareto optimal points are weak and proper Kuhn–Tucker points, respectively, and strict local Pareto optimal points of order one are weak, proper and strong Kuhn–Tucker points, respectively. The constructions are based on an appropriate generalization of polarity to pairs of matrices and vectors.

Device association and trajectory planning for UAV-assisted MEC in IoT: a matching theory-based approach

Unmanned aircraft vehicles (UAVs)-enabled mobile edge computing (MEC) can enable Internet of Things devices (IoTD) to offload computing tasks to them. Considering this, we study how multiple aerial service providers (ASPs) compete with each other to provide edge computing services to multiple ground network operators (GNOs). An ASP owning multiple UAVs aims to achieve the maximum profit from providing MEC service to the GNOs, while a GNO operating multiple IoTDs aims to seek the computing service of a certain ASP to meet its performance requirements. To this end, we first quantify the conflicting interests of the ASPs and GNOs by using different profit functions. Then, the UAV scheduling and resource allocation is formulated as a multi-objective optimization problem. To address this problem, we first solve the UAV trajectory planning and resource allocation problem between one ASP and one GNO by using the Lagrange relaxation and successive convex optimization (SCA) methods. Based on the obtained results, the GNOs and ASPs are then associated in the framework based on the matching theory, which results in a weak Pareto optimality. Simulation results show that the proposed method achieves the considerable performance.

Inconsistency indices for pairwise comparisons and the Pareto dominance principle

We consider the problem of quantifying inconsistency in pairwise comparisons and valued-preferences. A wide range of indices have been proposed in the literature to perform this task, and two sets of conditions have been introduced to validate such indices. We summarize some criticisms from the literature and we add more evidence to show that neither of the two systems is adequate in its current formulation. Thanks to the widely accepted concept of weak Pareto dominance, we formulate a new property. We argue that a simple regularity condition and this new property can overcome the shortcomings of the two axiomatic systems, and represents a significantly simpler framework. Finally, we claim that, if we had resorted to strict Pareto dominance, we would have needed just one axiom.

Strong dictatorship via ratio-scale measurable utilities: a simpler proof

Tsui and Weymark (Econ Theory 10:241–256, 1997, https://doi.org/10.1007/s001990050156) have shown that the only continuous social welfare orderings on the whole Euclidean space which satisfy the weak Pareto principle and are invariant to individual-specific similarity transformations of utilities are strongly dictatorial. Their proof relies on functional equation arguments which are quite complex. This note provides a simpler proof of their theorem.

Risk-neutral multiobjective optimal control of random Volterra integral equations

We analyze risk-neutral multiobjective optimal control problems, governed by Volterra integral equations with random inputs, and subject to expectation-type final/pure state constraints and mixed pointwise control-state constraints. Moreover, controls are unbounded, and inclusion-form pure state as well as control-state constraints are given by measurable set-valued mappings, whose images are nonempty closed subsets of infinite dimensional spaces. With such constraints, the multiplier corresponding to pure state constraints belongs to the dual space of the Banach space-valued continuous functions. We set up a decomposition result for this dual, and develop a Lagrange multiplier theorem for an abstract multi-criteria optimization model, in which a Robinson-type constraint qualification is applied. Then, making use of the obtained outcomes, exploiting higher integrability and regularity of the random field control/state variables, and employing Banach space-valued integration techniques, we derive Fritz-John necessary optimality conditions, with integrable functions and (countably additive) measures singular with respect to the Lebesgue measure as multipliers, for local weak Pareto solutions of studied problems.

When is weak Pareto equivalent to strong Pareto?

When is weak Pareto equivalent to strong Pareto?

Multiobjective Optimization Model of Quality Education Reform in Research and Learning Activities

In order to solve the problem of quality education reform in research activities, a multiobjective optimization model for the optimal performance distribution of college teachers is proposed. We mainly use the multiobjective optimization method to study the allocation of optimal performance. On the basis of considering individual differences, a multiobjective optimization model is established with the goal of maximizing the total satisfaction of all assessment objects and balancing the satisfaction as much as possible by introducing the score conversion function and the satisfaction function. The results show that the variation trends of the weak Pareto fronts at five different confidence levels are basically the same; that is, different confidence levels have little effect on the results. Conclusion. The use of a multiobjective optimization model can improve the optimal performance distribution of college teachers, so as to effectively promote the reform of quality education in research activities.

A Kernel-Based Indicator for Multi/Many-Objective Optimization

How to evaluate Pareto front approximations generated by multi/many-objective optimizers is a critical issue in the field of multiobjective optimization. Currently, there exist two types of comprehensive quality indicators (i.e., volume-based and distance-based indicators). Distance-based indicators, such as inverted generational distance (IGD), are usually computed by summing up the distance of each reference point to its nearest solution. Their high computational efficiency leads to their prevalence in many-objective optimization. However, in the existing distance-based indicators, the distributions of the solution sets are usually neglected, leading to their lack of ability to well distinguish between different solution sets. This phenomenon may become even more severe in high-dimensional space. To address such an issue, a kernel-based indicator (KBI) is proposed as a comprehensive indicator. Different from other distance-based indicators, a kernel-based maximum mean discrepancy is adopted in KBI for directly measuring the difference that can characterize the convergence, spread, and uniformity of two sets, i.e., the solution set and reference set, by embedding them in reproducing kernel Hilbert space (RKHS). As a result, KBI not only reflects the distance between the solution set and the reference set but also can reflect the distribution of the solution set itself. In addition, to maintain the desirable weak Pareto compliance property of KBI, a nondominated set reconstruction approach is also proposed to shift the original solution set. The detailed theoretical and experimental analysis of KBI is provided in this article. The properties of KBI have also been analyzed by the optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> -distribution.

Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness is illustrated by solving a collection of multiobjective test problems.

Inexact proximal point algorithm for quasiconvex optimization problems on Hadamard manifolds

ABSTRACT In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution.

Approximations for Pareto and Proper Pareto solutions and their KKT conditions

In this article, we view the Pareto and weak Pareto solutions of the multiobjective optimization by using an approximate version of KKT type conditions. In one of our main results Ekeland’s variational principle for vector-valued maps plays a key role. We also focus on an improved version of Geoffrion proper Pareto solutions and it’s approximation and characterize them through saddle point and KKT type conditions.

Designing an optimization model for green closed-loop supply chain network of heavy tire by considering economic pricing under uncertainty.

The conditions of global competition and environmental sensitivities have made organizations and factories to collect returned products, in such a way that these organizations have tried to rehabilitate, recycle, or destroy these products in order to protect the environment. This paper propose a mathematical model for the green closed-loop supply chain network of heavy tire by considering the economic pricing of its products under conditions of uncertainty, which economically determines the price and leads to more profitability. In addition, the relevant model is a two-objective fuzzy model, the first objective of which is to minimize costs and maximize profits, and the second objective is to minimize environmental issues. The proposed model can also determine the optimal location of each center based on potential locations, the optimal amount of production, distribution, collection, recycling, as well as the reproduction of products. The ε-constraint method is used to solve the model with two objective functions; this method ensures strong Pareto optimal answers and prevents weak Pareto answers. Independent two-sample t-test is used to verify the results of certain and uncertain models in the studied model. In order to evaluate the effectiveness and profitability of the proposed method, a case study in the field of heavy tires is finally used, through which very useful results are obtained.

An improved proximal method with quasi-distance for nonconvex multiobjective optimization problem

Abstract Multiobjective optimization is the optimization with several conflicting objective functions. However, it is generally tough to find an optimal solution that satisfies all objectives from a mathematical frame of reference. The main objective of this article is to present an improved proximal method involving quasi-distance for constrained multiobjective optimization problems under the locally Lipschitz condition of the cost function. An instigation to study the proximal method with quasi distances is due to its widespread applications of the quasi distances in computer theory. To study the convergence result, Fritz John’s necessary optimality condition for weak Pareto solution is used. The suitable conditions to guarantee that the cluster points of the generated sequences are Pareto–Clarke critical points are provided.

Strong and weak conditions of regularity and optimality

Nondegenerate optimality conditions for Pareto and weak Pareto optimal solutions to multiobjective optimization problems with inequality and multi-equality constraints determined by Fréchet differentiable functions are established. First, weak and strong