Local Pareto Optimality Research Articles

MAGIKS: Fair Multi-Resource Allocation Game Induced by Kalai-Smorodinsky Bargaining Solution

We study the problem of fair and efficient mechanism design for allocating multiple resources in multiple servers among a set of users with Leontief utilities. This problem is motivated by a mobile edge computing environment where each mobile user cannot establish a wireless connection simultaneously to multiple edge servers. Each user is a selfish utility maximizing agent that chooses a single server for its job execution. When a server is shared by multiple users, a resource allocation rule decides the utility that each user must receive. Our goal is to design a mechanism that always admits a Nash Equilibrium (NE), i.e., a state where no user has incentive to change its server, that (1) can be reached in polynomial time and (2) provides fair and efficient resource allocation. We propose the Multi-resource Allocation Game Induced by Kalai-Smorodinsky bargaining solution (MAGIKS) and prove that under discrete resource demands it finds an NE in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}({\textrm {poly}(n)})$ </tex-math></inline-formula> moves for any fixed server configuration, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is the number of users. Furthermore, MAGIKS satisfies envy-freeness, sharing incentive, and Pareto optimality on each server. Regarding fairness among users in different servers, we prove that MAGIKS satisfies 2-approximate envy-freeness and maximin share guarantee. Moreover, we show that 2-approximate envy-freeness is the best that any mechanism that satisfies local Pareto optimality can achieve at its NEs.

Global and Local Pareto Optimality in Coevolution for Solving Carpool Service Problem With Time Windows

In metropolitan areas, drivers share their vehicles with people who commute daily via carpooling. In this paper, we first defined a multiobjective carpool service problem with time windows (MOCSPTW) by considering four optimized objectives; subsequently, we propose a coevolutionary algorithm for two solution sets, population and archive, using objective-wise local search and set-based simulated binary operation in order to address the MOCSPTW. In the evolution module of the proposed algorithm, three different methods, namely, objective-wise local search in an archive, set-based simulated binary operation in a population, and set-based simulated binary operation both in a population and in an archive, were used to generate the offspring. Meanwhile, the $\epsilon $ -domination and normal domination in the update module were adopted to control the convergence and diversity of the population and archive. In the experimental result, 48 sets of tests, including three moving patterns in a metropolitan area for the MOCSPTW, were prepared. The results of the quantitative comparison and objective visualization showed that the proposed algorithm can obtain superior Pareto-optimal solutions regarding convergence and diversity compared with a fast nondominated sorting genetic algorithm.

A Double Learning Models-Based Multi-Objective Estimation of Distribution Algorithm

The recently developed regularity model-based multi-objective estimation of distribution algorithm (RM-MEDA) and inverse models-based multi-objective evolutionary algorithm (IM-MOEA) have been shown to be two effective methods for solving some complex multi-objective optimization problems (MOPs). However, RM-MEDA and IM-MOEA are still challenged when solving MOPs with many local Pareto fronts, and usually generate poor solutions when the population has no obvious regularity. In order to overcome these limits, an ensemble of RM-MEDA and IM-MOEA, denoted as RM-IM-EDA, is proposed in this paper. This ensemble is based on a dynamic mixture of the sampling in the decision space by the regularity-based learning model and the sampling in the objective space using the inverse learning models. In addition, a sequence-based deterministic initialization method is introduced to identify the properties of fitness landscape. The objective behind this scheme is to reduce the probability of sinking into the local Pareto optimum. For the comparison purposes, the proposed RM-IM-EDA is tested on 32 benchmark problems. Experiment results statistically affirm the efficiency of the proposed approach to obtain better results compared with each individual algorithm and other four state-of-the-art MEDAs.

Multi-objective retrospective optimization using stochastic zigzag search

We propose a new retrospective optimization (RO) method for multi-objective simulation optimization (MOSO) problems. RO algorithms generate a sequence of sample-path (SP) problems and solve these SP problems iteratively using a nonlinear optimizer. In this study, a stochastic zigzag search algorithm is chosen in the RO framework to solve SP problems. The key idea of zigzag search is searching around the Pareto front by applying an efficient local-search procedure using the gradients of the objective functions. Many continuous MOSO problems have smooth objective functions and their non-dominated objective function values form a smooth surface in the image space. This fact motivates developing the zigzag search method embedded in RO for such relatively well-posed MOSO problems. A numerical implementation of this method—multi-objective retrospective optimization using zigzag search (MOROZS)—is presented particularly for continuous bi-objective simulation optimization (BOSO) problems with well-connected Pareto optimal solutions. MOROZS is designed for BOSO problems in which a simulation oracle returns both objective function values and gradients. Due to the local nature of zigzag search, MOROZS can only guarantee the asymptotic convergence to local Pareto optimality. The efficiency of MOROZS is studied using three BOSO problems with noisy objective functions and is compared to that of Genetic Algorithms based NSGA-II and a recently developed method MO-COMPASS.

Joint Subcarrier Assignment and Power Allocation in Full-Duplex OFDMA Networks

Recent advances in the physical layer have demonstrated the feasibility of in-band wireless full-duplex which enables a node to transmit and receive simultaneously on the same frequency band. While the full-duplex operation can ideally double the spectral efficiency, the network-level gain of full-duplex in large-scale networks remains unclear due to the complicated resource allocation in multi-carrier and multi-user environments. In this paper, we consider a single-cell full-duplex OFDMA network which consists of one full-duplex base station (BS) and multiple full-duplex mobile nodes. Our goal is to maximize the sum-rate performance by jointly optimizing subcarrier assignment and power allocation considering the characteristics of full-duplex transmissions. We develop an iterative solution that achieves local Pareto optimality in typical scenarios. Through extensive simulations, we demonstrate that our solution empirically achieves near-optimal performance and outperforms other resource allocation schemes designed for half-duplex networks. Also, we reveal the impact of various factors such as the channel correlation, the residual self-interference, and the distance between the BS and nodes on the full-duplex gain.

Necessary and sufficient conditions for local Pareto optimality on time scales

We study a multiobjective variational problem on time scales. For this problem, necessary and sufficient conditions for weak local Pareto optimality are given. We also prove a necessary optimality condition for the isoperimetric problem with multiple constraints on time scales.

A new multiobjective genetic algorithm with heterogeneous population for solving flowshop scheduling problems

The present paper discusses the application of a new genetic algorithm (GA) featuring heterogeneous population to solve multiobjective flowshop scheduling problems. Many GAs have been developed to solve multiobjective scheduling problems, but they used a non-heterogeneous population approach, which could lead to premature convergence and local Pareto-optimum solutions. Our experiments with a 20-job and 20-machine benchmark problem given in Taillard (1993) show that the heterogeneous multiobjective genetic algorithm (hMGA) developed in this research outperforms NSGA-II (Deb 2001) one of the widely used algorithms with non-heterogeneous population. Moreover, in this paper we also present the comparison of hMGA with another meta-heuristic method, i.e. multi-objective simulated annealing (MOSA), proposed by Varadharajan and Rajendran (2005). This research concludes that hMGA developed in this work is promising as it can produce a new set of Pareto-optimum solutions that have not been found by MOSA before.

Characteruzation of pareto optimal points in problems with mult-eqality constraints

In the paper the following optimization problem is considered: Let Xbe a Banach space is a vector performance index. We are looking for a point such that where U(x) denotes some neighbourhood of x. Making use of the results of Walczak [10] and Censor [1] we generalize the well-known Dubovicki-Milutin Theorem and them we apply it to the problem (*) obtaining a necessary condition for local Pareto optimum. For convex problems a local Pareto optimum in also a global Pareto one. If additonal assumptions are fulfilled for convex problems i.e. are continuous and Ponstein convex and the so-called Slater's condition is satisfied, then the Euler-Lagrange equation gives a sufficient condition for global Pareto optimum. In the paper an example of a Pareto optimal control problem is also given. A dynamical cascade system is described by two partial differential equations of parabolic type in a Sobolev space.

Pareto optima, non-convexities and regulated market equilibria

In a differentiable setting without convexity assumptions, a typical local Pareto optimum can be sustained by a combination of a price system and a system of market regulations. Some consumers are required to consume fixed amounts of some basic commodity bundles, and some producers are required to produce fixed amounts of other basic commodity bundles. The agents are otherwise free to maximize utility and profit (locally). No more than l–1 agents are subject to regulation, where l is the number of commodities.

A note on local pareto optima

This note gives an elementary proof of the sufficient second order conditions for a local Pareto optimum.

On the structure and stability of local Pareto optima in a pure exchange economy

On the structure and stability of local Pareto optima in a pure exchange economy

Accessibility of local pareto optima with constraints

Accessibility of local pareto optima with constraints

On the algebraic criteria for local Pareto optima. II

Constrained vector optimization problems in the large are studied in this paper. Fixing any constraints so that the feasible set is a manifold with corners, we prove that the set of local Pareto optima for typical vector-valued functions admit Whitney prestratifications. Furthermore, these prestratifications persist under small perturbation of vector-valued functions. The main tools used here are a variant of Thom’s transversality theorem and the stratification theory of semialgebraic sets.

Pareto optimality in multiobjective problems

In this study, the optimization theory of Dubovitskii and Milyutin is extended to multiobjective optimization problems, producing new necessary conditions for local Pareto optima. Cones of directions of decrease, cones of feasible directions and a cone of tangent directions, as well as, a new cone of directions of nonincrease play an important role here. The dual cones to the cones of direction of decrease and to the cones of directions of nonincrease are characterized for convex functionals without differentiability, with the aid of their subdifferential, making the optimality theorems applicable. The theory is applied to vector mathematical programming, giving a generalized Fritz John theorem, and other applications are mentioned. It turns out that, under suitable convexity and regularity assumptions, the necessary conditions for local Pareto optima are also necessary and sufficient for global Pareto optimum. With the aid of the theory presented here, a result is obtained for the, so-called, “scalarization” problem of multiobjective optimization.

Multiple-objective problems: Pareto-optimal solutions by method of proper equality constraints

The method of proper equality constraints for obtaining the set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> of all Pareto-optimal solutions of a general multiple-objective problem and the set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> of all performance index vectors attainable by the Pareto-optimal solutions is presented in this paper. These sets are very useful for those designers, performance analyzers, control agents, and decision makers who are faced with multiple objectives to make appropriate compromises, tradeoffs or choices. That this method is effective for general applications is clearly demonstrated by examples with various kinds of complexity. New necessary and sufficient conditions for Pareto optimality, local Pareto optimality, and local maximal index vectors are also presented. These conditions and the method are useful for both computation and analysis. New observations on a generalized classical Pareto-optimum problem are thereby obtained.

Comments on "Cooperative games and vector-valued criteria problems"

A theorem in the above paper,[1] giving sufficient conditions for the local Pareto optimality, is shown to be incorrect by a counter-example and a revision of its proof. Moreover, an alternative formulation of this theorem is suggested.

Characterization of optima in smooth Pareto economic systems

Simple techniques of calculus and geometry are used to study and characterize the optima of pure exchange economies in which the utility functions are smooth but not necessarily convex. It is also shown how one can reduce the problem of optimizing p functions on the manifold of states to that of maximizing a single function on a submanifold of this space. Two models are described: one in which a person cannot trade to an optimum unless he starts at one; and one in which a person cannot even get near a local Pareto optimum along continuous ‘trade curves’ from most initial distributions. Finally, the set of optima is described for a generic set of utility mappings.

On local Pareto optima

This paper is concerned with the local problem of optimizing several functions at once. The first-order (necessary) conditions for a local Pareto Optimum are well-known. In this paper we obtain the best second-order criteria for local Pareto Optima in an invariant form.