Approximation in multiobjective optimization

Some results of approximation type for multiobjective optimization problems with a finite number of objective functions are presented. Namely, for a sequence of multiobjective optimization problems P _n which converges in a suitable sense to a limit problem P, properties of the sequence of approximate Pareto efficient sets of the P _n 's, are studied with respect to the Pareto efficient set of P. The exterior penalty method as well as the variational approximation method appear to be particular cases of this framework.     

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.     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Smooth approximations without critical points

In any separable Banach space containing c ₀ which admits a C ^k-smooth bump, every continuous function can be approximated by a C ^k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.     

Two-Sided Pareto Front Approximations

A new approach to derive Pareto front approximations with evolutionary computations is proposed here. At present, evolutionary multiobjective optimization algorithms derive a discrete approximation of the Pareto front (the set of objective maps of efficient solutions) by selecting feasible solutions such that their objective maps are close to the Pareto front. However, accuracy of such approximations is known only if the Pareto front is known, which makes their usefulness questionable. Here we propose to exploit also elements outside feasible sets to derive pairs of such Pareto front approximations that for each approximation pair the corresponding Pareto front lies, in a certain sense, in-between. Accuracies of Pareto front approximations by such pairs can be measured and controlled with respect to distance between elements of a pair. A rudimentary algorithm to derive pairs of Pareto front approximations is presented and the viability of the idea is verified on a limited number of test problems.     

Finding preferred subsets of Pareto optimal solutions

Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.     

A fast steady-state <Emphasis Type="Italic">ε</Emphasis>-dominance multi-objective evolutionary algorithm

Multi-objective evolutionary algorithms (MOEAs) have become an increasingly popular tool for design and optimization tasks in real-world applications. Most of the popular baseline algorithms are pivoted on the use of Pareto-ranking (that is empirically inefficient) to improve the convergence to the Pareto front of a multi-objective optimization problem. This paper proposes a new ε-dominance MOEA (EDMOEA) which adopts pair-comparison selection and steady-state replacement instead of the Pareto-ranking. The proposed algorithm is an elitist algorithm with a new preservation technique of population diversity based on the ε-dominance relation. It is demonstrated that superior results could be obtained by the EDMOEA compared with other algorithms: NSGA-II, SPEA2, IBEA, ε-MOEA, PESA and PESA-II on test problems. The EDMOEA is able to converge to the Pareto optimal set much faster especially on the ZDT test functions with a large number of decision variables.     

On Some Extremal Properties of Lagrange Interpolatory Polynomials

In this paper, we will show that Lagrange interpolatory polynomials are optimal for solving some approximation theory problems concerning the finding of linear widths.In particular, we will show that

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, where _n is a set of the linear operators with finite rank n+1 defined on −1,1], and where _n+1 denotes the set of polynomials p=∑_i=0ⁿ⁺¹a_ixⁱ of degreen+1 such that a_n+11. The infimum is achieved for Lagrange interpolatory polynomial for nodes .     

Randomized algorithm for the k-server problem on decomposable spaces

We study the randomized k-server problem on metric spaces consisting of widely separated subspaces. We give a method which extends existing algorithms to larger spaces with the growth rate of the competitive quotients being at most O(logk). This method yields o(k)-competitive algorithms solving the randomized k-server problem for some special underlying metric spaces, e.g. HSTs of “small” height (but unbounded degree). HSTs are important tools for probabilistic approximation of metric spaces.     

How to Allocate Network Centers

This paper deals with the issue of allocating and utilizing centers in a distributed network, in its various forms. The paper discusses the significant parameters of center allocation, defines the resulting optimization problems, and proposes several approximation algorithms for selecting centers and for distributing the users among them. We concentrate mainly on balanced versions of the problem, i.e., in which it is required that the assignment of clients to centers be as balanced as possible. The main results are constant ratio approximation algorithms for the balanced κ-centers and balanced κ-weighted centers problems, and logarithmic ratio approximation algorithms for the ρ-dominating set and the k-tolerant set problems.     

Hybrid adaptive methods for approximating a nonconvex multidimensional Pareto frontier

New hybrid methods for approximating the Pareto frontier of the feasible set of criteria vectors in nonlinear multicriteria optimization problems with nonconvex Pareto frontiers are considered. Since the approximation of the Pareto frontier is an ill-posed problem, the methods are based on approximating the Edgeworth-Pareto hull (EPH), i.e., the maximum set having the same Pareto frontier as the original feasible set of criteria vectors. The EPH approximation also makes it possible to visualize the Pareto frontier and to estimate the quality of the approximation. In the methods proposed, the statistical estimation of the quality of the current EPH approximation is combined with its improvement based on a combination of random search, local optimization, adaptive compression of the search region, and genetic algorithms.     

The M/G/c queue in light traffic

Under light traffic, we investigate the quality of a well‐known approximation for first‐moment performance measures for an M/G/c queue, and, in particular, conditions under which the approximation is either an upper or a lower bound. The approach is to combine known relationships between quantities such as average delay and time‐average work in system with direct sample‐path comparisons of system operation under two modes of operation: conventional FIFO and a version of preemptive LIFO. We then use light traffic limit theorems to show an inequality between time‐average work of the M/G/c queue and that of the approximation. In the process, we obtain new and improved approximations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fluid approximations for a processor-sharing queue

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Pareto fitness genetic algorithm: Test function study

Evolutionary algorithms have shown some success in solving multiobjective optimization problems. The methods of fitness assignment are mainly based on the information about the dominance relation between individuals. We propose a Pareto fitness genetic algorithm (PFGA) in which we introduce a modified ranking procedure and a promising way of sharing; a new fitness function based on the rank of the individual and its density value is designed. This is considered as our main contribution. The performance of our algorithm is evaluated on six multiobjective benchmarks with different Pareto front features. Computational results (quality of the approximation of the Pareto optimal set and the number of fitness function evaluations) proving its efficiency are reported.     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part I. Optimality

In this paper, a generalization of convexity, namely G-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. The modified Karush-Kuhn-Tucker necessary optimality conditions for a certain class of multiobjective programming problems are established. To prove this result, the Kuhn-Tucker constraint qualification and the definition of the Bouligand tangent cone for a set are used. The assumptions on (weak) Pareto optimal solutions are relaxed by means of vector-valued G-invex functions.     

Randomized k-server algorithms for growth-rate bounded graphs

The k-server problem is a fundamental online problem where k mobile servers should be scheduled to answer a sequence of requests for points in a metric space as to minimize the total movement cost. While the deterministic competitive ratio is at least k, randomized k-server algorithms have the potential of reaching o(k)-competitive ratios. Prior to this work only few specific cases of this problem were solved. For arbitrary metric spaces, this goal may be approached by using probabilistic metric approximation techniques. This paper gives the first results in this direction, obtaining o(k)-competitive ratio for a natural class of metric spaces, including d-dimensional grids, and wide range of k.     

LU factorizations, q = 0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q = 0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas. Dedicated to Dick Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—33D45, 33D80 Work done at KdV Institute, Amsterdam and supported by NWO, project number 613.006.573.     

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L_∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O_P(n^−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the B _π characters, which can be viewed as the “π-modular” characters in G, such that the B _p′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs’ work, Slattery has developed some Brauer’s ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer’s three main theorems to the π-blocks. In this paper, depending on Isaacs’ and Slattery’s work, we will extend the first main theorem for π-blocks.     

The Dirichlet problem for Baire-one functions

Let X be a compact convex set and let ext X stand for the set of all extreme points of X. We characterize those bounded function defined on ext X which can be extended to an affine Baire-one function on the whole set X.