The Edgeworth-Pareto principle in terms of a fuzzy choice function

The Edgeworth-Pareto Principle, the simplest version of which has been known since the 19th century, is stated in general terms of a fuzzy choice function. The application of the principle is justified for a wide class of fuzzy multicriteria choice problems described by certain axioms of the rational behavior. These results bring the axiomatic substantiation of the Edgeworth-Pareto principle performed previously by the author to the most general form and make it possible to reveal the boundaries of that class of multicriteria choice problems for which the application of this principle is required. Based on scalarization methods as applied to multicriteria problems, upper bounds are derived for the unknown set of selected vectors.     

On the selection of the most adequate radial basis function

Radial basis function (RBF) methods can provide excellent interpolants for a large number of poorly distributed data points. For any finite data set in any Euclidean space, one can construct an interpolation of the data by using RBFs. However, RBF interpolant trends between and beyond the data points depend on the RBF used and may exhibit undesirable trends using some RBFs while the trends may be desirable using other RBFs. The fact that a certain RBF is commonly used for the class of problems at hand, previous good behavior in that (or other) class of problems, and bibliography, are just some of the many valid reasons given to justify a priori selection of RBF. Even assuming that the justified choice of the RBF is most likely the correct choice, one should nonetheless confirm numerically that, in fact, the most adequate RBF for the problem at hand is the RBF chosen a priori. The main goal of this paper is to alert the analyst as to the danger of a priori selection of RBF and to present a strategy to numerically choose the most adequate RBF that better captures the trends of the given data set. The wing weight data fitting problem is used to illustrate the benefits of an adequate choice of RBF for each given data set.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

Solution of a periodic optimal control problem by asymptotic series

In order to understand the numerical behavior of a certain class of periodic optimal control problems, a relatively simple problem is posed. The complexity of the extremal paths is uncovered by determining an analytic approximation to the solution by using the Lindstedt-Poincaré asymptotic series expansion. The key to obtaining this series is in the proper choice of the expansion parameter. The resulting expansion is essentially a harmonic series in which, for small values of the expansion parameter and a few terms of the series, excellent agreement with the numerical solution is obtained. A reasonable approximation of the solution is achieved for a relatively large value of the expansion parameter.This work was sponsored partially by the National Science Foundation, Grant No. ECS-84-13745.     

Pigeonhole and Choice Principles

We shall investigate certain set‐theoretic pigeonhole principles which arise as generalizations of the usual (finitary) pigeonhole principle; and we shall show that many of them are equivalent to full AC. We discuss also several restricted cases and variations of those principles and relate them to restricted choice principles. In this sense the pigeonhole principle is a rich source of weak choice principles. It is shown that certain sequences of restricted pigeonhole principles form implicational hierarchies with respect to ZF. We state also several open problems in order to indicate the extent to which the whole subject of pigeonhole principles requires further exploration.     

Heuristic Parameter Choice Rules for Tikhonov Regularization with Weakly Bounded Noise

We study the choice of the regularization parameter for linear ill-posed problems in the presence of noise that is possibly unbounded but only finite in a weaker norm, and when the noise-level is unknown. For this task, we analyze several heuristic parameter choice rules, such as the quasi-optimality, heuristic discrepancy, and Hanke-Raus rules and adapt the latter two to the weakly bounded noise case. We prove convergence and convergence rates under certain noise conditions. Moreover, we analyze and provide conditions for the convergence of the parameter choice by the generalized cross-validation and predictive mean-square error rules.     

Tractability through increasing smoothness

We prove that some multivariate linear tensor product problems are tractable in the worst case setting if they are defined as tensor products of univariate problems with logarithmically increasing smoothness. This is demonstrated for the approximation problem defined over Korobov spaces and for the approximation problem of certain diagonal operators. For these two problems we show necessary and sufficient conditions on the smoothness parameters of the univariate problems to obtain strong polynomial tractability. We prove that polynomial tractability is equivalent to strong polynomial tractability, and that weak tractability always holds for these problems. Under a mild assumption, the Korobov space consists of periodic functions. Periodicity is crucial since the approximation problem defined over Sobolev spaces of non-periodic functions with a special choice of the norm is not polynomially tractable for all smoothness parameters no matter how fast they go to infinity. Furthermore, depending on the choice of the norm we can even lose weak tractability.     

Concise complexity analyses for trust region methods

Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity bounds. In addition, a new update strategy for the trust region radius is proposed that offers a second-order complexity bound.     

Relaxed Steepest Descent and Cauchy-Barzilai-Borwein Method

The negative gradient direction to find local minimizers has been associated with the classical steepest descent method which behaves poorly except for very well conditioned problems. We stress out that the poor behavior of the steepest descent methods is due to the optimal Cauchy choice of steplength and not to the choice of the search direction. We discuss over and under relaxation of the optimal steplength. In fact, we study and extend recent nonmonotone choices of steplength that significantly enhance the behavior of the method. For a new particular case (Cauchy-Barzilai-Borwein method), we present a convergence analysis and encouraging numerical results to illustrate the advantages of using nonmonotone overrelaxations of the gradient method.     

A family of multistep methods to integrate orbits on spheres

Summary A trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution, rather than in finding the actual correspondence between the values of the parameter and the points on that curve. This paper introduces a family of multi-stage, multi-step numerical methods to integrate trajectory problems whose solution is on a spherical surface. It has been shown that this kind of algorithms has good numerical properties: consistency, stability, convergence and others that are not standard. The latest ones make them a better choice for certain problems.     

Myopia,amnesia, and consistent intertemporal choice

Social scientists often explain observed streams of behavior as the outcome of constrained choice at successive dates, with actors choosing among planned sequences of actions extending into the future. This paper presents four axioms for consistent intertemporal choice in such models. Within this framework, myopia and amnesia are defined to be choice processes in which future feasible actions and past decisions, respectively, play no role in current choice. Each of these modeling strategies is argued to have some plausibility on the basis of bounded rationality considerations. It is shown that an anticipatory process reduces to a behaviorally equivalent myopic process if and only if binary choices are lexically organized. Amnesia, or the absence of endogenous preference formation, is characterized by the condition that binary choices be ‘ahistorical’. The paper closes with some remarks on the methodological role of the consistency axioms, and the problems posed by inconsistent choice processes.     

A framework for solving network revenue management problems with customer choice behavior

Revenue management is the process of understanding, anticipating and influencing consumer behavior in order to maximize revenue. Network revenue management models attempt to maximize revenue when customers buy bundles of multiple resources. The dependence among the resources in such cases is created by customer demand. Network revenue management can be formulated as a stochastic dynamic programming problem whose exact solution is computationally intractable. Solutions are based on approximations of various types. Customer choice behavior modeling has been gaining increasing attention in the revenue management. A framework for solving network revenue management problems with customer choice behavior is proposed. The modeling and solving framework is composed from three inter-related network structures: basic network model, Petri net, and neural net.     

A parameter choice strategy for (iterated) tikhonov regularization of iii-posed problems leading to superconvergence with optimal rates

For ordinary and iterated Tikhonov regularization of linear ill-posed problems, we propose a parameter choice strategy that leads to optimal (super-) convergence rates for certain linear functionals of the regularized solution. It is not necessary to know the smoothness index of the exact solution; approximate knowledge of the smoothness index for the linear functional suffices     

A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.     

Adaptive approaches to stochastic programming

Economists have found a need to model agents who behave in ways that are not consistent with the traditional notions of rational behavior under uncertainty but that are oriented in some looser manner toward achieving good outcomes. Adaptation over time in a myopic manner, rather that forward-looking optimization, has been proposed as one such model of behavior that displays bounded rationality. This paper investigates the relationship between adaptation as a model of behavior and as an algorithmic approach that has been used in computing solutions to optimization problems. It describes a specific adaptive model of behavior in discrete choice problems, one that is closely related to adaptive algorithms for optimization, and shows that this model can be fruitfully applied in studying several economic issues.     

A characterization of limiting distributions of regular estimates

Summary We consider a sequence of estimates in a sequence of general estimation problems with a k-dimensional parameter. Under certain very general conditions we prove that the limiting distribution of the estimates, if properly normed, is a convolution of a certain normal distribution, which depends only of the underlying distributions, and of a further distribution, which depends on the choice of the estimate. As corollaries we obtain inequalities for asymptotic variances and for asymptotic probabilities of certain sets, generalizing so some results of J. Wolfowitz (1965), S. Kaufman (1966), L. Schmetterer (1966) and G. G. Roussas (1968).     

Convexity cuts for multiple choice problems

The purpose of this note is to show that the convexity (or intensection) cut ideas can be exploited to special advantage for integer programs with certain structures. In particular, we show how to obtain now cuts for “multiple choice” and other related “combinatorial” problems. Our approach can usefully be applied to problems involving logical alternatives without having to reformulate (and enlarge) such problems by the addition of 0–1 variables and associated linear constraints.     

On the Consistency of Prony's Method and Related Algorithms

Abstract

Modifications of Prony's classical technique for estimating rate constants in exponential fitting problems have many contemporary applications. In this article the consistency of Prony's method and of related algorithms based on maximum likelihood is discussed as the number of observations n → ∞ by considering the simplest possible models for fitting sums of exponentials to observed data. Two sampling regimes are relevant, corresponding to transient problems and problems of frequency estimation, each of which is associated with rather different kinds of behavior. The general pattern is that the stronger results are obtained for the frequency estimation problem. However, the algorithms considered are all scaling dependent and consistency is not automatic. A new feature that emerges is the importance of an appropriate choice of scale in order to ensure consistency of the estimates in certain cases. The tentative conclusion is that algorithms referred to as Objective function Reweighting Algorithms (ORA's) are superior to their exact maximum likelihood counterparts, referred to as Gradient condition Reweighting Algorithms (GRA's), especially in the frequency estimation problem. This conclusion does not extend to fitting other families of functions such as rational functions.     

The ease of generating chaotic behavior in economics

Supported by the wealth of models demonstrating chaotic behavior in economics, it becomes appropriate to examine whether this phenomenon is restricted to concocted models or is central to economics. After showing why this behavior is prevalent, aspects of economics causing this behavior are identified. Surprisingly, the basic source of problems from choice theory also generate some of the random motion of economics.     

On the choice of the smoothing parameter in kernel density estimation

Algorithms for the automatic choice of the smoothing parameter in the kernel density estimation problems are proposed. These algorithms use the behavior of the total variation of the estimate.