Speed-up for the expectation-maximization algorithm for clustering categorical data

In model-based cluster analysis, the expectation-maximization (EM) algorithm has a number of desirable properties, but in some situations, this algorithm can be slow to converge. Some variants are proposed to speed-up EM in reducing the time spent in the E-step, in the case of Gaussian mixture. The main aims of such methods is first to speed-up convergence of EM, and second to yield same results (or not so far) than EM itself. In this paper, we compare these methods from categorical data, with the latent class model, and we propose a new variant that sustains better results on synthetic and real data sets, in terms of convergence speed-up and number of misclassified objects.     

A stochastic analysis of the NFD bin-packing algorithm

We consider the Next-Fit-Decreasing (NFD) algorithm for the one-dimensional bin packing problem. The input consists of n pieces of i.i.d. sizes. Bounds on the first two moments of the number of bins used in the packing are established for arbitrary size distributions. These bounds are asymptotically tight. For the uniform distribution we also exhibit numerical results and examine their dependence on the support of the size distribution.     

Data envelopment analysis with stochastic data

Data envelopment analysis (DEA) has proven to be a useful technique in evaluating the efficiency of decision making units that produce multiple-outputs using multiple-inputs. However, the ability to estimate efficiency reliably is hampered in the presence of measurement error and other statistical noise. A main and legitimate criticism of all deterministic models is the inability to separate out measurement error from inefficiency, both of which are unobserved. In this paper, we consider panel data models of efficiency estimation. One DEA model that has been used averages cross-sectional efficiency estimates across time and has been shown to work relatively well. In this paper, it is shown that this approach leads to biased efficiency estimates and provide an alternative model that corrects this problem. The approaches are compared using simulated data for illustrative purposes.     

A stochastic algorithm for optimization problems with continua of inequalities

Optimization algorithms for solving mathematical programming problems involving continua of inequalities are presented. The algorithms use an outer-approximation method, by which they attempt to approximate, at each point, the maxima of sets of inequality constraints. They do so by performing random experiments, resulting in a finite number of points, over which the maximum is taken. They use constraint-dropping schemes, by which they eliminate points from the constraint set at hand, which are felt to be irrelevant. At each point that the algorithms construct, they evaluate a measure of optimality, which indicates the distance of the point from the set of solutions of the optimization problem. They use this measure to determine the number of random experiments performed. Thus, the number of such experiments tends to be small initially, when the points at hand are far from optimal, and they tend to increase when an optimal point is approached.     

A regularized stochastic decomposition algorithm for two-stage stochastic linear programs

In this paper a regularized stochastic decomposition algorithm with master programs of finite size is described for solving two-stage stochastic linear programming problems with recourse. In a deterministic setting cut dropping schemes in decomposition based algorithms have been used routinely. However, when only estimates of the objective function are available such schemes can only be properly justified if convergence results are not sacrificed. It is shown that almost surely every accumulation point in an identified subsequence of iterates produced by the algorithm, which includes a cut dropping scheme, is an optimal solution. The results are obtained by including a quadratic proximal term in the master program. In addition to the cut dropping scheme, other enhancements to the existing methodology are described. These include (i) a new updating rule for the retained cuts and (ii) an adaptive rule to determine when additional reestimation of the cut associated with the current solution is needed. The algorithm is tested on problems from the literature assuming both descrete and continuous random variables.A majority of this work is part of the author's Ph.D. dissertation prepared at the University of Arizona in 1990.     

New estimating equation approaches with application in lifetime data analysis

Estimating equation approaches have been widely used in statistics inference. Important examples of estimating equations are the likelihood equations. Since its introduction by Sir R. A. Fisher almost a century ago, maximum likelihood estimation (MLE) is still the most popular estimation method used for fitting probability distribution to data, including fitting lifetime distributions with censored data. However, MLE may produce substantial bias and even fail to obtain valid confidence intervals when data size is not large enough or there is censoring data. In this paper, based on nonlinear combinations of order statistics, we propose new estimation equation approaches for a class of probability distributions, which are particularly effective for skewed distributions with small sample sizes and censored data. The proposed approaches may possess a number of attractive properties such as consistency, sufficiency and uniqueness. Asymptotic normality of these new estimators is derived. The construction of new estimation equations and their numerical performance under different censored schemes are detailed via Weibull distribution and generalized exponential distribution.     

A stochastic quasigradient algorithm with variable metric

This paper deals with a new variable metric algorithm for stochastic optimization problems. The essence of this is as follows: there exist two stochastic quasigradient algorithms working simultaneously — the first in the main space, the second with respect to the matrices that modify the space variables. Almost sure convergence of the algorithm is proved for the case of the convex (possiblynonsmooth) objective function.     

A branch-and-price algorithm for the capacitated vehicle routing problem with stochastic demands

This article introduces a new exact algorithm for the capacitated vehicle routing problem with stochastic demands (CVRPSD). The CVRPSD can be formulated as a set partitioning problem and it is shown that the associated column generation subproblem can be solved using a dynamic programming scheme. Computational experiments show promising results.     

A stochastic probing algorithm for global optimization

Recently, simulated annealing methods have proven to be a valuable tool for global optimization. We propose a new stochastic method for locating the global optimum of a function. The proposed method begins with the subjective specification of a probing distribution. The objective function is evaluated at a few points sampled from this distribution, which is then updated using the collected information. The updating mechanism is based on the entropy of a move selecting distribution and is loosely connected to some notions in statistical thermodynamics. Examples of the use of the proposed method are presented. These indicate its superior performance as compared with simulated annealing. Preliminary considerations in applying the method to discrete problems are discussed.     

A smoothing stochastic algorithm for quantile estimation

In this paper, we provide the almost-sure convergence and the asymptotic normality of a smooth version of the Robbins–Monro algorithm for the quantile estimation. A Monte Carlo simulation study shows that our proposed method works well within the framework of a data stream.     

A descent algorithm for constrained stochastic extrema

In situations where it is not feasible to find an optimal feedback control law for a stochastic system, an open-loop law can often be derived by optimization. This article presents a method of finding the extremum of certain stochastic functionals analogous to the steepest descent method. Necessary conditions for the convergence of the algorithm are given. Two examples illustrate the use of the algorithm.This research was supported by the Office of Naval Research, Contract No. Nonr-1866 (16) and by the National Aeronautics and Space Administration, Grant No. NGR-22-007-068.     

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the inequalities, and we establish necessary and sufficient conditions which guarantee that the inequalities are facet-defining. A separation heuristic for inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. This research has been supported in part by the National Science Foundation under Award number DMII-0121495.     

On optimal operating conditions for a data processing system: A stochastic approach

In this paper, we consider the problem of determining optimal operating conditions for a data processing system. The system is burned‐in for a fixed burn‐in time before it is put into field operation and, in field operation, it has a work size and follows an age‐replacement policy. Assuming that the underlying lifetime distribution of the system has a bathtub‐shaped failure rate function, the properties of optimal burn‐in time, optimal work size and optimal age‐replacement policy will be derived. It can be seen that this model is a generalization of those considered in the previous works, and it yields a better optimal operating conditions. This paper presents an analytical method for three‐dimensional optimization problem. An algorithm for determining optimal operating conditions is also given. Copyright © 2007 John Wiley & Sons, Ltd.     

A stochastic local search algorithm with adaptive acceptance for high-school timetabling

Automating high school timetabling is a challenging task. This problem is a well known hard computational problem which has been of interest to practitioners as well as researchers. High schools need to timetable their regular activities once per year, or even more frequently. The exact solvers might fail to find a solution for a given instance of the problem. A selection hyper-heuristic can be defined as an easy-to-implement, easy-to-maintain and effective ‘heuristic to choose heuristics’ to solve such computationally hard problems. This paper describes the approach of the team hyper-heuristic search strategies and timetabling (HySST) to high school timetabling which competed in all three rounds of the third international timetabling competition. HySST generated the best new solutions for three given instances in Round 1 and gained the second place in Rounds 2 and 3. It achieved this by using a fairly standard stochastic search method but significantly enhanced by a selection hyper-heuristic with an adaptive acceptance mechanism.     

A policy iteration algorithm for zero-sum stochastic games with mean payoff

We give a policy iteration algorithm to solve zero-sum stochastic games with finite state and action spaces and perfect information, when the value is defined in terms of the mean payoff per turn. This algorithm does not require any irreducibility assumption on the Markov chains determined by the strategies of the players. It is based on a discrete nonlinear analogue of the notion of reduction of a super-harmonic function. To cite this article: J. Cochet-Terrasson, S. Gaubert, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Hierarchical multi-innovation stochastic gradient algorithm for Hammerstein nonlinear system modeling

This paper decomposes a Hammerstein nonlinear system into two subsystems, one containing the parameters of the linear dynamical block and the other containing the parameters of the nonlinear static block, and presents a hierarchical multi-innovation stochastic gradient identification algorithm for Hammerstein systems based on the hierarchical identification principle. The proposed algorithm is simple in principle and easy to implement on-line. A simulation example is provided to test the effectiveness of the proposed algorithm.     

A stochastic gradient type algorithm for closed-loop problems

We focus on the numerical solution of closed-loop stochastic problems, and propose a perturbed gradient algorithm to achieve this goal. The main hurdle in such problems is the fact that the control variables are infinite-dimensional, due to, e.g., the information constraints. Alternatively said, control variables are feedbacks, i.e., functions. Such controls have hence to be represented in a finite way in order to solve the problem numerically. In the same way, the gradient of the criterion is itself an infinite-dimensional object. Our algorithm replaces this exact (and unknown) gradient by a perturbed one, which consists of the product of the true gradient evaluated at a random point and a kernel function which extends this gradient to the neighbourhood of the random point. Proceeding this way, we explore the whole space iteration after iteration through random points. Since each kernel function is perfectly known by a small number of parameters, say N, the control at iteration k is perfectly known as an infinite-dimensional object by at most N × k parameters. The main strength of this method is that it avoids any discretization of the underlying space, provided that we can sample as many points as needed in this space. Moreover, our algorithm can take into account the possible measurability constraints of the problem in a new way. Finally, the randomized strategy implemented by the algorithm causes the most probable parts of the space to be the most explored ones, which is a priori an interesting feature. In this paper, we first prove two convergence results of this algorithm in the strongly convex and convex cases, and then give some numerical examples showing the interest of this method for practical stochastic optimization problems. In Memoriam: Jean-Sébastien Roy passed away July 04, 2007. He was 33 years old.     

A multicut algorithm for two-stage stochastic linear programs

Outer linearization methods, such as Van Slyke and West's L-shaped method for stochastic linear programs, generally apply a single cut on the nonlinear objective at each major iteration. The structure of stochastic programs allows for several cuts to be placed at once. This paper describes a multicut algorithm to carry out this procedure. It presents experimental and theoretical justification for reductions in major iterations.     

A successive approximation algorithm for stochastic control problems

We consider optimal stochastic control problems in which the state variables are governed by Itô equations. A successive approximation algorithm for optimal stochastic control is obtained. This algorithm, together with the existing numerical methods for parabolic or elliptic PDEs, provides numerical schemes for the solution of Bellman equations.