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Random binary search trees, b-ary search trees, median-of-(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.

Abstract. Improving Hit-and-Run is a random search algorithm for global optimization that at each iteration generates a candidate point for improvement that is uniformly distributed along a randomly chosen direction within the feasible region. The candidate point is accepted as the next iterate if it offers an improvement over the current iterate. We show that for positive definite quadratic programs, the expected number of function evaluations needed to arbitrarily well approximate the optimal solution is at most O(n 5~2) where n is the dimension of the problem. Improving Hit-and-Run when applied to global optimization problems can therefore be expected to converge polynomially fast as it approaches the global optimum. Key words. Random search, Monte Carlo optimization, algorithm complexity, global optimization. 1.

Statistical model validation is treated for a class of parametric uncertainty models and also for a more general class of nonparametric uncertainty models. We show that, in many cases of interest, this problem reduces to computing relative weighted volumes of convex sets in R N (where N is the number of uncertain parameters) for parametric uncertainty models, and to computing the limit of a sequence (Vk ) 1 1 of relative weighted volumes of convex sets in R k for nonparametric uncertainty models. We then present and discuss a randomized algorithm based on gas kinetics for probable approximate computation of these volumes. We also review the existing Hit-and-Run family of algorithms for this purpose. Finally, we introduce the notion of testability to describe uncertainty models that can be statistically validated with arbitrary reliability using input-output data records of sufficient (finite) length. It is then shown that some common nonparametric uncertainty models, such as thos...

In this paper we formulate a particular statistical model validation problem in which we wish to determine the probability that a certain hypothesized parametric uncertainty model is consistent with a given input-output data record. Using a Bayesian approach and ideas from the field of hypothesis testing, we show that in many cases of interest this problem reduces to computing relative weighted volumes of convex sets in R N (where N is the number of uncertain parameters). We also present and discuss a randomized algorithm based on gas kinetics, as well as the existing Hit-and-Run family of algorithms, for probable approximate computation of these volumes. 1 Introduction Motivated by the desire to produce identified models that are compatible with modern robust control design methodologies, many researchers have recently been working in the area of control-oriented system identification (see for example [5, 6, 7, 11, 12, 16, 17, 19, 23, 24, 25, 27, 28] and the references cited therein...

Abstract. Given a basic compact semi-algebraic set K ⊂ R n, we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, and so permits to approximate the integral on K of any given polynomial; extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed. Key words. Computational geometry; volume; integration; K-moment problem; semidefinite programming AMS subject classifications. 14P10, 11E25, 12D15, 90C25 1. Introduction. Computing

We consider the problem of sampling a point from an arbitrary distribution π over an arbitrary subset S of an integer hyper-rectangle. Neither the distribution π nor the support set S are assumed to be available as explicit mathematical equations but may only be defined through oracles and in particular computer programs. This problem commonly occurs in black-box discrete optimization as well as counting and estimation problems. The generality of this setting and high-dimensionality of S precludes the application of conventional random variable generation methods. As a result, we turn to Markov Chain Monte Carlo (MCMC) sampling, where we execute an ergodic Markov chain that converges to π so that the distribution of the point delivered after sufficiently many steps can be made arbitrarily close to π. Unfortunately, classical Markov chains such as the nearest neighbor random walk or the co-ordinate direction random walk fail to converge to π as they can get trapped in isolated regions of the support set. To surmount this difficulty, we propose Discrete Hit-and-Run (DHR), a Markov chain motivated by the Hit-and-Run algorithm known to be the most efficient method for sampling from log-concave distributions over convex bodies in Rn. We prove that the limiting distribution of DHR is π as desired, thus enabling us to sample approximately from π by delivering the last iterate of a sufficiently

Hit-and-Run algorithms are Monte Carlo procedures for generating points that are asymptotically distributed according to general absolutely continuous target distributions G over open bounded regions S. Applications include nonredundant constraint identification, global optimization, and Monte Carlo integration. These algorithms are reversible random walks which commonly apply uniformly distributed step directions. We investigate nonuniform direction choice and show that under minimal restrictions on the region S and target distribution G, there exists a unique direction choice distribution, characterized by necessary and sufficient conditions depending on S and G, which optimizes a bound on the rate of convergence. We provide computational results demonstrating greatly accelerated convergence for this optimizing direction choice and We consider the Monte Carlo problem of generating a sample of points according to a given probability distribution G over an open, bounded region S in ℜn. After motivating the problem through several applications, this section discusses the limitations of exact sampling

Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to “conventional methods”, especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratioof-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension.

A simulation model is successful if it leads to policy action, i.e., if it is implemented. Studies show that for a model to be implemented, it must have good correspondence with the mental model of the system held by the user of the model. The user must feel confident that the simulation model corresponds to this mental model. An understanding of how the model works is required. Simulation models for implementation must be developed step by step, starting with a simple model, the simulation prototype. After this has been explained to the user, a more detailed model can be developed on the basis of feedback from the user. Software for simulation prototyping is discussed, e.g., with regard to the ease with which models and output can be explained and the speed with which small models can be written.