James M. Calvin

On the convergence of the P-algorithm for one-dimensional global optimization of smooth functions

The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.

One-Dimensional P-Algorithm with Convergence Rate O(n−3+δ) for Smooth Functions

Algorithms based on statistical models compete favorably with other global optimization algorithms as shown by extensive testing results. A theoretical inadequacy of previously used statistical models for smooth objective functions was eliminated by the authors who, in a recent paper, have constructed a P-algorithm for a statistical model for smooth functions. In the present paper, a modification of that P-algorithm with an improved convergence rate is described.

Using permutations in regenerative simulations to reduce variance

We propose a new estimator for a large class of performance measures obtained from a regenerative simulation of a system having two distinct sequences of regeneration times. To construct our new estimator, we first generate a sample path of a fixed number of cycles based on one sequence of regeneration times, divide the path into segments based on the second sequence of regeneration times, permute the segments, and calculate the performance on the new path using the first sequence of regeneration times. We average over all possible permutations to construct the new estimator. This strictly reduces variance when the original estimator is not simply an additive functional of the sample path. To use the new estimator in practice, the extra computational effort is not large since all permutations do not actually have to be computed as we derive explicit formulas for our new estimators. We examine the small-sample behavior of our estimators. In particular, we prove that for any fixed number of cycles from the first regenerative sequence, our new estimator has smaller mean squared error than the standard estimator. We show explicitly that our method can be used to derive new estimators for the expected cumulative reward until a certain set of states is hit and the time-average variance parameter of a regenerative simulation.

Simulation output analysis using integrated paths

This article considers the steady-state simulation output analysis problem for a process that satisfies a functional central limit theorem. We construct an estimator for the time-average variance constant that is based on iterated integrations of the sample path. When the observations are batched, the method generalizes the method of batch means. One advantage of the method is that it can be used without batching the observations; that is, it can allow for the process variance to be estimated at any time as the simulation runs without waiting for a fixed time horizon to complete. When used in conjunction with batching, the method can improve efficiency (the reciprocal of work times mean-squared error) compared with the standard method of batch means. In numerical experiments, efficiency improvement ranged from a factor of 1.5 (for the waiting time sequence in an M/M/1 queueing system with a single integrated path) up to a factor of 14 (for an autoregressive process and 19 integrated paths).

The semi-regenerative method of simulation output analysis

We develop a class of techniques for analyzing the output of simulations of a semi-regenerative process. Called the semi-regenerative method, the approach is a generalization of the regenerative method, and it can increase efficiency. We consider the estimation of various performance measures, including steady-state means, expected cumulative reward until hitting a set of states, derivatives of steady-state means, and time-average variance constants. We also discuss importance sampling and a bias-reduction technique. In each case, we develop two estimators: one based on a simulation of a single sample path, and the other a type of stratified estimator in which trajectories are generated in an independent and identically distributed manner. We establish a central limit theorem for each estimator so confidence intervals can be constructed.

Permuted Standardized Time Series for Steady-State Simulations

We describe an extension procedure for constructing new standardized time series procedures from existing ones. The approach is based on averaging over sample paths obtained by permuting path segments. Analytical and empirical results indicate that permuting improves standardized time series methods. We compare permuting to an alternative extension procedure known as batching. We demonstrate the permuting method by applying it to estimators based on the maximum and the area of a normalized path.

A One-Dimensional Optimization Algorithm and Its Convergence Rate under the Wiener Measure

In this paper we describe an adaptive algorithm for approximating the global minimum of a continuous function on the unit interval, motivated by viewing the function as a sample path of a Wiener process. It operates by choosing the next observation point to maximize the probability that the objective function has a value at that point lower than an adaptively chosen threshold. The error converges to zero for any continuous function. Under the Wiener measure, the error converges to zero at rate e?n?n, where {?n} (a parameter of the algorithm) is a positive sequence converging to zero at an arbitrarily slow rate.

Central Limit Theorems for Permuted Regenerative Estimators

We prove strong laws of large numbers and central limit theorems for some permuted estimators from regenerative simulations. These limit theorems provide the basis for constructing asymptotically valid confidence intervals for the permuted estimators.

Lower bound on complexity of optimization of continuous functions

This paper considers the problem of approximating the minimum of a continuous function using a fixed number of sequentially selected function evaluations. A lower bound on the complexity is established by analyzing the average case for the Brownian bridge.

Accelerated Regeneration for Markov Chain Simulations

This paper describes a generalization of the classical regenerative method of simulation output analysis. Instead of blocking a generated sample path on returns to a fixed return state, a more general scheme to randomly decompose the path is used. In some cases, this decomposition scheme results in regeneration times that are a supersequence of the classical regeneration times. This “accelerated” regeneration is advantageous in several simulation contexts. It is shown that when this decomposition scheme accelerates regeneration relative to the classical regenerative method, it also yields a smaller asymptotic variance of the regenerative variance estimator than the classical method. Several other contexts in which increased regeneration frequency is beneficial are also discussed.