A Bayesian Approach to Bioassay.

Optimization problems arising in practice involve random parameters. For the computation of robust optimal solutions, i.e., optimal solutions being insensitive with respect to random parameter variations, deterministic substitute problems are needed. Based on the distribution of the random data, and using decision theoretical concepts, optimization problems under stochastic uncertainty are converted into deterministic substitute problems. Due to the occurring probabilities and expectations, approximative solution techniques must be applied. Deterministic and and stochastic approximation methods and their analytical properties are provided: Taylor expansion, regression and response surface methods, probability inequalities, First Order Reliability Methods, convex approximation/deterministic descent directions/efficient points, stochastic approximation methods, differentiation of probability and mean value functions. Convergence results of the resulting iterative solution procedures are given.

Optimization in Simulation is an important problem often encountered in system behavior investigation; however, the existing methods such as response surface methodology and stochastic approximation method are inefficient. This paper presents a modification of a quasi‐Newton method, in which the parameters are determined from some numerical experiments. To demonstrate the validity of the devised method, two examples resembling the M/M/1 queueing problem are solved. The closeness of the converged solutions to the optimal solutions and a comparison with two stochastic approximation methods indicate that the modified quasi‐Newton method as devised in this paper is a robust and efficient method for solving optimization problems in simulation.

23 Stochastic approximation

Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by $$\mathop {\min }\limits_{x \in D} F(x)withF(x): = Ef(\omega ,x).$$ (1) Here,f=f(ω,x) is a function on ℛr depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ℛr. Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.

Efficient recursive estimation; application to estimating the parameters of a covariance function

We apply the stochastic approximation method to construct a large class of recursive kernel estimators of a distribution function. We study the properties of these estimators and compare them with Nadaraya’s distribution estimator. It turns out that, with an adequate choice of the stepsize of the proposed algorithm, the MWISE (Mean Weighted Integrated Squared Error) of the proposed estimator is smaller than that of Nadaraya’s estimator. We corroborate these theoretical results by simulations and a real dataset.

This paper is a continuation of our papers [9], [10] and [11] and is concerned with a Robbins-Monro type stochastic approximation method when a sequence of dependently distributed random vectors is given. The method of stochastic approximation has been first proposed by H. Robbins and S. Monro ([5]) and its modifications have been thereafter given by many authors. A typical one of them is as follows. Suppose that an RN-valued random vector Y„(x) can be observed at xERN and each instant n, and the expected value of Y n(X), denoted by E[Y„(x)]=1/17,(x), is unknown to us. Assuming that the equation itin(x)=0 has a solution x=8,, for each n=1, 2, • , it is desire to estimate 0,, for sufficiently large n on the basis of observed values Y1(X0), Y2(Xi), Y.+I(X,i), ••• at the points X„ Xi, •-• , X„, ••• which are produced by the following recurrence relation,

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

Noncoherent multiuser detection for nonlinear modulation was previously studied and the idea of phase-independent noncoherent decorrelation was introduced and three post-decorrelative detectors were obtained and analyzed. However, their implementation requires the knowledge of the signature waveforms of all the users, which may be available only for centralized implementation. In this paper, we obtain a blind adaptive noncoherent decorrelative detector for nonlinear modulation that is suitable for distributed implementation with the knowledge of only the normalized signals of the desired user and the additive noise variance. This detector is based on the stochastic approximation method and does not require the overhead of any kind of "training." Two adaptive algorithms are developed, one guided by every signal in the desired user's signal set individually, and the other by the user's entire signal space. While this paper focuses on the particular problem of blind adaptive noncoherent decorrelative detection, it addresses a more general adaptation issue, namely, that of improving the convergence properties of an adaptive scheme by effectively using all the information that is known, and adapting only to the part of the desired solution that is truly unknown. Convergence is shown in the mean squared error sense for both the fixed step-size and time-varying step-size versions of the two algorithms.

This chapter is concerned with decentralized and asynchronous forms of the stochastic approximation algorithms, a relatively new area of research. Compared with the rapid progress and extensive literature in stochastic approximation methods, the study of parallel stochastic approximations is still in its infancy. Perhaps the first work on the subject was [14], which dealt with a very particular class of algorithms, where similar computations were done by several processors and convex combinations were taken. The general ideas of weak convergence theory were applied to a fairly broad class of realistic algorithms in [111, 112]. The general ideas presented there and in [105, 171] form the basis of this chapter. Analogously to the problems in Chapter 8, those methods can handle correlated and state dependent noise, delays in communication, and asynchronous and distributed network forms. Various examples are given in Section 1. In the basic model, there are several processors; each one is responsible for the updating of only a part of the parameter vector. There might be overlaps in that several processors contribute to the updating of the same component of the parameter. Such models were treated in [105, 171, 172]. For a similar model, the problem of finding zeros of a nonlinear function with noisy observations via parallel processing methods and with random truncation bounds was treated in [201]. An attempt to get real-time implementable procedures via pipelining (see Section 1.2) of communication and computation for algorithms with “delayed” observations was in [203].

Analytical gradient based non-rigid image registration methods,\nusing intensity based similarity measures (e.g. mutual information),\nhave proven to be capable of accurately handling\nmany types of deformations. While their versatility is largely\nin part to their high degrees of freedom, the computation of\nthe gradient of the similarity measure with respect to the many\nwarp parameters becomes very time consuming. Recently, a\nsimple stochastic approximation method using a small random\nsubset of image pixels to approximate this gradient has\nbeen shown to be effective. We propose to use importance\nsampling to improve the accuracy and reduce the variance of\nthis approximation by preferentially selecting pixels near image\nedges. Initial empirical results show that a combination of\nstochastic approximation methods and importance sampling\ngreatly improves the rate of convergence of the registration\nprocess while preserving accuracy.

To alleviate the excessive computational burden on topology optimizations, an efficient methodology on topology optimization is proposed. In the proposed methodology, a multi-scale ON/OFF method is designed to avoid using infinite number of decision parameters; and a stochastic approximation method is employed to minimize the computational costs when computing the sensitivity information. Numerical results are reported to showcase the merits of the proposed methodology.

On the estimation of the adjustment coefficient in risk theory by means of stochastic approximation procedures

In this paper, we consider the convex stochastic conic optimization defined by expectations. An augmented Lagrangian stochastic approximation (ALSAcp) method is proposed to solve this convex stochastic conic optimization problem and the regrets of this method are analyzed. Under mild conditions, we show that this method exhibits ${\rm~O}(T^{-1/2})$ regret for both objective reduction and constraint violation if parameters in the algorithm are properly chosen, where $T$ denotes the number of iterations. Moreover, we show that, with at least $1-1/T$ probability, the method has no more than ${\rm~O}(\log~(T)/\sqrt{T})$ for both objective descent regret and constraint violation regret.

By focusing on computational aspects, this work is concerned with numerical methods for stock selling decision using stochastic approximation methods. Concentrating on the class of decisions depending on threshold values, an optimal stopping problem is converted to a parametric stochastic optimization problem. The algorithms are model free and are easily implementable on‐line. Convergence of the algorithms is established, second moment bound of estimation error is obtained, and escape probability from a neighborhood of the true parameter is also derived. Numerical examples using both daily closing prices and intra‐day data are provided to demonstrate the performance of the algorithms.