Stochastic ultimate load analysis:models and solution methods 1

The stochastic ultimate load analysis model used in the safety analysis of engineering structures can be treated as a special case of chance-constrained problems (CCP) which minimize a stochastic cost function subject to some probabilistic constraints. Some special cases (such as a deterministic cost function with probabilistic constraints or deterministic constraints with a random cost function) for ultimate load analysis have airady been investigated by various researchers. In this paper, a generai probabilistic approach to stochastic ultimate load analysis is given. In doing so, some approximation techniques are needed due to the fact that the problems at hand are too complicated to evaluate precisely. We propose two extensions of the SQP method in which the variables appear in the algorithms inexactly. These algorithms are shown to be globally convergent for all models and locally superlinearly convergent for some special cases     

求解阵列天线自适应滤波问题的一种调比随机逼近算法

提出了求解阵列天线自适应滤波问题的一种调比随机逼近算法.每一步迭代中,算法选取调比的带噪负梯度方向作为新的迭代方向.相比已有的其他随机逼近算法,这个算法不需要调整稳定性常数,在一定程度上解决了稳定性常数选取难的问题.数值仿真实验表明,算法优于已有的滤波算法,且比经典Robbins-Monro （RM）算法具有更好的稳定性.     

Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms

Using ideas and results from polynomial time approximation and exact computation we design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular two paradigmatic problems, max independent set and min vertex cover.     

Convergence of stochastic search algorithms to finite size pareto set approximations

In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of ε-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.      

Validation analysis of mirror descent stochastic approximation method

The main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation (SA) type algorithms. To this end we show that while running a Mirror Descent Stochastic Approximation procedure one can compute, with a small additional effort, lower and upper statistical bounds for the optimal objective value. We demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method, while their computational cost is considerably smaller.     

Online stochastic reservation systems

This paper considers online stochastic reservation problems, where requests come online and must be dynamically allocated to limited resources in order to maximize profit. Multi-knapsack problems with or without overbooking are examples of such online stochastic reservations. The paper studies how to adapt the online stochastic framework and the consensus and regret algorithms proposed earlier to online stochastic reservation systems. On the theoretical side, it presents a constant sub-optimality approximation of multi-knapsack problems, leading to a regret algorithm that evaluates each scenario with a single mathematical programming optimization followed by a small number of dynamic programs for one-dimensional knapsacks. It also proposes several integer programming models for handling cancellations and proves their equivalence. On the experimental side, the paper demonstrates the effectiveness of the regret algorithm on multi-knapsack problems (with and without overbooking) based on the benchmarks proposed earlier.     

Approximate nonlinear programming algorithms for solving stochastic programs with recourse

This paper summarizes the main results on approximate nonlinear programming algorithms investigated by the author. These algorithms are obtained by combining approximation and nonlinear programming algorithms. They are designed for programs in which the evaluation of the objective functions is very difficult so that only their approximate values can be obtained. Therefore, these algorithms are particularly suitable for stochastic programming problems with recourse.Project supported by the National Natural Science Foundation of China.     

Fitting Social Network Models Using Varying Truncation Stochastic Approximation MCMC Algorithm

The exponential random graph model (ERGM) plays a major role in social network analysis. However, parameter estimation for the ERGM is a hard problem due to the intractability of its normalizing constant and the model degeneracy. The existing algorithms, such as Monte Carlo maximum likelihood estimation (MCMLE) and stochastic approximation, often fail for this problem in the presence of model degeneracy. In this article, we introduce the varying truncation stochastic approximation Markov chain Monte Carlo (SAMCMC) algorithm to tackle this problem. The varying truncation mechanism enables the algorithm to choose an appropriate starting point and an appropriate gain factor sequence, and thus to produce a reasonable parameter estimate for the ERGM even in the presence of model degeneracy. The numerical results indicate that the varying truncation SAMCMC algorithm can significantly outperform the MCMLE and stochastic approximation algorithms: for degenerate ERGMs, MCMLE and stochastic approximation often fail to produce any reasonable parameter estimates, while SAMCMC can do; for nondegenerate ERGMs, SAMCMC can work as well as or better than MCMLE and stochastic approximation. The data and source codes used for this article are available online as supplementary materials.     

An optimal method for stochastic composite optimization

This paper considers an important class of convex programming (CP) problems, namely, the stochastic composite optimization (SCO), whose objective function is given by the summation of general nonsmooth and smooth stochastic components. Since SCO covers non-smooth, smooth and stochastic CP as certain special cases, a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming. Note however that the optimization algorithms that can achieve this lower bound had never been developed. In this paper, we show that the simple mirror-descent stochastic approximation method exhibits the best-known rate of convergence for solving these problems. Our major contribution is to introduce the accelerated stochastic approximation (AC-SA) algorithm based on Nesterov’s optimal method for smooth CP (Nesterov in Doklady AN SSSR 269:543–547, 1983; Nesterov in Math Program 103:127–152, 2005), and show that the AC-SA algorithm can achieve the aforementioned lower bound on the rate of convergence for SCO. To the best of our knowledge, it is also the first universally optimal algorithm in the literature for solving non-smooth, smooth and stochastic CP problems. We illustrate the significant advantages of the AC-SA algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems.     

Inexact subgradient methods with applications in stochastic programming

In many instances, the exact evaluation of an objective function and its subgradients can be computationally demanding. By way of example, we cite problems that arise within the context of stochastic optimization, where the objective function is typically defined via multi-dimensional integration. In this paper, we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods. We refer to this new class of methods as inexact subgradient algorithms. With relatively mild conditions imposed on the approximations, we show that the inexact subgradient algorithms inherit properties associated with their traditional (i.e., exact) counterparts. Within the context of stochastic optimization, the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi-gradient methods. Additionally, we study methods in which steplengths may be defined adaptively, in a manner that reflects the improvement in the objective function approximations as the iterations proceed. We illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs.This work was supported by Grant Nos. NSF-DDM-89-10046 and NSF-DDM-9114352 from the National Science Foundation.     

Three-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss three-stage stochastic Runge–Kutta (SRK) methods with strong order 1.0 for a strong solution of Stratonovich stochastic differential equations (SDEs). Higher deterministic order is considered. Two methods, a three-stage explicit (E3) method and a three-stage semi-implicit (SI3) method, are constructed in this paper. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of several standard test problems.     

Stochastic second-order-cone complementarity problems: expected residual minimization formulation and its applications

This paper considers a class of stochastic second-order-cone complementarity problems (SSOCCP), which are generalizations of the noticeable stochastic complementarity problems and can be regarded as the Karush–Kuhn–Tucker conditions of some stochastic second-order-cone programming problems. Due to the existence of random variables, the SSOCCP may not have a common solution for almost every realization . In this paper, motivated by the works on stochastic complementarity problems, we present a deterministic formulation called the expected residual minimization formulation for SSOCCP. We present an approximation method based on the Monte Carlo approximation techniques and investigate some properties related to existence of solutions of the ERM formulation. Furthermore, we experiment some practical applications, which include a stochastic natural gas transmission problem and a stochastic optimal power flow problem in radial network.     

A new hybrid stochastic approximation algorithm

In this paper, we consider optimizing the performance of a stochastic system that is too complex for theoretical analysis to be possible, but can be evaluated by using simulation or direct experimentation. To optimize the expected performance of such system as a function of several input parameters, we propose a hybrid stochastic approximation algorithm for finding the root of the gradient of the response function. At each iteration of the hybrid algorithm, alternatively, either an average of two independent noisy negative gradient directions or a scaled noisy negative gradient direction is selected. The almost sure convergence of the hybrid algorithm is established. Numerical comparisons of the hybrid algorithm with two other existing algorithms in a simple queueing system and five nonlinear unconstrained stochastic optimization problems show the advantage of the hybrid algorithm.     

An anytime multistep anticipatory algorithm for online stochastic combinatorial optimization

The one-step anticipatory algorithms (1s-AA) is an online algorithm making decisions under uncertainty by ignoring the non-anticipativity constraints in the future. It was shown to make near-optimal decisions on a variety of online stochastic combinatorial problems in dynamic fleet management and reservation systems. Here we consider applications in which 1s-AA is not as close to the optimum and propose Amsaa, an anytime multi-step anticipatory algorithm. Amsaa combines techniques from three different fields to make decisions online. It uses the sampling average approximation method from stochastic programming, search algorithms for Markov decision processes from artificial intelligence, and discrete optimization algorithms. Amsaa was evaluated on a stochastic project scheduling application from the pharmaceutical industry featuring endogenous observations of the uncertainty. The experimental results show that Amsaa significantly outperforms state-of-the-art algorithms on this application under various time constraints.     

Hybrid Population-Based Algorithms for the Bi-Objective Quadratic Assignment Problem

We present variants of an ant colony optimization (MO-ACO) algorithm and of an evolutionary algorithm (SPEA2) for tackling multi-objective combinatorial optimization problems, hybridized with an iterative improvement algorithm and the robust tabu search algorithm. The performance of the resulting hybrid stochastic local search (SLS) algorithms is experimentally investigated for the bi-objective quadratic assignment problem (bQAP) and compared against repeated applications of the underlying local search algorithms for several scalarizations. The experiments consider structured and unstructured bQAP instances with various degrees of correlation between the flow matrices. We do a systematic experimental analysis of the algorithms using outperformance relations and the attainment functions methodology to asses differences in the performance of the algorithms. The experimental results show the usefulness of the hybrid algorithms if the available computation time is not too limited and identify SPEA2 hybridized with very short tabu search runs as the most promising variant. This research was mainly done while Luís Paquete and Thomas Stützle were with the Intellectics Group at the Computer Science Department of Darmstadt University of Technology, Germany     

Approximating the minimum clique cover and other hard problems in subtree filament graphs

Subtree filament graphs are the intersection graphs of subtree filaments in a tree. This class of graphs contains subtree overlap graphs, interval filament graphs, chordal graphs, circle graphs, circular-arc graphs, cocomparability graphs, and polygon-circle graphs. In this paper we show that, for circle graphs, the clique cover problem is NP-complete and the h-clique cover problem for fixed h is solvable in polynomial time. We then present a general scheme for developing approximation algorithms for subtree filament graphs, and give approximation algorithms developed from the scheme for the following problems which are NP-complete on circle graphs and therefore on subtree filament graphs: clique cover, vertex colouring, maximum k-colourable subgraph, and maximum h-coverable subgraph.     

Pathwise convergence rates for numerical solutions of Markovian switching stochastic differential equations

This work develops numerical approximation algorithms for solutions of stochastic differential equations with Markovian switching. The existing numerical algorithms all use a discrete-time Markov chain for the approximation of the continuous-time Markov chain. In contrast, we generate the continuous-time Markov chain directly, and then use its skeleton process in the approximation algorithm. Focusing on weak approximation, we take a re-embedding approach, and define the approximation and the solution to the switching stochastic differential equation on the same space. In our approximation, we use a sequence of independent and identically distributed (i.i.d.) random variables in lieu of the common practice of using Brownian increments. By virtue of the strong invariance principle, we ascertain rates of convergence in the pathwise sense for the weak approximation scheme.     

Random-Direction Optimization Algorithms with Applications to Threshold Controls

This work develops a class of stochastic optimization algorithms. It aims to provide numerical procedures for solving threshold-type optimal control problems. The main motivation stems from applications involving optimal or suboptimal hedging policies, for example, production planning of manufacturing systems including random demand and stochastic machine capacity. The proposed algorithm is a constrained stochastic approximation procedure that uses random-direction finite-difference gradient estimates. Under fairly general conditions, the convergence of the algorithm is established and the rate of convergence is also derived. A numerical example is reported to demonstrate the performance of the algorithm.     

Approximating maximum independent sets by excluding subgraphs

An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known toO(n/(logn)²). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strongsimultaneous performance guarantee for the clique and coloring problems.The framework ofsubgraph-excluding algorithms is presented. We survey the known approximation algorithms for the independent set (clique), coloring, and vertex cover problems and show how almost all fit into that framework. We show that among subgraph-excluding algorithms, the ones presented achieve the optimal asymptotic performance guarantees.A preliminary version of this paper appeared in [9].Supported in part by National Science Foundation Grant CCR-8902522 and PYI Award CCR-9057488.Research done at Rutgers University. Supported in part by Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) fellowship.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par