Adaptive random search for continuous simulation optimization

We present, analyze, and compare three random search methods for solving stochastic optimization problems with uncountable feasible regions. Our adaptive search with resampling (ASR) approach is a framework for designing provably convergent algorithms that are adaptive and may consequently involve local search. The deterministic and stochastic shrinking ball (DSB and SSB) approaches are also convergent, but they are based on pure random search with the only difference being the estimator of the optimal solution [the DSB method was originally proposed and analyzed by Baumert and Smith]. The three methods use different techniques to reduce the effects of noise in the estimated objective function values. Our ASR method achieves this goal through resampling of already sampled points, whereas the DSB and SSB approaches address it by averaging observations in balls that shrink with time. We present conditions under which the three methods are convergent, both in probability and almost surely, and provide a limited computational study aimed at comparing the methods. Although further investigation is needed, our numerical results suggest that the ASR approach is promising, especially for difficult problems where the probability of identifying good solutions using pure random search is small. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Optimal power control in a wireless network using a model with stochastic link coefficients

This paper addresses optimal power allocation in a wireless communication network under uncertainty. The paper introduces a framework for optimal transmit power allocation in a wireless network where both the useful and interference coefficients are random. The new approach to power control is based on a stochastic programming formulation with probabilistic SIR constraints. This allows to state the power allocation problem as a convex optimization problem assuming normally or log‐normally distributed communication link coefficients. Numerical examples illustrate the performance of the optimal stochastic power allocation. A distributed algorithm for the decentralized solution of the stochastic power allocation problem is discussed. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Averaging frameworks for simulation optimization with applications to simulated annealing

We present two frameworks for designing random search methods for discrete simulation optimization. One of our frameworks is very broad (in that it includes many random search methods), whereas the other one considers a special class of random search methods called point‐based methods, that move iteratively between points within the feasible region. Our frameworks involve averaging, in that all decisions that require estimates of the objective function values at various feasible solutions are based on the averages of all observations collected at these solutions so far. Also, the methods are adaptive in that they can use information gathered in previous iterations to decide how simulation effort is expended in the current iteration. We show that the methods within our frameworks are almost surely globally convergent under mild conditions. Thus, the generality of our frameworks and associated convergence guarantees makes the frameworks useful to algorithm developers wishing to design efficient and rigorous procedures for simulation optimization. We also present two variants of the simulated annealing (SA) algorithm and provide their convergence analysis as example application of our point‐based framework. Finally, we provide numerical results that demonstrate the empirical effectiveness of averaging and adaptivity in the context of SA. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Statistical decision analysis of stochastic linear programming problems

This paper presents a statistical decision analysis of a one-stage linear programming problem with deterministic constraints and stochastic criterion function. Procedures for obtaining numerical results are given which are applicable to any problem having this general form. We begin by stating the statistical decision problems to be considered, and then discuss the expected value of perfect information and the expected value of sample information. In obtaining these quantities, use is made of the distribution of the optimal value of the linear programming problem with stochastic criterion function, and so we discuss Monte Carlo and numerical integration procedures for estimating the mean of this distribution. The case in which the random criterion vector has a multivariate Normal distribution is discussed separately, and more detailed methods are offered. We discuss dual problems, including some relationships of this work with other work in probabilistic linear programming. An example is given in Appendix A showing application of the methods to a sample problem. In Appendix B we consider the accuracy of a procedure for approximating the expected value of information.     

A finite algorithm for concave minimization over a polyhedron

We present a new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron. The algorithm is of the branch-and-bound type. It finds a globally optimal extreme point solution for this problem in a finite number of steps. One of the major advantages of the algorithm is that the linear programming subproblems solved during the branch-and-bound search each have the same feasible region. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some preliminary computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several bilinear programming problems with the code.     

Subspace dynamic‐simplex linear interpolation search for mixed‐integer black‐box optimization problems

Design and management of complex systems with both integer and continuous decision variables can be guided using mixed‐integer optimization models and analysis. We propose a new mixed‐integer black‐box optimization (MIBO) method, subspace dynamic‐simplex linear interpolation search (SD‐SLIS), for decision making problems in which system performance can only be evaluated with a computer black‐box model. Through a sequence of gradient‐type local searches in subspaces of solution space, SD‐SLIS is particularly efficient for such MIBO problems with scaling issues. We discuss the convergence conditions and properties of SD‐SLIS algorithms for a class of MIBO problems. Under mild conditions, SD‐SLIS is proved to converge to a stationary solution asymptotically. We apply SD‐SLIS to six example problems including two MIBO problems associated with petroleum field development projects. The algorithm performance of SD‐SLIS is compared with that of a state‐of‐the‐art direct‐search method, NOMAD, and that of a full space simplex interpolation search, Full‐SLIS. The numerical results suggest that SD‐SLIS solves the example problems efficiently and outperforms the compared methods for most of the example cases. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 305–322, 2017     

Augmented nested sampling for stochastic programs with recourse and endogenous uncertainty

We propose a novel simulation‐based approach for solving two‐stage stochastic programs with recourse and endogenous (decision dependent) uncertainty. The proposed augmented nested sampling approach recasts the stochastic optimization problem as a simulation problem by treating the decision variables as random. The optimal decision is obtained via the mode of the augmented probability model. We illustrate our methodology on a newsvendor problem with stock‐dependent uncertain demand both in single and multi‐item (news‐stand) cases. We provide performance comparisons with Markov chain Monte Carlo and traditional Monte Carlo simulation‐based optimization schemes. Finally, we conclude with directions for future research.     

A risk function for the stochastic modeling of electric capacity expansion

We present a stochastic optimization model for planning capacity expansion under capacity deterioration and demand uncertainty. The paper focuses on the electric sector, although the methodology can be used in other applications. The goals of the model are deciding which energy types must be installed, and when. Another goal is providing an initial generation plan for short periods of the planning horizon that might be adequately modified in real time assuming penalties in the operation cost. Uncertainty is modeled under the assumption that the demand is a random vector. The cost of the risk associated with decisions that may need some tuning in the future is included in the objective function. The proposed scheme to solve the nonlinear stochastic optimization model is Generalized Benders' decomposition. We also exploit the Benders' subproblem structure to solve it efficiently. Computational results for moderate‐size problems are presented along with comparison to a general‐purpose nonlinear optimization package. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:662–683, 2001     

Using stochastic programming to solve an outpatient appointment scheduling problem with random service and arrival times

We study a stochastic outpatient appointment scheduling problem (SOASP) in which we need to design a schedule and an adaptive rescheduling (i.e., resequencing or declining) policy for a set of patients. Each patient has a known type and associated probability distributions of random service duration and random arrival time. Finding a provably optimal solution to this problem requires solving a multistage stochastic mixed‐integer program (MSMIP) with a schedule optimization problem solved at each stage, determining the optimal rescheduling policy over the various random service durations and arrival times. In recognition that this MSMIP is intractable, we first consider a two‐stage model (TSM) that relaxes the nonanticipativity constraints of MSMIP and so yields a lower bound. Second, we derive a set of valid inequalities to strengthen and improve the solvability of the TSM formulation. Third, we obtain an upper bound for the MSMIP by solving the TSM under the feasible (and easily implementable) appointment order (AO) policy, which requires that patients are served in the order of their scheduled appointments, independent of their actual arrival times. Fourth, we propose a Monte Carlo approach to evaluate the relative gap between the MSMIP upper and lower bounds. Finally, in a series of numerical experiments, we show that these two bounds are very close in a wide range of SOASP instances, demonstrating the near‐optimality of the AO policy. We also identify parameter settings that result in a large gap in between these two bounds. Accordingly, we propose an alternative policy based on neighbor‐swapping. We demonstrate that this alternative policy leads to a much tighter upper bound and significantly shrinks the gap.     

On parallel machine replacement problems with general replacement cost functions and stochastic deterioration

The parallel machine replacement problem consists of finding a minimum cost replacement policy for a finite population of economically interdependent machines. In this paper, we formulate a stochastic version of the problem and analyze the structure of optimal policies under general classes of replacement cost functions. We prove that for problems with arbitrary cost functions, there can be optimal policies where a machine is replaced only if all machines in worse states are replaced (Worse Cluster Replacement Rule). We then show that, for problems with replacement cost functions exhibiting nonincreasing marginal costs, there are optimal policies such that, in any stage, machines in the same state are either all kept or all replaced (No‐Splitting Rule). We also present an example that shows that economies of scale in replacement costs do not guarantee optimal policies that satisfy the No‐Splitting Rule. These results lead to the fundamental insight that replacement decisions are driven by marginal costs, and not by economies of scale as suggested in the literature. Finally, we describe how the optimal policy structure, i.e., the No‐Splitting and Worse Cluster Replacement Rules, can be used to reduce the computational effort required to obtain optimal replacement policies. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A descent approach to solving the complementary programming problem

In recent years, much attention has focused on mathematical programming problems with equilibrium constraints. In this article we consider the case where the constraints are complementarity constraints. Problems of this type arise, for instance, in the design of traffic networks. We develop here a descent algorithm for this problem that will converge to a local optimum in a finite number of iterations. The method involves solving a sequence of subproblems that are linear programs. Computational tests comparing our algorithm with the branch-and-bound algorithm in [7] bear out the efficacy of our method. When solving large problems, there is a definite advantage to coupling both methods. A local optimum incumbent provided by our algorithm can significantly reduce the computational effort required by the branch-and-bound algorithm.     

Sensor placement in active multistatic sonar networks

The idea of deploying noncollocated sources and receivers in multistatic sonar networks (MSNs) has emerged as a promising area of opportunity in sonar systems. This article is one of the first to address point coverage problems in MSNs, where a number of points of interest have to be monitored in order to protect them from hostile underwater assets. We consider discrete “definite range” sensors as well as various diffuse sensor models. We make several new contributions. By showing that the convex hull spanned by the targets is guaranteed to contain optimal sensor positions, we are able to limit the solution space. Under a definite range sensor model, we are able to exclude even more suboptimal solutions. We then formulate a nonlinear program and an integer nonlinear program to express the sensor placement problem. To address the nonconvex single‐source placement problem, we develop the Divide Best Sector (DiBS) algorithm, which quickly provides an optimal source position assuming fixed receivers. Starting with a basic implementation of DiBS, we show how incorporating advanced sector splitting methods and termination conditions further improve the algorithm. We also discuss two ways to use DiBS to find multiple source positions by placing sensors iteratively or simultaneously. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 287–304, 2017     

混合遗传算法及其在运载火箭最优上升轨道设计中的应用

运载火箭最优上升轨道设计问题是一类终端时刻未定、终端约束苛刻的最优控制问题,经典算法求解这类问题时收敛性差、局部收敛等问题表现得比较突出。针对上述问题,将具有良好全局收敛性的遗传算法应用到运载火箭最优上升段设计问题求解中,为了提高遗传算法的收敛速度和克服早熟问题,结合遗传算法和单纯型算法的优点,设计了两种混合遗传算法。计算结果表明,所设计的混合遗传算法是求解复杂问题的有效全局优化方法,可以成功地解决一类终端时刻可变飞行器最优控制问题。     

The interactive fixed charge linear programming problem

In many decision-making situations, each activity that can be undertaken may have associated with it both a fixed and a variable cost. Recently, we have encountered serveral practical problems in which the fixed cost of undertaking an activity depends upon which other activities are also undertaken. To our knowledge, no existing optimization model can accomodate such a fixed cost structure. To do so, we have therefore developed a new model called the interactive fixed charge linear programming problem (IFCLP). In this paper we present and motivate problem (IFCLP), study some of its characteristics, and present a finite branch and bound algorithm for solving it. We also discuss the main properties of this algorithm.     

Optimal service and arrival rates in Jackson queueing networks

In this paper we present an algorithm for solving a class of queueing network design problems. Specifically, we focus on determining both service and arrival rates in an open Jackson network of queueing stations. This class of problems has been widely studied and used in a variety of applications, but not well solved due to the difficulty of the resulting optimization problems. As an example, consider the classic application in computer network design which involves determining the minimum cost line capacities and flow assignments while satisfying a queueing performance measure such as an upper limit on transmission delay. Other application areas requiring the selection of both service and arrival rates in a network of queues include the design of communication, manufacturing, and health care systems. These applications yield optimization problems that are difficult to solve because typically they are nonconvex, which means they may have many locally optimal solutions that are not necessarily globally optimal. Therefore, to obtain a globally optimal solution, we develop an efficient branch and bound algorithm that takes advantage of the problem structure. Computational testing on randomly generated problems and actual problems from a health care organization indicate that the algorithm is able to solve realistic sized problems in reasonable computing time on a laptop computer. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 1–17, 2000     

Approximating nondominated sets in continuous multiobjective optimization problems

Many important problems in Operations Research and Statistics require the computation of nondominated (or Pareto or efficient) sets. This task may be currently undertaken efficiently for discrete sets of alternatives or for continuous sets under special and fairly tight structural conditions. Under more general continuous settings, parametric characterisations of the nondominated set, for example through convex combinations of the objective functions or ε‐constrained problems, or discretizations‐based approaches, pose several problems. In this paper, the lack of a general approach to approximate the nondominated set in continuous multiobjective problems is addressed. Our simulation‐based procedure only requires to sample from the set of alternatives and check whether an alternative dominates another. Stopping rules, efficient sampling schemes, and procedures to check for dominance are proposed. A continuous approximation to the nondominated set is obtained by fitting a surface through the points of a discrete approximation, using a local (robust) regression method. Other actions like clustering and projecting points onto the frontier are required in nonconvex feasible regions and nonconnected Pareto sets. In a sense, our method may be seen as an evolutionary algorithm with a variable population size. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

基于近似策略的联合优化方法

针对优化中收敛速度和优化解全局性的问题,提出了一种联合优化方法:构造原问题的近似模型,使用全局优化方法对近似函数进行优化,得到优化点作为初值,再使用局部优化方法对原问题进行优化.为了获得对原问题更好的近似,改进了径向基插值方法,以优化误差的方法来选择参数.利用临近空间机翼模型的优化对算法进行了测试,结果表明,优化参数的...     

Optimal joint preventive maintenance and production policies

We study joint preventive maintenance (PM) and production policies for an unreliable production‐inventory system in which maintenance/repair times are non‐negligible and stochastic. A joint policy decides (a) whether or not to perform PM and (b) if PM is not performed, then how much to produce. We consider a discrete‐time system, formulating the problem as a Markov decision process (MDP) model. The focus of the work is on the structural properties of optimal joint policies, given the system state comprised of the system's age and the inventory level. Although our analysis indicates that the structure of optimal joint policies is very complex in general, we are able to characterize several properties regarding PM and production, including optimal production/maintenance actions under backlogging and high inventory levels, and conditions under which the PM portion of the joint policy has a control‐limit structure. In further special cases, such as when PM set‐up costs are negligible compared to PM times, we are able to establish some additional structural properties. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Minimizing the makespan in open‐shop scheduling problems with a convex resource consumption function

We consider open‐shop scheduling problems where operation‐processing times are a convex decreasing function of a common limited nonrenewable resource. The scheduler's objective is to determine the optimal job sequence on each machine and the optimal resource allocation for each operation in order to minimize the makespan. We prove that this problem is NP‐hard, but for the special case of the two‐machine problem we provide an efficient optimization algorithm. We also provide a fully polynomial approximation scheme for solving the preemptive case. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

A stochastic approximation algorithm to compute bid prices for joint capacity allocation and overbooking over an airline network

In this article, we develop a stochastic approximation algorithm to find good bid price policies for the joint capacity allocation and overbooking problem over an airline network. Our approach is based on visualizing the total expected profit as a function of the bid prices and searching for a good set of bid prices by using the stochastic gradients of the total expected profit function. We show that the total expected profit function that we use is differentiable with respect to the bid prices and derive a simple expression that can be used to compute its stochastic gradients. We show that the iterates of our stochastic approximation algorithm converge to a stationary point of the total expected profit function with probability 1. Our computational experiments indicate that the bid prices computed by our approach perform significantly better than those computed by standard benchmark strategies and the performance of our approach is relatively insensitive to the frequency with which we recompute the bid prices over the planning horizon. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011