Path Optimization for the Resource-Constrained Searcher

Naval Research LogisticsWe formulate and solve a discrete-time path-optimization problem where a single searcher, operating in a discretized 3-dimensional airspace, looks for a moving target in a finite set of cells. The searcher is constrained by maximum limits on the consumption of several resources such as time, fuel, and risk along any path. We develop a special- ized branch-and-bound algorithm for this problem that utilizes several network reduction procedures as well as a new bounding technique based on Lagrangian relaxation and net- work expansion. The resulting algorithm outperforms a state-of-the-art algorithm for solving time-constrained problems and also is the first algorithm to solve multi-constrained problems

On Solving Large-Scale Finite Minimax Problems using Exponential Smoothing

Journal of Optimization Theory and Applications, Vol. 148, No. 2, pp. 390-421

On Sample Size Control in Sample Average Approximations for Solving Smooth Stochastic Programs

Computational Optimization and Applications, Volume 55, Issue 2, Page 265-309

On Optimality Functions in Stochastic Programming and Applications

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate point and a local minimizer. We construct confidence intervals for the value of the optimality function at a candidate point and, hence, provide a quantitative measure of solution quality. Based on sample average approximations, we develop two algorithms for classes of stochastic programs that include CVaR-problems and utilize optimality functions to select sample sizes as well as “active” sample points in an active-set strategy. Numerical tests illustrate the procedures

Optimality Functions in Stochastic Programming

Mathematical Programming, Vol. 135, No. 1-2, pp. 293-321

Extensions of Stochastic Optimization Results from Problems with Simple to Problems with Complex Failure Probability Functions

We derive an implementable algorithm for solving nonlinear stochastic optimization problems with failure probability constraints using sample average approximations. The paper extends prior results dealing with a failure probability expressed by a single measure to the case of failure probability expressed in terms of multiple performance measures. We also present a new formula for the failure probability gradient. A numerical example addressing the optimal design of a reinforced concrete highway bridge illustrates the algorithm.This work was sponsored by the Research Associateship Program, National Research Council

Efficient Sample Sizes in Stochastic Nonlinear Programming

Journal of Computational and Applied Mathematics, Vol. 217, pp. 301-310.We consider a class of stochastic nonlinear programs for which an approximation to a locally optimal solution is speci_ed in terms of a fractional reduction of the initial cost error. We show that such an approximate solution can be found by approximately solving a sequence of sample average approximations. The key issue in this approach is the determination of the required sequence of sample average approximations as well as the number of iterations to be carried out on each sample average approximation in this sequence. We show that one can express this requirement as an idealized optimization problem whose cost function is the computing work required to obtain the required error reduction. The speci_cation of this idealized optimization problem requires the exact knowledge of a few problems and algorithm parameters. Since the exact values of these parameters are not known, we use estimates, which can be updated as the computation progresses. We illustrate our approach using two numerical examples from structural engineering design

Optimized Routing of Unmanned Aerial Systems for the Interdiction of Improvised Explosive Devices

Military Operations Research, Vol. 14, No. 4, pp. 5-19.The paper describes an optimization-based tool for selecting routes that will best employ unmanned aerial systems (UASs) for the purpose of detecting improvised explosive devices (IEDs) or related activity. The routing tool uses preprocessing procedures, an integer linear program, and an IED prediction model to direct UASs to sectors of the area of operations with high IED activity, while accounting for factors such as winds, aircraft de-confliction, and blue force activity. Initial evaluation of the routing tool through field experiments with actual UASs suggests that the tool produces realistic routes, which can be flown in the allocated amount of time, even under windy conditions

Implementable Algorithm for Stochastic Optimization Using Sample Average Approximations

We develop an implementable algorithm for stochastic optimization problems involving probability functions. Such problems arise in the design of structural and mechanical systems. The algorithm consists of a nonlinear optimization algorithm applied to sample average approximations and a precision-adjustment rule. The sample average approximations are constructed using Monte Carlo simulations or importance sampling techniques. We prove that the algorithm converges to a solution with probability one and illustrate its use by an example involving a reliability-based optimal design.Research Associateship Program at the National Research CouncilTaisei Chair in Civil Engineering at UC BerkeleyNational Science Foundation under Grant ECS-9900985Research Associateship Program at the National Research CouncilTaisei Chair in Civil Engineering at UC BerkeleyNational Science Foundation under Grant ECS-990098

On Buffered Failure Probability in Design and Optimization of Structures

Reliability Engineering & System SafetyIn reliability engineering focused on the design and optimization of structures, the typical measure of reliability is the probability of failure of the structure or its individual components relative to specific limit states. However, the failure probability has troublesome properties that raise several theoretical, practical, and computational issues. This paper explains the seriousness of these issues in the context of design optimization and goes on to propose a new alternative measure, the buffered failure probability, which offers significant advantages. The buffered failure probability is handled with relative ease in design optimization problems, accounts for the degree of violation of a performance threshold, and is more conservative than the failure probability