Johannes O. Royset

Reliability-based optimal structural design by the decoupling approach

Abstract A decoupling approach for solving optimal structural design problems involving reliability terms in the objective function, the constraint set or both is discussed and extended. The approach employs a reformulation of each problem, in which reliability terms are replaced by deterministic functions. The reformulated problems can be solved by existing semi-infinite optimization algorithms and computational reliability methods. It is shown that the reformulated problems produce solutions that are identical to those of the original problems when the limit-state functions defining the reliability problem are affine. For nonaffine limit-state functions, approximate solutions are obtained by solving series of reformulated problems. An important advantage of the approach is that the required reliability and optimization calculations are completely decoupled, thus allowing flexibility in the choice of the optimization algorithm and the reliability computation method.

Solving the Bi-Objective Maximum-Flow Network-Interdiction Problem

We describe a new algorithm for computing the efficient frontier of the “bi-objective maximum-flow network-interdiction problem.” In this problem, an “interdictor” seeks to interdict (destroy) a set of arcs in a capacitated network that are Pareto-optimal with respect to two objectives, minimizing total interdiction cost and minimizing maximum flow. The algorithm identifies these solutions through a sequence of single-objective problems solved using Lagrangian relaxation and a specialized branch-and-bound algorithm. The Lagrangian problems are simply max-flow min-cut problems, while the branch-and-bound procedure partially enumerates s-t cuts. Computational tests reveal the new algorithm to be one to two orders of magnitude faster than an algorithm that replaces the specialized branch-and-bound algorithm with a standard integer-programming solver.

On buffered failure probability in design and optimization of structures

In 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.

On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems

We present an approach for the solution of a class of generalized semi-infinite optimization problems. Our approach uses augmented Lagrangians to transform generalized semi-infinite min-max problems into ordinary semi-infinite min-max problems, with the same set of local and global solutions as well as the same stationary points. Once the transformation is effected, the generalized semi-infinite min-max problems can be solved using any available semi-infinite optimization algorithm. We illustrate our approach with two numerical examples, one of which deals with structural design subject to reliability constraints.

On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing

This paper focuses on finite minimax problems with many functions, and their solution by means of exponential smoothing. We conduct run-time complexity and rate of convergence analysis of smoothing algorithms and compare them with those of SQP algorithms. We find that smoothing algorithms may have only sublinear rate of convergence, but as shown by our complexity results, their slow rate of convergence may be compensated by small computational work per iteration. We present two smoothing algorithms with active-set strategies and novel precision-parameter adjustment schemes. Numerical results indicate that the algorithms are competitive with other algorithms from the literature, and especially so when a large number of functions are nearly active at stationary points.

Probabilistic search optimization and mission assignment for heterogeneous autonomous agents

This paper presents an algorithmic framework for conducting search and identification missions using multiple heterogeneous agents. Dynamic objects of type “neutral” or “target” move through a discretized environment. Probabilistic representation of the current level of situational awareness - knowledge or belief of object locations and identities - is updated with imperfect observations. Optimization of search is formulated as a mixed-integer program to maximize the expected number of targets found and solved efficiently in a receding horizon approach. The search effort is conducted in tandem with object identification and target interception tasks, and a method for assignment of these missions among agents is developed. The proposed framework is demonstrated in simulation studies, and an implementation of its decision support capabilities in a recent field experiment is reported.

Adaptive Approximations and Exact Penalization for the Solution of Generalized Semi-infinite Min-Max Problems

We develop an implementable algorithm for the solution of a class of generalized semi-infinite min-max problems. To this end, first we use exact penalties to convert a generalized semi-infinite min-max problem into a finite family of semi-infinite min-max-min problems. Second, the inner min-function is smoothed and the semi-infinite max part is approximated, using discretization, to obtain a three-parameter family of finite min-max problems. Under a calmness assumption, we show that when the penalty is sufficiently large the semi-infinite min-max-min problems have the same solutions as the original problem, and that when the smoothing and discretization parameters go to infinity the solutions of the finite min-max problems converge to solutions of the original problem, provided the penalty parameter is sufficiently large. Our algorithm combines tests for adjusting the penalty, the smoothing, and the discretization parameters and makes use of a min-max algorithm as a subroutine. In effect, the min-max algorithm is applied to a sequence of gradually better-approximating min-max problems, with the penalty parameter eventually stopping to increase, but with the smoothing and discretization parameters driven to infinity. A numerical example demonstrates the viability of the algorithm.

Superquantile regression with applications to buffered reliability, uncertainty quantification, and conditional value-at-risk

The paper presents a generalized regression technique centered on a superquantile (also called conditional value-at-risk) that is consistent with that coherent measure of risk and yields more conservatively fitted curves than classical least-squares and quantile regression. In contrast to other generalized regression techniques that approximate conditional superquantiles by various combinations of conditional quantiles, we directly and in perfect analog to classical regression obtain superquantile regression functions as optimal solutions of certain error minimization problems. We show the existence and possible uniqueness of regression functions, discuss the stability of regression functions under perturbations and approximation of the underlying data, and propose an extension of the coefficient of determination R-squared for assessing the goodness of fit. The paper presents two numerical methods for solving the error minimization problems and illustrates the methodology in several numerical examples in the areas of uncertainty quantification, reliability engineering, and financial risk management.

ENGINEERING DECISIONS UNDER RISK-AVERSENESS ∗

AbstractEngineering decisions are invariably made under substantial uncertainty about current and future system cost and response, including cost and response associated with low-probability, high-consequence events. A risk-neutral decision maker would rely on expected values when comparing designs, while a risk-averse decision maker might adopt nonlinear utility functions or failure probability criteria. The paper shows that these models for making decisions are related to a framework of risk measures that includes many possibilities. The authors describe how risk measures provide an expanded set of models for handling risk-averse decision makers. General recommendations for selecting risk measures lead to decision models for risk-averse decision making that comprehensively represent risks in engineering systems, avoid paradoxes, and accrue substantial benefits in subsequent risk, reliability, and cost optimization. The paper provides an overview of the framework of decision making based on risk measures.

Optimal Budget Allocation for Sample Average Approximation

The sample average approximation approach to solving stochastic programs induces a sampling error, caused by replacing an expectation by a sample average, as well as an optimization error due to approximating the solution of the resulting sample average problem. We obtain estimators of an optimal solution and the optimal value of the original stochastic program after executing a finite number of iterations of an optimization algorithm applied to the sample average problem. We examine the convergence rate of the estimators as the computing budget tends to infinity, and we characterize the allocation policies that maximize the convergence rate in the case of sublinear, linear, and superlinear convergence regimes for the optimization algorithm.