Gradient-based simulation optimization under probability constraints

• Published in: European journal of operational research Vol. 212; no. 2; pp. 345 - 351
• Main Authors: Andrieu, Laetitia, Cohen, Guy, Vázquez-Abad, Felisa J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16.07.2011; Elsevier; Elsevier Sequoia S.A
• Series: European Journal of Operational Research
• Subjects: Applied sciences; Computer simulation; Confidence intervals; Convergence; Decision theory. Utility theory; Estimation bias; Estimators; Exact sciences and technology; Failure; Mathematical analysis; Mathematics; Operational research and scientific management; Operational research. Management science; Optimization; Optimization algorithms; Optimization and Control; Parameter estimation; Portfolio theory; Probability constraints; Probability constraints Stochastic gradient algorithm Stochastic approximation; Probability distribution; Simulation; Solvers; Stochastic approximation; Stochastic gradient algorithm; Studies; Stochastic approximation; Probability constraints; Stochastic gradient algorithm; Gradient; Probabilistic approach; Bias; Finance; Rupture; Statistical simulation; Failures; Search algorithm; Confidence interval; Constrained optimization; Convolution; Convergence rate; Observation data; Reliability; Gradient method; Numerical convergence; Finite difference method
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2011.01.049

We study optimization problems subject to possible fatal failures. The probability of failure should not exceed a given confidence level. The distribution of the failure event is assumed unknown, but it can be generated via simulation or observation of historical data. Gradient-based simulation–optimization methods pose the difficulty of the estimation of the gradient of the probability constraint under no knowledge of the distribution. In this work we provide two single-path estimators with bias: a convolution method and a finite difference, and we provide a full analysis of convergence of the Arrow–Hurwicz algorithm, which we use as our solver for optimization. Convergence results are used to tune the parameters of the numerical algorithms in order to achieve best convergence rates, and numerical results are included via an example of application in finance.