An Efficient Direct Search Method for Simulation Optimization With Conditional-Expectation- Based Objectives

Abstract

In order to generalize the applicability of Conditional Value at Risk, one of the most widely used measurements used in financial risk management, we develop a solution methodology for the conditional expectation (CE)-based simulation optimization problems. To optimize CE-based objective functions in a highly generalized context, we propose a gradient-free, direct search optimization method, called SNM-CE, which inherits the search framework of Stochastic Nelder-Mead (SNM) Simplex Method but further incorporates effective mechanisms designed for handling problems with CE-based objective functions. As we assume the underlying problem is complicated enough that no closed-form expression can represent the objective function, stochastic simulation is applied to estimate CE. We apply Importance Sampling (IS) as a variance reduction technique, which, combined with a newly-developed methodology, called SOCBA-mn, ensures that simulation resources are used with great efficiency. We show that SNM-CE can converge to the true global optimum with probability one (w.p.1) like SNM. An extensive numerical study and a communication system-based empirical study are both conducted to demonstrate the effectiveness, efficiency and viability of this research in both theoretical and practical settings. Note to Practitioners—This paper develops a direct search algorithm, called SNM-CE, used for solving conditional expectation-based simulation optimization problems. By tackling conditional expectation-based problems, SNM-CE fills a gap in the stochastic optimization literature which has traditionally focused on expectation- or quantile-based objective functions. SNM-CE possesses a high degree of flexibility and generalizability which users can benefit from in solving real-world applications. For example, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> value in the CE-based simulation optimization formulation can be freely adjusted. In other words, the quantile value above which the CE is estimated and optimized can be set according to the needs of the practitioner and/or characteristics of the problem to be solved. Also, SNM-CE is designed such that instead of setting the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> value, the user may choose to utilize a particular numerical value of the objective function as the threshold for the CE estimation and optimization. SNM-CE is also simple in terms of implementation and, being a direct search method, does not impose many assumptions about the structure of the underlying objective function and does not necessitate the use of gradient information in the search process. As one application example, SNM-CE can be utilized to select the operational parameters which minimize the average delay in the manufacturing of semiconductors given that the delay exceeds the 90th percentile of simulated delay times.