Abstract
For many real world optimization problems, the objective function is stochastic. When optimizing a stochastic function f, one has to deal with the problem of varying outputs f(x, C) for the same input x due to the effects of a random variable C. One possibility for optimizing f is considering the expectation and standard deviation of f(x, C) and choosing x such that the expected value of f(x, C) is optimal, e.g. minimal and the standard deviation of f(x, C) is minimal. This turns the optimization of f into a biobjective optimization problem. We investigate the optimization of expensive stochastic black box functions f(x, C) with \(x \in \mathbb {R}\) and C being a one dimensional random variable. Because f is an expensive function, we want to evaluate it seldom. Therefore, we use a surrogate model \(\hat{f}\) of f and numerical integration to estimate the expectation \(\mathrm {E}(f(x, C))\) and the standard deviation \(\mathrm {S}(f(x, C))\). We perform a simulation study to analyze how well our approach works and compare it to a classic method. Our approach enables us to estimate \(\mathrm {E}(f(x, C))\) and \(\mathrm {S}(f(x, C))\) for each feasible x-value with a comparably high quality and yields a good approximation of the true Pareto set at the cost of requiring that C is observable.
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Bommert, M., Rudolph, G. (2019). Reliable Biobjective Solution of Stochastic Problems Using Metamodels. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_46
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DOI: https://doi.org/10.1007/978-3-030-12598-1_46
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