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Simulation-Based Optimization for Convex Functions Over Discrete Sets

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  • Eunji Lim

Abstract

We propose a new iterative algorithm for finding a minimum point of f_*-X \subset \mathbb{R}^d \rightarrow \mathbb{R}, when f_* is known to be convex, but only noisy observations of f_*(\textbf{x}) are available at \textbf{x} \in X for a finite set X. At each iteration of the proposed algorithm, we estimate the probability of each point \textbf{x} \in X being a minimum point of f_* using the fact that f_* is convex, and sample r points from X according to these probabilities. We then make observations at the sampled points and use these observations to update the probability of each point \textbf{x} \in X being a minimum point of f_*. Therefore, the proposed algorithm not only estimates the minimum point of f_* but also provides the probability of each point in X being a minimum point of f_*. Numerical results indicate the proposed algorithm converges to a minimum point of f_* as the number of iterations increases and shows fast convergence, especially in the early stage of the iterations.

Suggested Citation

  • Eunji Lim, 2021. "Simulation-Based Optimization for Convex Functions Over Discrete Sets," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(5), pages 1-31, September.
  • Handle: RePEc:ibn:ijspjl:v:10:y:2021:i:5:p:31
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    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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