Abstract
A random search algorithm for unconstrained local nonsmooth optimization is described. The algorithm forms a partition on \(\mathbb{R}^{n}\) using classification and regression trees (CART) from statistical pattern recognition. The CART partition defines desirable subsets where the objective function f is relatively low, based on previous sampling, from which further samples are drawn directly. Alternating between partition and sampling phases provides an effective method for nonsmooth optimization. The sequence of iterates {z k } is shown to converge to an essential local minimizer of f with probability one under mild conditions. Numerical results are presented to show that the method is effective and competitive in practice.
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We acknowledge the helpful comments of two anonymous referees which led to an improved version of the paper.
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Appendix
Appendix
A nonsmooth version of the Cosine Mixture problem [3] is
where x 0=(0,0,…,0). The cases n=4 and n=6 were studied, where f ∗=−4.4 and f ∗=−6.6 respectively.
A nonsmooth version of the Exponential problem [3] is
where x 0=(1,1,…,1) and f ∗=−1. The cases n=6 and n=8 were studied.
The discontinuous versions of the Rosenbrock function are defined as follows.
Each problem uses x 0=(−1.2,1) and has an essential local minimizer at x ∗=(1,1) with f ∗=0.
The discontinuous versions of the Beale function are defined as follows. Let
Each problem uses x 0=(1,1) and has an essential local minimizer at x ∗=(3,0.5) with f ∗=0.
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Robertson, B.L., Price, C.J. & Reale, M. CARTopt: a random search method for nonsmooth unconstrained optimization. Comput Optim Appl 56, 291–315 (2013). https://doi.org/10.1007/s10589-013-9560-9
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DOI: https://doi.org/10.1007/s10589-013-9560-9