Recursive Modified Pattern Search on High-Dimensional Simplex : A Blackbox Optimization Technique
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Recursive Modified Pattern Search on High-Dimensional Simplex : A Blackbox Optimization Technique Priyam Das Harvard Medical School, Boston, USA
Abstract In this paper, a novel derivative-free pattern search based algorithm for Black-box optimization is proposed over a simplex constrained parameter space. At each iteration, starting from the current solution, new possible set of solutions are found by adding a set of derived step-size vectors to the initial starting point. While deriving these step-size vectors, precautions and adjustments are considered so that the set of new possible solution points still remain within the simplex constrained space. Thus, no extra time is spent in evaluating the (possibly expensive) objective function at infeasible points (points outside the unit-simplex space); which being the primary motivation of designing a customized optimization algorithm specifically when the parameters belong to a unit-simplex. While minimizing any objective function of m parameters, within each iteration, the objective function is evaluated at 2m new possible solution points. So, upto 2m parallel threads can be incorporated which makes the computation even faster while optimizing expensive objective functions over high-dimensional parameter space. Once a local minimum is discovered, in order to find a better solution, a novel ‘re-start’ strategy is considered to increase the likelihood of finding a better solution. Unlike existing pattern search based methods, a sparsity control parameter is introduced which can be used to induce sparsity in the solution in case the solution is expected to be sparse in prior. A comparative study of the performances of the proposed algorithm and other existing algorithms are shown for a few low, moderate and high-dimensional optimization problems. Upto 338 folds improvement in computation time is achieved using the proposed algorithm over Genetic algorithm along with better solution. The proposed algorithm is used to estimate the simultaneous quantiles of North Atlantic Hurricane velocities during 1981–2006 by maximizing a non-closed form likelihood function with (possibly) multiple maximums. AMS (2000) subject classification. Primary 90C26; Secondary 90C56, 90C59. Keywords and phrases. Simplex, Constrained optimization, Pattern search, Convex optimization, Blackbox optimization.
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P. Das
1 Introduction Black-box can be described as a device, system or an object which can be observed only in terms of inputs and outputs. However, the ongoing process within it, is considered unknown. Black-box objective function can be considered similar to any other Blackbox device (Fig.1) where for any given input of the values of the parameters, only the value of the objective function is observed without any further knowledge about the structure, continuity or differentiability of that objective function. Now, consider the minimization problem minimize : subject to :
f (p), where p = (p1 , · · · , pm ) m pi = 1, p ∈ Rm , pi ≥ 0, 1 ≤ i ≤ m,
(1.1)
i=1
where f (p) might have points of
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