Abstract
We analyze the convergence and performance of a novel direct search algorithm for identifying at least a local minimum of unconstrained mixed integer nonlinear optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a randomized pattern search, which is modified by two main operations for finding at least a local minimum of our problem, namely: moving operation and shrinking operation. The convergence properties of the MIRPSA are here analyzed from a Markov chain viewpoint, which is represented by an infinite countable set of states {d(q)}∞q=0, where each state d(q) is defined by a measure of the qth randomized pattern search Hq, for all q ∈ N. According to the algorithm, when a moving operation is carried out on a qth randomized pattern search Hq, the MIRPSA Markov chain holds its state. Meanwhile, if the MIRPSA carries out a shrinking operation on a qth randomized pattern search Hq, the algorithm will then visit the next (q + 1)th state. Since the MIRPSA Markov chain never goes back to any visited state, we therefore say that the MIRPSA yields a birth and miscarriage Markov chain.
Highlights
Consider the following unconstrained mixed integer nonlinear problem: minimize f (x, y),(x,y)∈Rn×Zm (1.1)where f (x, y) : Rn × Zm → R is a nonlinear objective function, for which an analytical and explicit mathematical expression cannot be obtained
The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA) starts with an initial guess 0th mixed integer randomized pattern search H0, which is used for locating an initial set of randomized trial points, in order to evaluate the objective function at each trial point for comparing with the center of the randomized pattern search function value f
We have examined some convergence properties of the MIRPSA based on the approach of Markov chains
Summary
Consider the following unconstrained mixed integer nonlinear problem: minimize f (x, y),. The MIRPSA starts with an initial guess 0th mixed integer randomized pattern search H0, which is used for locating an initial set of randomized trial points, in order to evaluate the objective function at each trial point for comparing with the center of the randomized pattern search function value f (ck). The MIRPSA is basically based on two main operations: moving operation and shrinking operation, on any kth mixed integer randomized pattern search Hk, which is used for locating a set of randomized trial points at each kth iteration around the center of the randomized pattern search ck. The pattern search center will remain unchanged, but pattern search range will shrink for both real and integer factors, if the objective function value at the best trial point does not improve the objective function at the center of the current pattern search This procedure is iteratively repeated until some stopping rule is satisfied.
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