LAUSR

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.