A Level-Value Estimation Algorithm and Its Stochastic Implementation for Global Optimization

In this paper, we propose a new method for finding global optimum of continuous optimization problems, namely Level-Value Estimation algorithm(LVEM). First we define the variance function v(c) and the mean deviation function m(c) with respect to a single variable (the level value c), and both of these functions depend on the optimized function f(x). We verify these functions have some good properties for solving the equation v(c) = 0 by Newton method. We prove that the largest root of this equation is equivalent to the global optimum of the corresponding optimization problem. Then we proposed LVEM algorithm based on using Newton method to solve the equation v(c) = 0, and prove convergence of LVEM algorithm. We also propose an implementable algorithm of LVEM algorithm, abbreviate to ILVEM algorithm. In ILVEM algorithm, we use importance sampling to calculate integral in the functions v(c) and m(c). And we use the main ideas of the cross-entropy method to update parameters of probability density function of sample distribution at each iteration. We verify that ILVEM algorithm satisfies the convergent conditions of (one-dimensional) inexact Newton method for solving nonlinear equation, and then we prove convergence of ILEVM algorithm. The numerical results suggest that ILVEM algorithm is applicable and efficient in solving global optimization problem.