(A) new global minimization method using quadratic underestimator

Abstract

In this study, a new global optimization method is proposed for an optimization problem with twice-differentiable objectives of a single variable. The method employs a difference of convex underestimator, that is a continuous piecewise concave quadratic function. The key idea of this research is to make the quadratic concave underestimator which does not need an iterative local optimizer to determine the lower bounding value of the objective function. The proposed method is proven to have a finite $\\\\epsilon$-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering methods. For multivariate NLPs, a new branch-and-bound algorithm was proposed, that utilized DC underestimator as a lower bounding function for a lower bounding rule. The proposed algorithm is successfully applied to unconstrained NLP minimization problems. Compared with $\\\\alpha$BB algorithm numerically, the propose algorithm requires less number of function evaluations and computational load, CPU time, since it dose not need iterative optimizer for obtaining lower bounding value of each subregion. Modified IDP which utilizes DCU as an optimization technique, is proposed for optimal control problems. The optimal control trajectory obtained by modified IDP can guarantee finite $\\\\epsilon$-convergence. For all optimal control problems, numerical solutions are obtained the similar policy compared with the solutions of other researchers. For engineering problems, fed-batch bioreactor problem and bifunctional catalyst blend optimal control problem, the optimal control trajectories and the objective function values show robust behaviors regarding random initial guesses.