Auxiliary Function Method For Constrained Nonlinear Optimization

With the development of the times and the progress of science and technology,Global optimization theory is widely used in various fields,such as natural sciences,economic management,transportation and so on.In simple terms,global optimization is a method for finding the global minimizer.It mainly includes two kinds of algorithms: deterministic algorithm and stochastic algorithm.This paper mainly studies the auxiliary function method of the global optimization deterministic algorithm.Its main idea is minimizing the objective function by any local optimization algorithm to find a local minimizer,then construct an auxiliary function containing the objective function and the local minimizer to jump out of the domain of the current local minimizer and find a better local minimizer.After a finite number of iterations,the global(or approximate)minimizer of the global optimization problem is finally found.This paper first introduces the background and related concepts of global optimization problems.And some preliminary knowledge for solving global optimization problems is reviewed.Several classical optimization algorithms to solve local optimization problems and some deterministic algorithms to solve global optimization problems are listed.The advantages and disadvantages of these algorithms are briefly analyzed.For unconstrained nonlinear optimization problems,a new continuously differentiable filled function with one parameter is constructed.Because this filled function only contains one parameter,it overcomes the shortcoming of two parameters which are highly dependent and not easy to adjust,so it can effectively improve the efficiency of the algorithm.Then we give the corresponding algorithm steps,and the results of numerical experiments of four numerical example are given.For nonlinear optimization problem of box constrained,a new filled-cross function is proposed.It is proved that this function has the properties of the filled function and the cross function,and has the same local minimizers as the objective function of optimization problem.The speed of the algorithm is greatly speeded up.The corresponding algorithm steps are given and numerical experiments are carried out for three examples.For constrained nonsmooth global optimization,a new filled function is proposed.By discussing its theoretical properties,the algorithm steps that are suitable for the filled function are given.Numerical experiments are given to prove the feasibility and effectiveness of the algorithm.Finally,the general conclusion of this paper is given,and the future research is prospected.