Research On Auxiliary Function Methods For Several Classes Of Global Optimization Problems

In social life and production practice, many practical problems can be at-tributed to the solution of global optimization problems, such as economic man-agement, engineering design, transportation, molecular biology, defense and mil-itary and so on. Since most of the practical problems have multiple local optimalsolutions, it is relatively difcult to solve global optimization problems. More-over, presently there is no unified method for solving these kinds of problemsefectively, which makes it important and necessary to put an efort on findingefective methods to solve the general global optimization problems. There aretwo major issues need to be addressed for global optimization problems: howto jump from a local minimizer to a better one and how to judge whether thecurrent minimizer is a global one.For diferent classes of global optimization problems, including unconstrainedoptimization, optimization with inequality constraints, integer programming and0-1programming, auxiliary functions with well properties are constructed respec-tively, based on which auxiliary function algorithms are proposed respectively.For unconstrained optimization problems, an auxiliary function with onlyone parameter is constructed; whose analytic properties are the same as those ofthe objective function. The proposed function can guarantee that a point whosefunction value is greater than that of the current minimize is not a minimizerof the auxiliary function, and that there must exist a minimizer of the auxiliaryfunction which falls into a basin lower than the current one. Moreover, an opti-mization algorithm based on the constructed auxiliary function is proposed andused to solve unconstrained optimization problems.For optimization problems with inequality constraints, a new auxiliary func-tion with only one parameter is presented. When the objective function and constraint functions are twice continuously diferentiable, the auxiliary functionis also twice continuously diferentiable. A point which is infeasible or whosefunction value is greater than that of the current minimizer is not a minimizer ofthe auxiliary function. There must exists a minimizer of the auxiliary function ina feasible region in which the objective function values of all points are less thanthat of the current minimize. Furthermore, an optimization algorithm based onthe auxiliary function is proposed and used to solve this kind of problems.For general integer programming problems, although the proposed auxiliaryfunction contains one parameter, when the parameter is taken as large as possible,the proposed function can guarantee that a point which is infeasible or whosefunction value is greater than that of the current minimize is not a minimizerof the auxiliary function, that a minimize better than the current minimizer ofthe original problem is also a minimizer of the auxiliary function, and that theminimizer of the auxiliary function is either a minimizer of the original problem ora vertex of the searching region. Furthermore, an optimization algorithm basedon the constructed auxiliary function is used to solve general integer programmingproblems.For0-1programming problems, a new auxiliary function without parametersis proposed. The minimizer of the auxiliary function is either better than thecurrent minimizer or be a point furthest away from the current minimizer. Sominimizing the auxiliary function can be able to jump from the current localminimizer. Moreover, a proposed algorithm based on the auxiliary function isused to solve the0-1programming problems.