Filled Function And Modified Tunnelling Function Methods For Global Optimization

Frequently,practitioners need to solve global optimization problems in manyfelds such as engineering design,fnancial management,bioengineering, and socialscience.So global optimization becomes a crucial computational task for researcherswhich discusses the characters of global optimal choice of multivariate nonlinearfunctions on a constrained region and constructs computing approaches to fndthe global optimal solution,as well as discusses the theoretical properties and cal-culation properties of the solutions.However,due to the existence of multiple localminimizers that difer from the global solution,we have to face two difculties:howto jump from a local minimizer to a smaller one and how to judge that the currentminimizer is a global one.Hence all these problems cannot be solved by classical non-linear programming techniques directly. Generally speaking,the global optimiza-tion methods can be divided into two types: stochastic methods and deterministicmethods.The flled function method,frst proposed for smooth optimization by Geand Qin(1987),is one of the efective deterministic global optimization methods forsettling the frst difculty.It modifes the objective function as a flled function,and then fnds a better local minimizer gradually by optimizing the flled functionconstructed on the minimizer previously found. The flled function method pro-vides us with a good idea to use the local optimization techniques to solve globaloptimization problems. The tunnelling function method, frst proposed by Levyand Montalco(1985), is also a efective deterministic global optimization methodsfor settling the frst difculty.The existing flled functions and tunnelling functions have some drawbacks suchas requiring that the objective function has only a fnite number of local minimizers,of the parameters of flled function heavily restricted by the minimal basin radiusof local minimizers,of requiring that the flled function has a minimizer on the line.All of these characteristics are strongly undesirable in numerical applications asthey are liable to the illness of computation. Therefore, further research is worthyof continuing on how we can construct flled functions and tunnelling functionswith simple forms, better properties and more efcient algorithms.Based on the current status of research scholars, for a number of outstandingissues, the aim of this paper is to develop the flled function with certain satisfactoryproperties and the modifed tunnelling function. This paper mainly consists of fourchapters.In chapter1, several basis concepts and methods on generally nonlinear pro-gramming are introduced. And some mainly methods for global optimizationproblems are briefy presented, including the flled function methods, the tunnellingfunction methods, the branch and bound methods and the integral methods.In chapter2, for general unconstrained global optimization problems,two meth-ods are presented, including the flled function method and the modifed tunnelling function method.In Section2.2, a new defnition of the flled function is given, it isdiferent from the primary defnition which was given by Ge in paper [21]. Basedon the defnition, a new flled function is proposed, and it has better properties. Analgorithm for unconstrained global optimization is developed from the new flledfunction. The implementation of the algorithms on several test problem is reportedwith satisfactory numerical results. In Section2.3, we give a defnition of the mod-ifed tunnelling function. Based on this defnition, an algorithm for unconstrainedglobal optimization problem is proposed, the algorithm overcomes these disadvan-tages of the tunnelling function algorithm. The implementation of the originalalgorithm on several test problem is reported with satisfactory numerical results.In chapter3, the new flled function method and the modifed tunnelling func-tion method are extended to inequalities constrained global optimization. Theimplementation of algorithms on several test problems is reported with satisfacto-ry numerical results.In chapter4, we extend the idea for unconstrained global optimization to equal-ities constrained global optimization.In section4.1, we proposes the flled functionand the corresponding flled function algorithm; and in section4.2, we proposes themodifed tunnelling function and the corresponding modifed tunnelling functionalgorithm.