A New Global Algorithm For Solving SIP Problems

The thesis presents a new global method for solving semi-infinite programming(SIP) problems,which organically combines adaptive discretization method with branch and bound scheme.Semi-infinite programming is an important branch in the field of nonlinear programming,which comprises finite decision variables with infinite constraints.It is widely applied in many fields such as economic equilibrium,optimal control,information technology and computer network.And there are many practical problems which may be treated by semi-infinite progamming techniques,including air pollution control,engineering design,moment problems and integration formulas,constr- ained Chebyshev approximation,and defect minimization methods for boundary and eigevalue problems.Especially as the development of science and technology,the popularity of computer applications,designing a reliable algorithm for solving semi-infinite programming problems has become a hot topic in research,and has very important practical significance.The solution of nonlinear SIP problems is more complex than that of general nonlinear programming problems.There existed methods include mainly Discretization method,Exchange method,Local descent method,SQP method and so on,among which SQP Method is mostly welcomed by researchers.In the SQP method,the nonlinear semi-infinite programming problem is first transformed into nonlinear programming problem by some discretization strategy and then to solve the latter by employing the SQP method for nonlinear programming.However the implement of this method is more difficult than solving the general nonlinear programming problems.Moreover,there exist many drawbacks in the existed SQP methods for nonlinear semi-infinite programming.In this thesis,we further study to solve nonlinear semi-infinite programming problems by using the welcomed discretization strategy.Firstly,we discrete the SIP problem,transform into the general nonlinear programming problem,then solve the discreted problem by using the methods of nonlinear programming.At last,the optimal value comes out.The discrete set should be enough large for making the discreted problem have a good approximation of the original problem when solving the discrete problem with traditional methods,which will lead to the numerical stability of solving the original problem.Here,we intoduce an adaptive discretization method,which greatly reduce the number of constraints and improve the computational efficiency during the discrete process.Due to needing solve a series of quadratic programming problems with quadratic constraints during the process of solving the problem,we must adopt a better,faster method to solve this kind of optimal problems.This thesis selects the accelerated branch and bound method,then we get a new global method organically combining adaptive discretization method with branch and bound scheme together.The full text consists of four chapters.In chapter 1,we introduce some basic theories and basic methods about semi-infinite programming problems.In the later three chapters,we propose a new global method for solving semi-infinite programming problems and a numerical experiment are given.