A New Homotopy Method For Solving Semi-infinite Programming And Nonlinear Programming Problems

Semi-infinite programming problem and nonlinear programming problem are classical and extremely important optimization problems in mathematical programming.They are widely used in many fields such as engineering design,information technology,financial management,etc.Therefore,how to solve semi-infinite programming problem and nonlinear programming problem is a very important research topic.The commonly used algorithms include the gradient descent algorithm,the exchange method,the Newton method and the quasi-newton method,the obstacle function method,the penalty function method and so on.However,the global convergence of these algorithms is either difficult to achieve or can only be achieved under strong conditions.Therefore,a new homotopy method is used to solve these two kinds of programming problems.Based on the existing combined homotopy method,a new homotopy method is proposed to solve semi-infinite programming problems and nonlinear programming problems.The research is conducted in the following three aspects:Firstly,on the basis of the existing combined homotopy method,a new non-interior homotopy method for solving the semi-infinite programming problem is constructed by using the constraint shifting combined homotopy method.This method no longer constrains the selection of the initial point in the feasible region.Thus,the range of the initial point is expanded,and the existence and convergence of the homotopy paths under the appropriate assumptions are proved.Secondly,based on the content of research,a generalized quadratic continuous differentiable mapping is introduced for the improved constraint shifting combined homotopy method,and its results for solving semi-infinite programming problems are extended to more general nonconvex sets.Thirdly,for the general nonlinear programming problem,based on the existing combined homotopy method,a generalized quadratic continuous differentiable mapping is introduced to solve the problem.First of all,the proper generalized weak normal cone condition is given and solved when the condition is established.This method extends the selection range of initial points and overcomes the constraints of nonconvex feasible regions.