A Class Of New Lagrange Multiplier Methods With NCP Function

The constrained nonlinear programming problems are a widely used discipline,which discusses constructs computing approaches to find the optimal solution. Most ofthe problems that abstracted from the society can be classified to the constrainedoptimization problems that finding optimal solution. so the constrained optimizationproblems have been a focus in the field for study of optimization. During these years,many new algorithms have been proposed, such as the penalty function methods、filter methods、trust-region methods、QP-free methods and Lagrange multipliermethods which we mainly discussed in this paper, etc.A class important methods that solve the constrained optimization problems areto turn constrained optimization problems into unconstrained optimization problems.Lagrange multiplier method is one of this methods, it is based on the solution of asequence of unconstrained optimization problems which are a reformulation of theprimal constrained problems. Technique of the method is to structure augmentedLagrange function S (x, λ, ω, C,D)and to turn constrained optimization problem intounconstrained optimization problems. There, C and D are positive parameter, IfC and D sufficient large, there is a good equivalence relation between theunconstrained optimization problems and the primal constrained problems. G. Di Pilloand Grippo proposed a class of augmented Lagrange function methods[19-20,22]whichhave nice equivalence relation between the unconstrained optimization and the primalconstrained problems and get good convergence properties of the related algorithm.However, a max function and min function which may be not differentiable at infinitenumbers of point is used for these methods. To overcome this shortcoming, in thisproper, a class of augmented Lagrange function with NCP function and Lagrangemultiplier methods are proposed for the solution of nonlinear programming problemswith equality constrains and inequality constrains. And his methods are implementand convergent. Chapter1is introduction, in this chapter, nonlinear programming problems andsome rudimentary knowledge are firstly reviewed, and then optimality condition areintroduced, the last of the chapter, some definitions and properties of NCP function areoutlined.In chapter2,A augmented Lagrange function with F-B NCP function is proposed,meanwhile, we discuss its properties,and then convergence of algorithm is proved.In chapter3,3-piecewise linear NCP function is firstly introduced，next, aaugmented Lagrange function which is introduced in chapter2is improved, and then anew augmented Lagrange function with3-piecewise linear NCP function isstructured. Meanwhile, a new Lagrange multiplier method corresponding with newaugmented Lagrange function is proposed. this method is implement table andconvergent.In chapter4,A property of NCP function is used to construct KKT condition,andthen, a new Lagrange multiplier method with4-piecewise linear NCP function isintroduced, according to prove, this method is implementable and convergent.In chapter5the major conclusions of this thesis are given, and the prospects ofaugmented Lagrange function methods are analysed.