A Lagrangian Algorithm For Solving Nonlinear Constrained Optimization Problem

The constrained nonlinear programming problems (NLP) is a very important compo-nent part in operations research, It has wide application in natural science,engineering andeconomics. The typical method for solving NLP have feasible direction method, penaltyfunction method, multiplier method, sequential quadratic programming and so on. In recentyears, the method that a local minimum of NLP is obtained by solving KKT-systems for NLPbecomes one of the efcient numerical methods for NLP. The main idea of the method is toreformulate the KKT-conditions for NLP as smooth nonlinear equations, then a KKT-pointfor NLP is obtained by using classical numerical methods for the problem. The methods inthis thesis a KKT-point is obtained, but the diference train of thought to solve the problem.The specific contents are as follows:1. The first we construct a new auxiliary function containing parameter, using the goodpropertys of this function.The constraint optimization problems with inequality constraintsinto equivalent conversion. This conversion not only ensure the KKT-condition of the newproblems and the KKT-condition of the original problems with the solution, but also theintroduction of parameters make the Lagrange function of the new problems with a penaltyfunction. Using this characteristic, we propose a new Lagrange function method and theglobal convergence of the method can also be proved. The numerical results show that thealgorithm has good adaptability and stability.2. The first equivalent to the equality constraints are converted into two inequalityconstraints, then using the method in the last chapter, turning them into new inequalityconstraints, According to the characteristics of the function, the two inequality constraintsfurther equivalently converted into an inequality constraint, so the contains equality con-strained problem into the Optimization problems which only have inequality constraints.Finally, the method of Lagrange function for general constrained optimization problems isgiven.