Spline Smoothing Methods For Nonlinear Programming

Nonlinear programming and minimax problem are classical and important mathemati-cal programming problems. They have important applications in many fields of science and engineering. For these two problems, people have already obtained plentiful theory, compu-tational and applied results, successful algorithms and software. But the effective methods for large scale problems with complex objectives and constraints are worthy studying for domestic and foreign scholars. In this dissertation, we study the more effective methods and theory about the two problems which are the nonlinear programming with a large number of complex constraints and the minimax problem with a large number of complex objec-tive functions. The main study about the two problems are the spline smoothing methods, including a spline smoothing Newton method and a spline smoothing homotopy method.In Chapter1, we give a brief introduction of the background of the nonlinear program-ming and minimax problem, spline smoothong approximation of the max-function and the homotopy methods.In Chapter2, Some properties of cubic spline and the formulas of computing its gradi-ent and Hessian are given. For the sake of using a cubic spline function and not a quartic spline, a parameterized version of the Sard’s theorem with Cr,1smoothness hypothesis is used. A spline smoothing homotopy method for nonconvex nonlinear programming is de-veloped by using cubic spline to smooth the max function of the constraints of nonlinear programming. The smooth spline approximation of the max-function of the constraints in-volves only few constraints, so it acts also as an active set technique, so it can improve the efficiency for nonlinear programming with large number of complicated constraints. The global convergence of spline smoothing homotopy under the weak normal cone condition is proven.In Chapter3, a constraint shifting spline smoothing homotopy method is proposed for solving nonlinear programming with both equality and inequality constraints. Not only does this homotopy method solve all problems that can be solved by previous homotopy methods but also the existence and global convergence of the homotopy path are proven under weaker conditions. Moreover, proposed method does not require starting from a feasible point or an interior point, so it is more convenient to be implemented. At the same time, the spline smoothing technique is applied to the homotopy method, so it can improve the efficiency for nonlinear programming with large number of complicated inequality constraints.In Chapter4, we propose a spline smoothing stabilized Newton method for large scale finite minimax problems, implement the method using the feedback precision-adjustment smoothing parameter rule which is proposed by Polak and the Newton inner iteration, prove the global convergence. The spline smoothing technique is applied to the method, it acts as an active set technique, so only few the components in the max function are computed at each iteration, hence it is more efficient for the large scale finite minimax problems.In Chapter5, the l∞distance regression with bound constraints is considered. To solve such optimization problems efficiently, we give a mixed smoothing Newton method which uses cubic spline and aggregate function to smooth two parts of max function. It is more efficient for the problem with large amount of measured data.