A Smoothing Newton Method For A Type Of Inverse Quadratic Programming Problems

In practice, we often meet the situations in which it is difficult to give the preciseparameters associated with decision variables in the objective function, even the programmingmodel has been established in a clear formulation, whereas we can find the optimal solutionby experience or experiments, we hope to get the satisfying result by adjusting the parametersas little as possible. This is just a so-called inverse problem. As the inverse problem theorycan be widely applied in many ways, it gradually attracts the attentions from lots of scholarsin recent years. But for continuous optimization, except for linear programming, we do notsee much deep study on their inverse problems. Therefore, our concern is an inverse quadraticprogramming problem, which is a specific inverse problem in continuous optimization. Usingthe smoothing Newton method, this paper solves the KKT system of the dual problem of theinverse quadratic programming problem. It gives the theoretic analysis of the smoothingNewton method, including the global convergence and local convergence rate. Furthermore itsolves the inverse quadratic programming problem in Matlab. The main results in this papercan begeneralized as follow:1. Chapter 2 gives some results from nonsmooth analysis which shall be used in ourconvergence analysis.2. Using the results from the literature, Chapter 3 gives the expressions of the inverse anddual quadratic programming problem, and establishes the KKT system of the dual quadraticprogramming problem. The KKT system is nonsmooth, actually it is a semismooth system.3. Chapter 4 solves the KKT system in Chapter 3 by using the smoothing Newton method.Introducing two smooth functions, it translates the KKT system to a smooth system. Then itconstructs an assistant function and designs an algorithm for solving the inverse quadraticprogramming problem.4. Chapter 5 demonstrates that the arithmetic in Chapter 4 has a good convergence characterin theory, including the global convergence and local quadratic rate.5. Chapter 6 reports some numerical experiments for the arithmetic in Chapter 4. Bycomparing with the semi-smooth Newton method, it indicates that the arithmetic is very effectin the inverse quadratic programming problems even it has big dimensions.