Numerical Comparisons Of Methods For A Type Of Inverse Quadratic Programming Problems

In an optimization model there are parameters associated with decision variables in theobjective function or in the constraint set. When solving the optimization problem, we usuallyassume that these parameter values are known and we need to find an optimal solution to it.However, there are many instances in the practice, in which we only know some estimates forparameter values, but we may have certain optimal solutions from experience, observations orexperiments. An inverse optimization problem is to find values of parameters which make theknown solutions optimal and which differ from the given estimates as little as possible. Thereare many important contributions to inverse optimization and a large number of inversecombinatorial optimization problems have been studied, but for continuous optimization, wedo not see much study on their inverse problem. In this paper, our concern is the numericalcomparison of different methods for solving a type of inverse quadratic programmingproblems introduced in[1].Chapter 1 introduces the inverse quadratic programming problem, which is aminimization problem with a positive semidefinite cone constraint.Chapter 2 derives the dual of the inverse quadratic programming problem, which is alinearly constrained semismoothly differentiable convex programming problem with fewervariables than the original one. Importantly, we adopt the augmented Lagrangian method,given in [1], for the dual problem with subproblems being solved by the Quasi-Newtonmethod and the Newton method, respectively. We focus on the comparison of numericalresults implemented by these two approaches. The numerical results show that the methodusing the Quasi-Newton method is much more effective than the one using the Newtonmethod.In chapter 3, we use the barrier function method for solving the inverse quadraticprogramming problems and the subproblems are solved by the Quasi-Newton method withArmijo line search. We report the numerical experiments that show the efficiency of thebarrier function method, but the barrier function method is less effective than the augmentedLagrangian method for solving this problem.