Conditioning theory of the equality constrained quadratic programming and its applications

Perturbation analysis of the equality constrained quadratic programming is considered. We present two different perturbation bounds to explore underlying factors for affecting the conditioning of equality constrained quadratic programming, and propose the condition numbers to give sharp forward error bounds. To improve the computational efficiency of condition numbers, some new compact forms and tight upper bounds of the condition numbers are introduced. Numerical examples are given to illustrate our theoretical results. As a special case of equality constrained quadratic programming, the rigorous perturbation analysis of Markowitz mean-variance model is also studied, which can be used to give a formal characterization of the roles of condition number and the smallest eigenvalue of the covariance matrix in bounding the forward errors. With respect to condition number and the smallest eigenvalue of the covariance matrix, numerical performances of two different covariance matrix estimators on optimal portfolio selection are also presented through simulations.