An alternating iterative method and its application in statistical inference

This paper studies non-convex programming problems. It is known that, in statistical inference, many constrained estimation problems may be expressed as convex programming problems. However, in many practical problems, the objective functions are not convex. In this paper, we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems. A two-step iterative algorithm called the alternating iterative method is proposed for finding solutions for such problems. The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al. (Biometrika, 72, 465–472 (1983)), Shi N. Z. (J. Multivariate Anal., 50, 282–293 (1994)) and El Barmi H. and Dykstra R. (Ann. Statist., 26, 1878–1893 (1998)).