Abstract :
New algorithms for minimax optimization are presented. First, a method using a local linearization of the functions at each step is developed. In contrast to existing methods, which use bounds on variable changes or scaled steps to make the linearization valid, the proposed method uses a bound on the error reduction at each step, and finds the smallest change in the variables to reach this bound. Quadratic final convergence is achieved for regular problems and one class of singular problems. Some comparisons with other methods are given. Next, a simple technique is developed for finding an approximation to the Hessians of the functions being optimized. Using this approximation, the linearized functions normally used at each step are replaced by quadratic relations. Assumptions necessary for the approximation to be useful are examined. Finally, examples are given to demonstrate the speed and utility of the proposed techniques.
