Convergence Properties of Optimization

The satissability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design eecient optimization algorithms for nding a solution for a satissable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed 31, 35, 34, 32]. Many optimization techniques, such as the steepest descent method, Newton's method, and the coordinate descent method, can be used to solve the Universal SAT problem. In this paper, we prove that, when the initial solution is suuciently close to the optimal solution, the steepest descent method has a linear convergence ratio < 1, Newton's method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1 ? =m) for the Universal SAT problem with m variables. An algorithm based on the coordinate descent method for the Universal SAT problem is also presented in this paper.