Convergence properties of optimization algorithms for the SAT problem

The satisfiability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design efficient algorithms to find a solution for a satisfiable conjunctive normal form (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 (J. Gu, 1988; 1992; 1994). 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. We prove that when the initial solution is sufficiently close to the optimal solution, the steepest descent method has a linear convergence ratio β<1, Newton´s method has a convergence ratio of order two, and the convergence ratio of the steepest descent method is approximately (1-β/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. Experimental results show that this algorithm is more efficient than some previous ones in finding a solution for certain classes of the satisfiable CNF formulas