The Modified Subgraident Method for Equality Constrained Nonconvex Optimization Problems

Abstract

In this paper we use a sharp Lagrangian function to construct a dual problem to the nonconvex minimization problem with equality constraints. By using the strong duality results we modify the subgradient method for solving a dual problem constructed. The algorithm proposed in this paper has some advantages. In contrast with the penalty or multiplier methods, for improving the value of the dual function, one need not to take the “penalty like parameter” to infinity in the new method. The value of the dual function strongly increases at each iteration. The subgradient of the dual function along which its value increases is calculated explicitly. In contrast, by using the primal-dual gap, the proposed algorithm possesses a natural stopping criteria. We do not use any convexity and differentiability conditions, and show that the sequence of the values of dual function converges to the optimal value. Finally, we demonstrate the presented method on numerical examples.