Local Convergence of the Proximal Point Algorithm and Multiplier Methods Without Monotonicity

Source abstract

This paper studies the convergence of the classical proximal point algorithm without assuming monotonicity of the underlying mapping. Practical conditions are given that guarantee the local convergence of the iterates to a solution of T(x) ∋ 0, where T is an arbitrary set-valued mapping from a Hilbert space to itself. In particular, when the problem is “nonsingular” in the sense that T -1 has a Lipschitz localization around one of the solutions, local linear convergence is obtained. This kind of regularity property of variational inclusions has been extensively studied in the literature under the name of strong regularity, and it can be viewed as a natural generalization of classical constraint qualifications in nonlinear programming to more general problem classes. Combining the new convergence results with an abstract duality framework for variational inclusions, the author proves the local convergence of multiplier methods for a very general class of problems. This gives as special cases new convergence results for multiplier methods for nonmonotone variational inequalities and nonconvex nonlinear programming.