Inexact versions of proximal point and cone-constrained augmented Lagrangians in Banach spaces
Rolando Garciga Otero
proximal point method | augmented Lagrangian | inexact solutions | maximal monotone operator | cone-constrained optimization
We extend to Banach spaces the hybrid proximal-extragradient and proximal-projection methods for finding zeroes of maximal monotone operators, recently proposed by Solodov and Svaiter in finite dimension and Hilbert spaces respectively. The generalization of the hybrid-projection method makes it possible the use of regularizations other than the quadratic by using an appropriate error criterion, which allows for bounded relative error, and a Bregman projection instead of the metric projection. Boundedness of the sequence generated by both methods and optimality of the weak accumulation points are established under suitable assumptions on the regularizing function, which hold for any power greater than one of the norm of any uniformly smooth and uniformly convex Banach space, without any assumption on the operator other than existence of zeroes. A variant of the error for the hybrid-projection method let us establish superlinear convergence even with inexact solutions of the proximal subproblem in Hilbert spaces. We show that the hybrid steps of the Proximal Point methods, allowing for constant relative errors, are necessary in order to ensure boundedness of the generated sequence, even in the optimization case. Moreover, we show that such conditions do not imply that the sequence of errors results summable ``a posteriori". We then transpose such methods to generate augmented Lagrangian methods for the following cone-constrained convex optimization problem in Banach spaces: $\min g(x)$ subject to $-G(x)\in K$, with $g:B_1\to\re$, $G:B_1 \to\B_2$, where $B_1$ and $B_2$ are real reflexive Banach spaces and $K$ is a nonempty closed convex cone in $B_2$. Two alternative procedures are developed which allow for inexact solutions of the primal subproblems. Boundedness of both the primal and the dual sequences, and optimality of primal and dual weak accumulation points, are then established, assuming only existence of Karush-Kuhn-Tucker pairs. Finally we add to the hybrid-extragradient method a penalization effect for solving variational inequality problems in Banach spaces, by introducing a boundary coercive condition on the regularizing function. We give examples of regularizing functions for the cases of the feasible set being closed balls and polyhedra. We get convergence results similar to those of the methods without penalization under the assumptions of pseudo- and paramonotonicity of the involved operator and existence of solutions.