A first-order augmented Lagrangian method for constrained minimax optimization

In this paper we study a class of constrained minimax problems. In particular, we propose a first-order augmented Lagrangian method for solving them, whose subproblems turn out to be a much simpler structured minimax problem and are suitably solved by a first-order method recently developed in [26] by the authors. Under some suitable assumptions, an \emph{operation complexity} of ${\cal O}(\varepsilon^{-4}\log\varepsilon^{-1})$, measured by its fundamental operations, is established for the first-order augmented Lagrangian method for finding an $\varepsilon$-KKT solution of the constrained minimax problems.

First-order penalty methods for bilevel optimization

In this paper we study a class of unconstrained and constrained bilevel optimization problems in which the lower-level part is a convex optimization problem, while the upper-level part is possibly a nonconvex optimization problem. In particular, we propose penalty methods for solving them, whose subproblems turn out to be a structured minimax problem and are suitably solved by a first-order method developed in this paper. Under some suitable assumptions, an \emph{operation complexity} of ${\cal O}(\varepsilon^{-4}\log\varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-7}\log\varepsilon^{-1})$, measured by their fundamental operations, is established for the proposed penalty methods for finding an $\varepsilon$-KKT solution of the unconstrained and constrained bilevel optimization problems, respectively. To the best of our knowledge, the methodology and results in this paper are new.

Accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient

In this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an $\varepsilon$-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose an first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for finding an $\varepsilon$-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are parameter-free or almost parameter-free except that the knowledge on convexity parameter is required. To the best of our knowledge, no prior studies were conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are entirely new.

Primal-dual extrapolation methods for monotone inclusions under local Lipschitz continuity with applications to variational inequality, conic constrained saddle point, and convex conic optimization problems

In this paper we consider a class of structured monotone inclusion (MI) problems that consist of finding a zero in the sum of two monotone operators, in which one is maximal monotone while another is locally Lipschitz continuous. In particular, we first propose a primal-dual extrapolation (PDE) method for solving a structured strongly MI problem by modifying the classical forward-backward splitting method by using a point and operator extrapolation technique, in which the parameters are adaptively updated by a backtracking line search scheme. The proposed PDE method is almost parameter-free, equipped with a verifiable termination criterion, and enjoys an operation complexity of ${\cal O}(\log \epsilon^{-1})$, measured by the amount of fundamental operations consisting only of evaluations of one operator and resolvent of another operator, for finding an $\epsilon$-residual solution of the structured strongly MI problem. We then propose another PDE method for solving a structured non-strongly MI problem by applying the above PDE method to approximately solve a sequence of structured strongly MI problems. The resulting PDE method is parameter-free, equipped with a verifiable termination criterion, and enjoys an operation complexity of ${\cal O}(\epsilon^{-1}\log \epsilon^{-1})$ for finding an $\epsilon$-residual solution of the structured non-strongly MI problem. As a consequence, we apply the latter PDE method to convex conic optimization, conic constrained saddle point, and variational inequality problems, and obtain complexity results for finding an $\epsilon$-KKT or $\epsilon$-residual solution of them under local Lipschitz continuity. To the best of our knowledge, no prior studies were conducted to investigate methods with complexity guarantees for solving the aforementioned problems under local Lipschitz continuity. All the complexity results obtained in this paper are entirely new.