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In this paper we develop accelerated first-order methods for convexoptimization with locally Lipschitz continuous gradient (LLCG), which is beyondthe well-studied class of convex optimization with Lipschitz continuousgradient. In particular, we first consider unconstrained convex optimizationwith LLCG and propose accelerated proximal gradient (APG) methods for solvingit. The proposed APG methods are equipped with a verifiable terminationcriterion 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 convexoptimization problem, respectively. We then consider constrained convexoptimization with LLCG and propose an first-order proximal augmented Lagrangianmethod for solving it by applying one of our proposed APG methods toapproximately solve a sequence of proximal augmented Lagrangian subproblems.The resulting method is equipped with a verifiable termination criterion andenjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log\varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ forfinding an $\varepsilon$-KKT solution of a constrained convex and stronglyconvex optimization problem, respectively. All the proposed methods in thispaper are parameter-free or almost parameter-free except that the knowledge onconvexity parameter is required. In addition, preliminary numerical results arepresented to demonstrate the performance of our proposed methods. To the bestof our knowledge, no prior studies were conducted to investigate acceleratedfirst-order methods with complexity guarantees for convex optimization withLLCG. All the complexity results obtained in this paper are new.

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