In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these Primal–Dual splitting methods. … Read more
We shed light on the structure of the “three-operator” version of the forward-Douglas–Rachford splitting algorithm for finding a zero of a sum of maximally monotone operators $A + B + C$, where $B$ is cocoercive, involving only the computation of $B$ and of the resolvent of $A$ and of $C$, separately. We show that it … Read more
We study the boundary structure of closed convex cones, with a focus on facially dual complete (nice) cones. These cones form a proper subset of facially exposed convex cones, and they behave well in the context of duality theory for convex optimization. Using the well-known and very commonly used concept of tangent cones in nonlinear … Read more
By reviewing the primal-dual hybrid algorithm (PDHA) proposed by He, You and Yuan (SIAM J. Imaging Sci. 2014;7(4):2526-2537), in this paper we introduce four improved schemes for solving a class of generalized saddle-point problems. By making use of the variational inequality, weaker conditions are presented to ensure the global convergence of the proposed algorithms, where … Read more
Under the strongly convex assumption, several recent works studied the global linear convergence rate of the proximal incremental aggregated gradient (PIAG) method for minimizing the sum of a large number of smooth component functions and a non-smooth convex function. In this paper, under the quadratic growth condition{a strictly weaker condition than the strongly convex assumption, … Read more
We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [1] by taking a different perspective, leading to an algorithm which is conceptually more natural, has notably improved convergence rates, and for which the analysis … Read more
The proximal point algorithm (PPA) is a fundamental method for convex programming. When PPA applied to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and in most cases, even admit closed form … Read more
The proximal ADMM which adds proximal regularizations to ADMM’s subproblems is a popular and useful method for linearly constrained separable convex problems, especially its linearized case. A well-known requirement on guaranteeing the convergence of the method in the literature is that the proximal regularization must be positive semidefinite. Recently it was shown by He et … Read more
The speed optimization problem over a path aims to find a set of speeds over each arc of the given path to minimize the total cost, while respecting the time-window constraint at each node and speed limits over each arc. In maritime transportation, the cost represents fuel cost or emissions, so study of this problem … Read more
We design and analyze a novel gradient-based algorithm for unconstrained convex optimization. When the objective function is $m$-strongly convex and its gradient is $L$-Lipschitz continuous, the iterates and function values converge linearly to the optimum at rates $\rho$ and $\rho^2$, respectively, where $\rho = 1-\sqrt{m/L}$. These are the fastest known guaranteed linear convergence rates for … Read more