We consider the problem of minimizing a convex objective which is the sum of a smooth part, with Lipschitz continuous gradient, and a nonsmooth part. Inspired by various applications, we focus on the case when the nonsmooth part is a composition of a proper closed convex function P and a nonzero affine map, with the … Read more
In this paper, we focus on the convergence analysis for the application of the Peaceman-Rachford splitting method to a convex minimization model whose objective function is the sum of a smooth and nonsmooth convex functions. The sublinear convergence rate in term of the worst-case O(1/t) iteration complexity is established if the gradient of the smooth … Read more
In this work we show that randomized (block) coordinate descent methods can be accelerated by parallelization when applied to the problem of minimizing the sum of a partially separable smooth convex function and a simple separable convex function. The theoretical speedup, as compared to the serial method, and referring to the number of iterations needed … Read more
We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the relaxation of complicating constraints with gradient and fast gradient methods based on inexact first order information. Moreover, since the exact solution of the augmented … Read more
Recently, in [17] we have showed the first possibility of combining the Douglas-Rachford alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving a convex minimization model with a general separable structure. This paper is a further study on theoretical aspects of this theme. We first derive a general algorithmic framework … Read more
In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an $\epsilon$-accurate solution with probability at least $1-\rho$ in at most $O(\tfrac{n}{\epsilon} \log \tfrac{1}{\rho})$ iterations, where $n$ is the number of blocks. For strongly convex functions … Read more
In this paper, we propose a simple variant of the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming problem (LP). Our variant executes a natural finite termination procedure at each iteration and it is easy to implement the algorithm. Our algorithm admits an objective-function free polynomial-time complexity when it is applied to LPs whose dual feasible region … Read more
It is well known that a wide-neighborhood interior-point algorithm for linear programming performs much better in implementation than those small-neighborhood counterparts. In this paper, we provide a unified way to enlarge the neighborhoods of predictor-corrector interior-point algorithms for linear programming. We prove that our methods not only enlarge the neighborhoods but also retain the so-far … Read more
We consider an augmented Lagrangian algorithm for minimizing a convex quadratic function subject to linear inequality constraints. Linear optimization is an important particular instance of this problem. We show that, provided the augmentation parameter is large enough, the constraint value converges {\em globally\/} linearly to zero. This property is viewed as a consequence of the … Read more