We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be inde nite which plays … Read more
The accelerated gradient algorithm is known to have non-monotonic, periodic convergence behavior in the high momentum regime. If important function parameters like the condition number are known, the momentum can be adjusted to get linear convergence. Unfortunately these parameters are usually not accessible, so instead heuristics are used for deciding when to restart. One of … Read more
In recent years, techniques based on convex optimization and real algebra that produce converging hierarchies of lower bounds for polynomial minimization problems have gained much popularity. At their heart, these hierarchies rely crucially on Positivstellens\”atze from the late 20th century (e.g., due to Stengle, Putinar, or Schm\”udgen) that certify positivity of a polynomial on an … Read more
An ADMM-based splitting method is proposed in [11] for solving convex minimization problems with linear constraints and multi-block separable objective functions; while a relatively large proximal parameter is required for theoretically ensuring the convergence. In this paper, we further study this method and find its optimal (smallest) proximal parameter. For succinctness, we focus on the … Read more
The alternating direction method of multipliers (ADMM) is being widely used in a variety of areas; its different variants tailored for different application scenarios have also been deeply researched in the literature. Among them, the linearized ADMM has received particularly wide attention from many areas because of its efficiency and easy implementation. To theoretically guarantee … Read more
In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. … Read more
This paper analyzes sequences generated by infeasible interior point methods. In convex and non-convex settings, we prove that moving the primal feasibility at the same rate as complementarity will ensure that the Lagrange multiplier sequence will remain bounded, provided the limit point of the primal sequence has a Lagrange multiplier, without constraint qualification assumptions. We … Read more
Given a non-convex optimization problem, we study conditions under which every Karush-Kuhn-Tucker (KKT) point is a global optimizer. This property is known as KT-invexity and allows to identify the subset of problems where an interior point method always converges to a global optimizer. In this work, we provide necessary conditions for KT-invexity in n-dimensions and … Read more
The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well-known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive … Read more
Stochastic user equilibrium models are fundamental to the analysis of transportation systems. Such models are typically developed under the assumption of route based choice models for the users. A class of link based models under a Markovian assumption on the route choice behavior of the users has been proposed to deal with the drawbacks of … Read more