Sublinear and Linear Convergence of Modified ADMM for Distributed Nonconvex Optimization

• In this article, we consider distributed nonconvex optimization over an undirected connected network. Each agent can only access to its own local nonconvex cost function and all agents collaborate to minimize the sum of these functions by using local information exchange. We first propose a modified alternating direction method of multipliers (ADMM) algorithm. We show that the proposed algorithm converges to a stationary point with the sublinear rate O(1/T) if each local cost function is smooth and the algorithm parameters are chosen appropriately. We also show that the proposed algorithm linearly converges to a global optimum under an additional condition that the global cost function satisfies the Polyak-Łojasiewicz condition, which is weaker than the commonly used conditions for showing linear convergence rates including strong convexity. We then propose a distributed linearized ADMM (L-ADMM) algorithm, derived from the modified ADMM algorithm, by linearizing the local cost function at each iteration. We show that the L-ADMM algorithm has the same convergence properties as the modified ADMM algorithm under the same conditions. Numerical simulations are included to verify the correctness and efficiency of the proposed algorithms. 

Subject headings

TEKNIK OCH TEKNOLOGIER  -- Elektroteknik och elektronik -- Reglerteknik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Electrical Engineering, Electronic Engineering, Information Engineering -- Control Engineering (hsv//eng)

Keyword

Alternating direction method of multipliers (ADMM)

distributed optimization

linear convergence

linearized ADMM

Polyak ojasiewicz condition

Costs

Iterative methods

Alternating directions method of multipliers

Communications networks

Condition

Convergence

Convex functions

Cost-function

Nonconvex optimization

Nonconvex-optimization

Sublinear

Symmetric matrices

Cost functions

Publication and Content Type

ref (subject category)

art (subject category)