Complexity of first-order inexact Lagrangian and penalty methods for conic convex programming. (4th March 2019)

• Record Type: Journal Article
• Authors: Necoara, I.; Patrascu, A.; Glineur, F.
• Is Part Of: Optimization methods and software. Volume 34:Number 2(2019)
• Page Start: 305
• Page End: 335
• Publication Date: 2019-03-04
• Subjects: conic convex problems -- smooth (augmented) dual functions -- penalty functions -- (augmented) dual first-order methods -- penalty fast gradient methods -- approximate primal solution -- overall computational complexity; 90C25 -- 90C46 -- 68Q25 -- 65K05; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2017.1380642
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : In this paper we present a complete iteration complexity analysis of inexact first-order Lagrangian and penalty methods for solving cone-constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal–dual first-order methods based on inexact information and augmented Lagrangian smoothing or Nesterov-type smoothing. For inexact (fast) gradient augmented Lagrangian methods, we derive an overall computational complexity ofO ( 1 / ϵ ) projections onto a simple primal set in order to attain an ε -optimal solution of the conic convex problem. For the inexact fast gradient method combined with Nesterov-type smoothing, we derive computational complexityO ( 1 / ϵ 3 / 2 ) projections onto the same set. Then, we assume that optimal Lagrange multipliers might not exist for the cone-constrained convex problem, and analyse the fast gradient method for solving penalty reformulations of the problem. For the fast gradient method combined with penalty framework, we also derive an overall computational complexity ofO ( 1 / ϵ 3 / 2 ) projections onto a simple primal set to attain an ε -optimal solution for the original problem.