Journal
MATHEMATICAL PROGRAMMING
Volume 155, Issue 1-2, Pages 511-547Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s10107-015-0861-x
Keywords
Penalty; First-order; Augmented Lagrangian method; Convex programming; Lagrange multiplier
Categories
Funding
- NSF [CCF-0808863, CMMI-1000347, CMMI-1254446, DMS-1319050]
- ONR [N00014-08-1-0033, N00014-13-1-0036]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1319050] Funding Source: National Science Foundation
- Div Of Civil, Mechanical, & Manufact Inn
- Directorate For Engineering [1300221] Funding Source: National Science Foundation
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This paper considers a special class of convex programming (CP) problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold. We present first-order methods for this class of problems based on an inexact version of the classical augmented Lagrangian (AL) approach, where the subproblems are approximately solved by means of Nesterov's optimal method. We then establish a bound on the total number of Nesterov's optimal iterations, i.e., the inner iterations, performed throughout the entire inexact AL method to obtain a near primal-dual optimal solution. We also present variants with possibly better iteration-complexity bounds than the original inexact AL method, which consist of applying the original approach directly to a perturbed problem obtained by adding a strongly convex component to the objective function of the CP problem.
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