Complexity of first-order inexact Lagrangian and penalty methods for conic convex programming

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 mu...

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Vydáno v:	Optimization methods & software Ročník 34; číslo 2; s. 305 - 335
Hlavní autoři:	Necoara, I., Patrascu, A., Glineur, F.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Abingdon Taylor & Francis 04.03.2019 Taylor & Francis Ltd
Témata:	(augmented) dual first-order methods approximate primal solution Complexity Computational geometry conic convex problems Convexity Iterative methods Lagrange multiplier Mathematical programming Methods overall computational complexity penalty fast gradient methods penalty functions smooth (augmented) dual functions Smoothing
ISSN:	1055-6788, 1029-4937
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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 of 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 complexity 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 of projections onto a simple primal set to attain an ε-optimal solution for the original problem.
AbstractList	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 of [Formula omitted.] projections onto a simple primal set in order to attain an [epsilon]-optimal solution of the conic convex problem. For the inexact fast gradient method combined with Nesterov-type smoothing, we derive computational complexity [Formula omitted.] 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 of [Formula omitted.] projections onto a simple primal set to attain an [epsilon]-optimal solution for the original problem. 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 of 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 complexity 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 of projections onto a simple primal set to attain an ε-optimal solution for the original problem.
Author	Necoara, I. Glineur, F. Patrascu, A.
Author_xml	– sequence: 1 givenname: I. surname: Necoara fullname: Necoara, I. email: ion.necoara@acse.pub.ro organization: Automatic Control and Systems Engineering Department, University Politehnica Bucharest – sequence: 2 givenname: A. surname: Patrascu fullname: Patrascu, A. organization: Automatic Control and Systems Engineering Department, University Politehnica Bucharest – sequence: 3 givenname: F. surname: Glineur fullname: Glineur, F. organization: Center for Operations Research and Econometrics, Catholic University of Louvain
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SubjectTerms	(augmented) dual first-order methods approximate primal solution Complexity Computational geometry conic convex problems Convexity Iterative methods Lagrange multiplier Mathematical programming Methods overall computational complexity penalty fast gradient methods penalty functions smooth (augmented) dual functions Smoothing
Title	Complexity of first-order inexact Lagrangian and penalty methods for conic convex programming
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