Iteration-complexity of first-order penalty methods for convex programming

This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for...

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Published in:	Mathematical programming Vol. 138; no. 1-2; pp. 115 - 139
Main Authors:	Lan, Guanghui, Monteiro, Renato D. C.
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer-Verlag 01.04.2013 Springer Nature B.V
Subjects:	Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Inverse Iterative methods Lagrange multiplier Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Optimization Perturbation methods Programming Projection Systems engineering Theoretical 90C20 Quadratic penalty method Lagrange multiplier 90C30 90C25 90C22 Convex programming
ISSN:	0025-5610, 1436-4646
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Abstract	This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iteration of Nesterov’s optimal method (or one of its variants) for approximately solving a smooth penalization subproblem, consists of one or two projections onto the simple convex set. Iteration-complexity bounds expressed in terms of the latter type of iterations are derived for two quadratic penalty based variants, namely: one which applies the quadratic penalty method directly to the original problem and another one which applies the latter method to a perturbation of the original problem obtained by adding a small quadratic term to its objective function.
AbstractList	This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iteration of Nesterovâ[euro](TM)s optimal method (or one of its variants) for approximately solving a smooth penalization subproblem, consists of one or two projections onto the simple convex set. Iteration-complexity bounds expressed in terms of the latter type of iterations are derived for two quadratic penalty based variants, namely: one which applies the quadratic penalty method directly to the original problem and another one which applies the latter method to a perturbation of the original problem obtained by adding a small quadratic term to its objective function.[PUBLICATION ABSTRACT] This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iteration of Nesterov's optimal method (or one of its variants) for approximately solving a smooth penalization subproblem, consists of one or two projections onto the simple convex set. Iteration-complexity bounds expressed in terms of the latter type of iterations are derived for two quadratic penalty based variants, namely: one which applies the quadratic penalty method directly to the original problem and another one which applies the latter method to a perturbation of the original problem obtained by adding a small quadratic term to its objective function.
Author	Lan, Guanghui Monteiro, Renato D. C.
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Cites_doi	10.1137/090753127 10.1137/S1052623403425629 10.1137/S1052623403427823 10.1007/s10107-008-0261-6 10.1007/s10107-006-0034-z 10.1007/s10107-004-0552-5 10.1007/978-1-4419-8853-9
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References	Nesterov (CR6) 1983; 269 Nesterov (CR9) 2007; 109 CR10 CR3 Nemirovski (CR5) 2005; 15 Nesterov (CR8) 2005; 103 Auslender, Teboulle (CR1) 2006; 16 Monteiro, Svaiter (CR4) 2010; 20 Nesterov (CR7) 2004 Lan, Lu, Monteiro (CR2) 2011; 126 A Nemirovski (588_CR5) 2005; 15 588_CR3 YE Nesterov (588_CR8) 2005; 103 RDC Monteiro (588_CR4) 2010; 20 G Lan (588_CR2) 2011; 126 YE Nesterov (588_CR7) 2004 A Auslender (588_CR1) 2006; 16 YE Nesterov (588_CR6) 1983; 269 YE Nesterov (588_CR9) 2007; 109 588_CR10
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SubjectTerms	Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Inverse Iterative methods Lagrange multiplier Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Methods Numerical Analysis Optimization Perturbation methods Programming Projection Systems engineering Theoretical
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Title	Iteration-complexity of first-order penalty methods for convex programming
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