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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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0588-x