@article{90a8e678d9bd4cc4a7461a2519630a95,

title = "Nonlinear rescaling and proximal-like methods in convex optimization",

abstract = "The nonlinear rescaling principle (NRP) consists of transforming the objective function and/or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides. A nonlinear transformation parameterized by a positive scalar parameter and based on a smooth scaling function is used to transform the constraints. The methods based on NRP consist of sequential unconstrained minimization of the classical Lagrangian for the equivalent problem, followed by an explicit formula updating the Lagrange multipliers. We first show that the NRP leads naturally to proximal methods with an entropy-like kernel, which is defined by the conjugate of the scaling function, and establish that the two methods are dually equivalent for convex constrained minimization problems. We then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy-like proximal methods for convex constrained optimization problems. Special cases of the nonlinear rescaling algorithm are presented. In particular a new class of exponential penalty-modified barrier functions methods is introduced.",

keywords = "Augmented lagrangians, Convex optimization, Modified barrier functions, Nonlinear rescaling, Proximal methods",

author = "Roman Polyak and Marc Teboulle",

note = "Funding Information: * Corresponding author. E-mail:

[email protected]. Partially supported by the National Science Foundation, under Grants DMS-9201297, and DMS-9401871. Currently on leave from Department of Mathematics and Statistics University of Maryland, Baltimore County, Baltimore, USA. {\textquoteright} E-mail:

[email protected]. Partially supported by NASA Grant NAG3-1397 and NSF Grant DMS-9403218.",

year = "1997",

month = feb,

day = "1",

doi = "10.1007/BF02614440",

language = "אנגלית",

volume = "76",

pages = "265--284",

journal = "Mathematical Programming",

issn = "0025-5610",

publisher = "Springer-Verlag GmbH and Co. KG",

number = "2",

}