, Volume 104, Issue 2-3, pp 609-633
Date: 14 Jul 2005

Newton methods for nonsmooth convex minimization: connections among /MediaObjects/s10107-005-0631-2flb1.gif -Lagrangian, Riemannian Newton and SQP methods

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Abstract

This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from http://static-content.springer.com/image/art%3A10.1007%2Fs10107-005-0631-2/MediaObjects/s10107-005-0631-2flb2.gif -Lagrangian theory and from Riemannian geometry. The Hessian based on the http://static-content.springer.com/image/art%3A10.1007%2Fs10107-005-0631-2/MediaObjects/s10107-005-0631-2flb2.gif -Lagrangian depends on the selection of a dual parameter g; by revealing the connection to Riemannian geometry, a natural choice of g emerges for which the two Newton directions coincide. This choice of g is also shown to be related to the least-squares multiplier estimate from a sequential quadratic programming (SQP) approach, and with this multiplier, SQP gives the same search direction as the Newton methods.