Newton methods for nonsmooth convex minimization: connections among **-Lagrangian, Riemannian Newton and SQP methods

Authors:
Scott A. Miller;Jérôme Malick
Affiliations:
Numerica Corp., P.O. Box 271246, 80527-1246, Ft. Collins, CO, USA;INRIA, P.O. Box 271246, 655 avenue de l'Europe, 38334, Saint Ismier Cedex, CO, France
Venue:
Mathematical Programming: Series A and B
Year:
2005

Citing 0
Cited 3

A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations

SIAM Journal on Matrix Analysis and Applications
Projection-like Retractions on Matrix Manifolds

SIAM Journal on Optimization
Weak Sharp Minima on Riemannian Manifolds

SIAM Journal on Optimization

Quantified Score

Hi-index	0.00

Visualization

Abstract

This paper studies Newton-type methods for minimization of partly smooth convex functions. Sequential Newton methods are provided using local parameterizations obtained from **-Lagrangian theory and from Riemannian geometry. The Hessian based on the **-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.