Mathematical Programming

, Volume 104, Issue 2, pp 609-633

First online:

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

  • Scott A. MillerAffiliated withNumerica Corp. Email author 
  • , Jérôme MalickAffiliated withINRIA

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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.