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On Affine Invariant Descent Directions

Yu-Hong Dai (dyh***at***lsec.cc.ac.cn)
Florian Jarre (jarre***at***hhu.de)
Felix Lieder (lieder***at***opt.uni-duesseldorf.de)

Abstract: This paper explores the existence of affine invariant descent directions for unconstrained minimization. While there may exist several affine invariant descent directions for smooth functions $f$ at a given point, it is shown that for quadratic functions there exists exactly one invariant descent direction in the strictly convex case and generally none in the nondegenerate indefinite case. These results can be generalized to smooth nonlinear functions and have implications regarding the initialization of minimization algorithms. They stand in contrast to recent works on constrained convex and nonconvex optimization for which there may exist an affine invariant framework that depends on the feasible set.

Keywords: Affine invariance, descent direction, Newton direction

Category 1: Nonlinear Optimization

Citation: submitted to Mathematical Programming August 29th, 2017

Download: [PDF]

Entry Submitted: 08/29/2017
Entry Accepted: 08/29/2017
Entry Last Modified: 08/13/2018

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