Abstract
We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also give the tight worst-case complexity bound for a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proofs are computer-assisted, and rely on the resolutions of semidefinite programming performance estimation problems as introduced in the paper (Drori and Teboulle, Math Progr 145(1–2):451–482, 2014).
Original language | English |
---|---|
Pages (from-to) | 1185–1199 |
Number of pages | 15 |
Journal | Optimization Letters |
Volume | 11 |
Issue number | 7 |
DOIs | |
Publication status | Published - 2016 |
Keywords
- Gradient method
- Steepest descent
- Semidefinite programming
- Performance estimation problem