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).
- gradient method
- steepest descent
- semidefinite programming
- performance estimation problem
de Klerk, E., Glineur, F., & Taylor, A. (2017). On the worst-case complexity of the gradient method with exact line search for smooth strongly convex functions. Optimization Letters, 11(7), 1185–1199. https://doi.org/10.1007/s11590-016-1087-4