Papers › A Globally Convergent Gradient Method with Momentum
A Globally Convergent Gradient Method with Momentum
Matteo Lapucci, Giampaolo Liuzzi, Stefano Lucidi, Davide Pucci, Marco Sciandrone
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In this work, we consider smooth unconstrained optimization problems and we deal with the class of gradient methods with momentum, i.e., descent algorithms where the search direction is defined as a linear combination of the current gradient and the preceding search direction. This family of algorithms includes nonlinear conjugate gradient methods and Polyak's heavy-ball approach, and is thus of high practical and theoretical interest in large-scale nonlinear optimization. We propose a general framework where the scalars of the linear combination defining the search direction are computed simultaneously by minimizing the approximate quadratic model in the 2 dimensional subspace. This strategy allows us to define a class of gradient methods with momentum enjoying global convergence guarantees and an optimal worst-case complexity bound in the nonconvex setting. Differently than all related works in the literature, the convergence conditions are stated in terms of the Hessian matrix of the bi-dimensional quadratic model. To the best of our knowledge, these results are novel to the literature. Moreover, extensive computational experiments show that the gradient method with momentum here presented is a solid choice to tackle some classes of nonconvex unconstrained problems.
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