Recent Advances in Variable Metric First-Order Methods

Minimization problems often occur in modeling phenomena dealing with real-life applications that nowadays handle large-scale data and require real-time solutions. For these reasons, among all possible iterative schemes, first-order algorithms represent a powerful tool in solving such optimization problems since they admit a relatively simple implementation and avoid onerous computations during the iterations. On the other hand, a well known drawback of these methods is a possible poor convergence rate, especially showed when an high accurate solution is required. Consequently, the acceleration of first-order approaches is a very discussed field which has experienced several efforts from many researchers in the last decades. The possibility of considering a variable underlying metric changing at each iteration and aimed to catch local properties of the starting problem has been proved to be effective in speeding up first-order methods. In this work we deeply analyze a possible way to include a variable metric in first-order methods for the minimization of a functional which can be expressed as the sum of a differentiable term and a nondifferentiable one. Particularly, the strategy discussed can be realized by means of a suitable sequence of symmetric and positive definite matrices belonging to a compact set, together with an Armijo-like linesearch procedure to select the steplength along the descent direction ensuring a sufficient decrease of the objective function.