Solving smooth minmin and minmax problems by mixed oracle algorithms
Abstract
In this paper, we consider two types of problems that have some similarity in their structure, namely, minmin problems and minmax saddlepoint problems. Our approach is based on considering the outer minimization problem as a minimization problem with inexact oracle. This inexact oracle is calculated via inexact solution of the inner problem, which is either minimization or a maximization problem. Our main assumptions are that the problem is smooth and the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem only zerothorder oracle is available. To solve the inner problem we use accelerated gradientfree method with zerothorder oracle. To solve the outer problem we use either inexact variant of Vaydya's cuttingplane method or a variant of accelerated gradient method. As a result, we propose a framework that leads to nonasymptotic complexity bounds for both minmin and minmax problems. Moreover, we estimate separately the number of first and zerothorder oracle calls which are sufficient to reach any desired accuracy.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2103.00434
 Bibcode:
 2021arXiv210300434G
 Keywords:

 Mathematics  Optimization and Control
 EPrint:
 doi:10.1007/9783030864330_2