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首页> 外文期刊>Journal of applied and industrial mathematics >Adaptive Mirror Descent Algorithms for Convex and Strongly Convex Optimization Problems with Functional Constraints
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Adaptive Mirror Descent Algorithms for Convex and Strongly Convex Optimization Problems with Functional Constraints

机译：具有功能约束的凸起和强凸优化问题的自适应镜像缩减算法

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摘要

Under consideration are some adaptive mirror descent algorithms for the problems of minimization of a convex objective functional with several convex Lipschitz (generally, nonsmooth) functional constraints. It is demonstrated that the methods are applicable to the objective functionals of various levels of smoothness: The Lipschitz condition holds either for the objective functional itself or for its gradient or Hessian (while the functional itself can fail to satisfy the Lipschitz condition). The main idea is the adaptive adjustment of the method with respect to the Lipschitz constant of the objective functional (its gradient or Hessian), as well as the Lipschitz constant of the constraint. The two types of methods are considered: adaptive (not requiring the knowledge of the Lipschitz constants neither for the objective functional nor for constraints, and partially adaptive (requiring the knowledge of the Lipschitz constant for constraints). Using the restart technique, some methods are proposed for strongly convex minimization problems. Some estimates of the rate of convergence are obtained for all algorithms under consideration in dependence on the level of smoothness of the objective functional. Numerical experiments are presented to illustrate the advantages of the proposed methods for some examples.
机译：正在考虑的是一些自适应镜面下降算法，用于最小化具有几个凸出的嘴唇尖端（通常，非光滑）功能约束的凸面物体功能的问题。结果证明，该方法适用于各种平滑度的目标函数：Lipschitz条件用于目标函数本身或其梯度或Hessian（而功能本身不能满足Lipschitz条件）。主要思想是对目标函数（其梯度或黑森州）的Lipschitz常数的方法的自适应调整，以及约束的嘴尖常数。考虑了两种类型的方法：自适应（不需要对Lipschitz常数的知识既不用于客观函数也不是约束，并且部分自适应（需要对约束的LipsChitz常数的知识）。使用重启技术，有些方法是提出了强烈凸起的最小化问题。用于根据目标官能团的平滑度所考虑的所有算法获得的一些估计。提出了数值实验以说明所提出的方法的一些实例的优点。

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