结构最优化问题求解方法的发展
Development of a method for solving structural optimization problems
摘要：
在实践中，优化任务具有某种结构，允许为每个问题开发具有更快收敛速度的新算法。利用优化任务的结构，我们可以为以下最优化问题提出具有更乐观收敛速度的算法：具有Holder连续梯度的函数、函数的叠加（最小-最大问题）、运输问题、选举模型聚类。在这项工作中，我们提出了统一的梯度型方法到一个方法使用一个特殊的概念不精确的模型，并开发了一系列的方法，可以解决广义最优化问题语句，并使用其结构与所提出的概念不精确模型的援助。我们构造了求解相对光滑问题的梯度法、原-对偶自适应梯度法和快速梯度法以及支持函数不精确模型的随机非自适应梯度法。此外，不精确模型的概念得到了最优化问题的不同实例的支持。
英文摘要：
In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence rates for the following optimization problems: functions with Holder continuous gradients, superposition of functions (min-max problems), transportation problems, clustering by electorial model. In this work, we propose the unification of gradient-type methods into one method using a special concept of inexact model and develop a series of methods that can solve generalized optimization problem statements and use its structure with the aid of the proposed concept of inexact model. We constructed the gradient method for problems with relative smoothness, the primal--dual adaptive gradient and fast gradient methods, and the stochastic nonadaptive gradient methods that support an inexact model of a function. Moreover, the concept of inexact model is supported by different examples of optimization problems.
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