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Line search fixed point algorithms based on nonlinear conjugate gradient directions: application to constrained smooth convex optimization

机译：基于非线性共轭梯度方向的线搜索定点算法：在约束光滑凸优化中的应用

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摘要

This paper considers the fixed point problem for a nonexpansive mapping on a real Hilbert space and proposes novel line search fixed point algorithms to accelerate the search. The termination conditions for the line search are based on the well-known Wolfe conditions that are used to ensure the convergence and stability of unconstrained optimization algorithms. The directions to search for fixed points are generated by using the ideas of the steepest descent direction and conventional nonlinear conjugate gradient directions for unconstrained optimization. We perform convergence as well as convergence rate analyses on the algorithms for solving the fixed point problem under certain assumptions. The main contribution of this paper is to make a concrete response to an issue of constrained smooth convex optimization; that is, whether or not we can devise nonlinear conjugate gradient algorithms to solve constrained smooth convex optimization problems. We show that the proposed fixed point algorithms include ones with nonlinear conjugate gradient directions which can solve constrained smooth convex optimization problems. To illustrate the practicality of the algorithms, we apply them to concrete constrained smooth convex optimization problems, such as constrained quadratic programming problems and generalized convex feasibility problems, and numerically compare them with previous algorithms based on the Krasnosel’skiĭ-Mann fixed point algorithm. The results show that the proposed algorithms dramatically reduce the running time and iterations needed to find optimal solutions to the concrete optimization problems compared with the previous algorithms.
机译：本文考虑了实希尔伯特空间上非膨胀映射的不动点问题，并提出了新颖的线搜索不动点算法来加速搜索。线搜索的终止条件基于众所周知的Wolfe条件，该条件用于确保无约束优化算法的收敛性和稳定性。通过使用最速下降方向和常规的非线性共轭梯度方向进行无约束优化，可以生成搜索固定点的方向。在某些假设下，我们对解决定点问题的算法进行收敛以及收敛速度分析。本文的主要贡献是对受约束的光滑凸优化问题做出具体的回应。也就是说，是否可以设计出非线性共轭梯度算法来解决约束光滑凸优化问题。我们表明，提出的定点算法包括具有非线性共轭梯度方向的算法，可以解决约束光滑凸优化问题。为了说明该算法的实用性，我们将其应用于具体的约束光滑凸优化问题，例如约束二次规划问题和广义凸可行性问题，并将其与基于Krasnosel'skiski-Mann不动点算法的先前算法进行数值比较。结果表明，与现有算法相比，所提出的算法大大减少了寻找具体优化问题最优解所需的运行时间和迭代次数。

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