新的梯度算法求解单位球笛卡尔积约束优化问题

New Gradient Algorithms for Optimization Problems Constrained by a Cartesian Product of Unit Balls

本文主要研究了单位球笛卡尔积作为约束的优化问题，给出了此类问题的最优性条件．同时将求解此问题的一些经典的梯度算法推广到了更加一般的形式，并证明了新算法的收敛性．随机二次规划问题和求解图像变分去噪模型的数值结果表明新算法并不弱于一些经典的算法，特别是在精度要求较高的情形下．

In this paper, we mainly study an optimization problem constrained by cartesian product of unit balls. Many efforts have been devoted to obtain efficient gradient descent schemes for this problem such as Chambolle algorithm or iterative shrinkage thresholding algorithm. The detail analysis of the optimal conditions of this problem are presented. Meanwhile, we extend some classic projected gradient algorithms to a general shrinking form. The new algorithm depends on two parameters, which are step length and shrinking factor. We give two types of shrinking algorithms based on different choices of step length and shrinking factor. What is more, the global convergence of the proposed algorithm is given in this paper. We conduct numerical experiments for solving rand test problems and total variation denoised model. Numerical results demonstrate that the proposed algorithm is competitive to some classic algorithms, especially when high accuracy is required.