一种非精确非光滑信赖域算法  

An Inexact Trust Region Algorithm for Nonsmooth Optimization

作　　者：李祉赟 王湘美 马德乐 LI Zhi-yun;WANG Xiang-mei;MA De-le(College of Mathematics and Statistics,Guizhou University,Guiyang,Guizhou,550025,China)

机构地区：[1]贵州大学数学与统计学院,贵州贵阳550025

出　　处：《新疆师范大学学报（自然科学版）》2024年第4期44-52,共9页Journal of Xinjiang Normal University(Natural Sciences Edition)

基　　金：国家自然科学基金项目(12161017);贵州省省级科技计划项目(ZK[2022]110)。

摘　　要：Aravkin等人提出了求解非光滑优化问题min_(x∈R^(d))f(x)+h(x)的非光滑信赖域算法(采用f的精确梯度),其中f是连续可微函数,h是邻近有界且下半连续的真函数。文章研究当该问题中f:=1/n ∑_(i=1)^(n)f_(i)(n很大且每个分量函数fi是连续可微)时,求解这类大规模可分离非光滑优化问题的有效算法。结合非精确算法和非光滑信赖域算法的思想,提出了用非精确梯度代替精确梯度的非精确非光滑信赖域算法。与非光滑信赖域算法(采用精确梯度)相比,该算法降低了每次迭代的计算量。在一定的假设条件下,证明了算法的迭代复杂度。Aravkin et al proposed the trust region algorithm(employing exact gradients of f)for solving the nonsmooth optimization problem min_(x∈R^(d))f(x)+h(x),where f is a continuously differentiable function and h is a lower semicontinuous and prox-bounded proper function.In the case when f:=1/n ∑_(i=1)^(n)f_(i)(n is quite big,and each component fi is continuously differentiable),the efficient algorithm for solving such kind of large-scale separable nonsmooth optimization problem is studied.Combining the concepts of the inexact algorithm and the above trust-region algorithm,it is proposed that the inexact trust-region algorithm replaces the exact gradients with the inexact gradients for solving this nonsmooth problem.Comparing with the trust-region algorithm(employing the exact gradients of f),the new algorithm can reduce the computational cost at each iteration.Under certain assumptions,the iteration complexity of this algorithm is established.

关 键 词：大规模可分离非光滑优化 非精确信赖域算法 邻近梯度算法 

分 类 号：O224[理学—运筹学与控制论]