线性规划的束方法
A version of bundle method with linear programming
摘要：
束方法已经被广泛研究用于求解凸和非凸最优化问题。在迄今为止开发的大多数束方法中，在每次迭代中至少需要求解一个二次规划（QP）子问题。在本文中，我们利用的可行性，开发一个聚束法，只解决线性子问题。我们从最小化的凸函数，并表明，序列的主要迭代收敛到一个极小。对于非凸函数，我们考虑在有界水平集上局部Lipschitz连续且正则的函数，并在具有无穷范数的置信域上极小化割平面模型。这些函数的仿凸性允许我们使用局部凸化模型及其凸性性质。在一定的条件和假设下，我们通过邻近映射的外连续性研究了算法的收敛性。令人鼓舞的初步数值实验的标准测试集的结果。
英文摘要：
Bundle methods have been intensively studied for solving both convex and nonconvex optimization problems. In most of the bundle methods developed thus far, at least one quadratic programming (QP) subproblem needs to be solved in each iteration. In this paper, we exploit the feasibility of developing a bundle algorithm that only solves linear subproblems. We start from minimization of a convex function and show that the sequence of major iterations converge to a minimizer. For nonconvex functions we consider functions that are locally Lipschitz continuous and prox-regular on a bounded level set, and minimize the cutting-plane model over a trust region with infinity norm. The para-convexity of such functions allows us to use the locally convexified model and its convexity properties. Under some conditions and assumptions, we study the convergence of the proposed algorithm through the outer semicontinuity of the proximal mapping. Encouraging results of preliminary numerical experiments on standard test sets are provided.
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