论文标题
可转换非凸优化的优化条件和可分解算法
Optimization Conditions and Decomposable Algorithms for Convertible Nonconvex Optimization
论文作者
论文摘要
本文定义了可转换的非凸功能（简称CN函数）和弱（强）均匀（可分解，精确）CN函数，证明了其全局解决方案的优化条件，并提出了用于解决可分解CN功能的无约束优化问题的算法。首先，为了说明某些非凸函数（非平滑或不连续的）实际上是弱均匀的CN函数的事实，给了一个示例。证明了CN功能的运行属性，包括加法，减法，乘法，除法和复合操作。其次，证明了全局最佳解决方案的优化条件，以弱均匀的CN函数进行不受约束的优化。基于可分解CN功能的无约束优化问题，通过其增强的Lagrangian惩罚函数提出了可分解的算法，并证明了其收敛性。数值结果表明，可以通过可分解算法获得近似的全局最佳解决方案，可通过CN函数获得不受限制的优化。可分解算法可以有效地减少可分解的CN函数解决无约束优化问题的规模。本文为解决无约束的非convex优化问题提供了一个新想法。
This paper defines a convertible nonconvex function(CN function for short) and a weak (strong) uniform (decomposable, exact) CN function, proves the optimization conditions for their global solutions and proposes algorithms for solving the unconstrained optimization problems with the decomposable CN function. First, to illustrate the fact that some nonconvex functions, nonsmooth or discontinuous, are actually weak uniform CN functions, examples are given. The operational properties of the CN functions are proved, including addition, subtraction, multiplication, division and compound operations. Second, optimization conditions of the global optimal solution to unconstrained optimization with a weak uniform CN function are proved. Based on the unconstrained optimization problem with the decomposable CN function, a decomposable algorithm is proposed by its augmented Lagrangian penalty function and its convergence is proved. Numerical results show that an approximate global optimal solution to unconstrained optimization with a CN function may be obtained by the decomposable algorithms. The decomposable algorithm can effectively reduce the scale in solving the unconstrained optimization problem with the decomposable CN function. This paper provides a new idea for solving unconstrained nonconvex optimization problems.
