论文标题
具有可转换非Convex功能的优化条件和优化算法
Optimization Condition and Algorithm of Optimization with Convertible Nonconvex Function
论文作者
论文摘要
本文介绍了几个用于求解非convex或非滑动优化问题的新概念，包括可转换的非Convex功能，确切的可转换非Convex功能和可区分可转换的非convex函数。此处证明了许多非凸函数或非滑动功能（或不连续的）函数实际上是可转换的非convex函数和可转换的非convex函数操作，例如加法，减法，乘法或除法导致可转换的非convex函数。事实证明，有足够的条件来判断全局最佳解决方案，以判断不受限制的可转换非convex功能的不受约束优化问题，这与Karush-Kuhn-Tucker（KKT）条件相当。通过相应定义的双重问题定义了两个可区分可转换非凸功能的Lagrange函数。证明了强双重定理，表明全局最佳解决方案的最佳目标值等于双重问题的最佳客观值，这等同于KKT条件。提出了增强的拉格朗日惩罚函数算法，并证明了其收敛性。因此，本文提供了一个新的想法，用于解决无约束的非凸或非平滑优化问题，并通过使用某些梯度搜索算法（例如梯度下降算法，牛顿算法等）来避免细分或平滑技术。
The paper introduces several new concepts for solving nonconvex or nonsmooth optimization problems, including convertible nonconvex function, exact convertible nonconvex function and differentiable convertible nonconvex function. It is proved herein many nonconvex functions or nonsmooth (or discontinuous) functions are actually convertible nonconvex functions and convertible nonconvex function operations such as addition, subtraction, multiplication or division result in convertible nonconvex functions. The sufficient condition for judging a global optimal solution to unconstrained optimization problems with differentiable convertible nonconvex functions is proved, which is equivalent to Karush-Kuhn-Tucker(KKT) condition. Two Lagrange functions of differentiable convertible nonconvex function are defined with their dual problems defined accordingly. The strong duality theorem is proved, showing that the optimal objective value of the global optimal solution is equal to the optimal objective value of the dual problem, which is equivalent to KKT condition. An augmented Lagrangian penalty function algorithm is proposed and its convergence is proved. So the paper provides a new idea for solving unconstrained nonconvex or non-smooth optimization problems and avoids subdifferentiation or smoothing techniques by using some gradient search algorithms, such as gradient descent algorithm, Newton algorithm and so on.
