【摘要】 KL不等式是解决优化分析、动态系统、偏微分方程和其他实际问题的一个重要研究工具,它在求解非凸非光滑优化问题的算法收敛性分析上扮演着重要的角色.基于Kurdyka-?ojasiewicz(KL)不等式讨论求解具有锥约束凸优化问题的邻近点方法.借助Lagrange函数,将有限维空间上的具有锥约束凸优化问题等价转化为无约束凸优化问题,讨论具有锥约束的凸优化问题的Lagrange函数的凸性.证明了满足二阶增长条件的正常的下半连续凸函数具有KL性质.构建了求解无约束等价问题的邻近点方法,并基于KL不等式分析了邻近点方法的收敛性.更多还原

【Abstract】 KL inequality is an important research tool for solving optimization analysis,dynamic systems,partial differential equations and other practical problems.It plays an important role in the analysis of algorithm convergence for solving non-convex and non-smooth optimization problems.Based on Kurdyka-?ojasiewicz(KL)inequality,proximal algorithm for solving cone-constrained convex optimization problem is discussed.With the help of Lagrange function,the cone-constrained convex optimization problem in the finite dimensional space is equivalently transformed into an unconstrained convex optimization problem.The convexity of the Lagrange function of the coneconstrained convex optimization problem is discussed.It is proved that proper and lower semi-continuous convex function satisfying second-order growth condition has KL property.Proximal algorithm for solving unconstrained equivalence problems is constructed,and the convergence of the proximal algorithm is analyzed based on the KL inequality.更多还原