Semismooth and Semiconvex Functions in Constrained Optimization
Semismooth and Semiconvex Functions in Constrained Optimization
复制标题
DOI：
10.1137/0315061
复制
发表时间：
1977-11
影响因子：
2.2
通讯作者：
R. Mifflin
中科院分区：
文献类型：
--
作者：
R. Mifflin
我们介绍半光滑和半凸函数，并讨论它们关于非光滑非凸约束优化问题的性质。这些函数是局部 Lipschitz 函数，因此具有广义梯度。作者给出了一种优化算法，该算法使用问题函数的广义梯度，如果函数是半光滑的，则收敛到驻点。如果函数是半凸的并且满足约束条件，那么我们表明驻点是最优点。我们证明，连续可微函数的紧凑族上的点状最大值或最小值是半光滑函数，半凸函数的紧凑族上的点状最大值是半凸函数。此外，我们证明半光滑函数的半光滑组合是半光滑的，并给出了广义梯度的链式法则。
We introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. The author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point. We show that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, we show that a semismooth composition of semismooth functions is semismooth and gives a type of chain rule for generalized gradients.