MONOTONE OPERATORS AND PROXIMAL POINT ALGORITHM
MONOTONE OPERATORS AND PROXIMAL POINT ALGORITHM
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DOI：
10.1137/0314056
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发表时间：
1976-01-01
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影响因子：
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通讯作者：
ROCKAFELLAR, RT
中科院分区：
文献类型：
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作者：
ROCKAFELLAR, RT
对于Hilbert空间上的下半连续真凸函数的极小化问题，精确形式的邻近点算法产生一个序列，取其中的极小值。这个算法有几个原因，但特别是因为它在某些基于对偶的计算方法中所起的作用，例如非线性规划中的乘子的Hestenes-Powell方法。这里以更一般的形式来研究它，其中在每次迭代时对精确最小化的要求被削弱，而次微分被任意的极大单调算子T所代替。趋同是在几个易于实施的标准下建立的。如果保持足够大，收敛速度是“典型的”线性的，有一个任意好的模数，实际上是超线性的。对……的案件作了更详细的处理。文中还给出了一个与极大极小问题相对应的应用实例。
For the problem of minimizing a lower semicontinuous proper convex functionfon a Hilbert space, the proximal point algorithm in exact form generates a sequenceby takingto be the minimizes of, where. This algorithm is of interest for several reasons, but especially because of its role in certain computational methods based on duality, such as the Hestenes-Powell method of multipliers in nonlinear programming. It is investigated here in a more general form where the requirement for exact minimization at each iteration is weakened, and the subdifferentialis replaced by an arbitrary maximal monotone operatorT. Convergence is established under several criteria amenable to implementation. The rate of convergence is shown to be “typically” linear with an arbitrarily good modulus ifstays large enough, in fact superlinear if. The case ofis treated in extra detail. Application is also made to a related case corresponding to minimax problems.