线性单调算子的自共轭表示
Autoconjugate representers for linear monotone operators
摘要：
单调算子在现代最优化和非线性分析中具有重要意义。他们的研究已经彻底改变了最近，由于菲茨帕特里克功能的系统使用。由Penot和Svaiter开创，最近的一个感兴趣的话题是由所谓的自共轭函数表示极大单调算子。有两个明确的结构被提出，第一个由Penot和Zalinescu在2005年提出，另一个由Bauschke和Wang在2007年提出。前者需要轻度约束规格，而后者基于近端平均值。证明了在自反空间上连续线性单调算子的这两个自共轭表示必须重合。连续性和线性假设都是必要的，例如不连续线性算子和次微分算子。此外，我们还构造了实数直线上的恒等算子的自共轭表示的无限族。
英文摘要：
Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions were proposed, the first by Penot and Zalinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we also construct an infinite family of autoconjugate representers for the identity operator on the real line.
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