论文标题
拉格朗日人和洛克菲尔人之间的二元性
Duality Between Lagrangians and Rockafellians
论文作者
论文摘要
Rockafellar在他的专着\ emph {结合二元性和优化}中提出了一种``扰动 +二重性''方法，以获取针对原始最小化问题的双重问题。 首先，将最小化问题嵌入到一个扰动问题的家族中（从而产生了所谓的扰动功能）。 最近，最小化的原始功能的扰动被称为Rockafellian。 其次，当扰动变量属于以双线性形式配对的原始矢量空间，带有双向矢量空间时，就会从摇滚乐中构建Lagrangian。 一个人还获得了所谓的双重函数（和双重问题）。 该方法已从Fenchel二元性扩展到广义的凸度：当扰动属于由耦合函数的原始集合时，带有二元组时，也会从Lagrangian构建Rockafellian。 遵循这些道路，我们重点介绍了拉格朗日人和洛克菲尔人之间的双重性。 上面提到的材料主要集中于从洛克菲利安人到拉格朗日，我们平等对待它们并显示出双向的公式。 我们提出了Lagrangian-Rockafellian夫妇的定义。 我们将后者表征为双重函数，相对于耦合以及广义凸函数。 扰动和双重功能之间的二元性并不那么明确。
In his monograph \emph{Conjugate Duality and Optimization}, Rockafellar puts forward a ``perturbation + duality'' method to obtain a dual problem for an original minimization problem. First, one embeds the minimization problem into a family of perturbed problems (thus giving a so-called perturbation function); the perturbation of the original function to be minimized has recently been called a Rockafellian. Second, when the perturbation variable belongs to a primal vector space paired, by a bilinear form, with a dual vector space, one builds a Lagrangian from a Rockafellian; one also obtains a so-called dual function (and a dual problem). The method has been extended from Fenchel duality to generalized convexity: when the perturbation belongs to a primal set paired, by a coupling function, with a dual set, one also builds a Rockafellian from a Lagrangian. Following these paths, we highlight a duality between Lagrangians and Rockafellians. Where the material mentioned above mostly focuses on moving from Rockafellian to Lagrangian, we treat them equally and display formulas that go both ways. We propose a definition of Lagrangian-Rockafellian couples. We characterize these latter as dual functions, with respect to a coupling, and also in terms of generalized convex functions. The duality between perturbation and dual functions is not as clear cut.
