On Symmetric Duality in Nonlinear Programming
On Symmetric Duality in Nonlinear Programming
复制标题
DOI：
10.1287/opre.21.1.1
复制
发表时间：
1973-02
期刊：
影响因子：
--
通讯作者：
M. Bazaraa;J. Goode
中科院分区：
文献类型：
--
作者：
M. Bazaraa;J. Goode
在这项研究中，我们对丹齐格、艾森伯格和科特尔提出的对称对偶形式进行了推广，使其包含不等式类型的约束通过闭凸锥及其极锥来定义的情况。新的形式保留了原始规划的对称性质。在适当的凸性/凹性假设下，我们推广了关于对称对偶的已知结果。还讨论了所涉及函数为强凸/强凹的情况，并推广了卡拉马迪安在这种情况下的结果。结果表明，任何强凸函数在任何闭凸锥上都在唯一一点处取得最小值。然后考虑了对称规划的一些特殊情况，从而推广了沃尔夫对偶以及二次规划和线性规划形式。
In this study we generalize the formulation of symmetric duality introduced by Dantzig, Eisenberg, and Cottle to include the case where the constraints of the inequality type are defined via closed convex cones and their polars. The new formulation retains the symmetric properties of the original programs. Under suitable convexity/concavity assumptions we generalize the known results about symmetric duality. The case where the function involved is strongly convex/strongly concave is also treated and Karamardian's result in this case is generalized. As a result, we show that every strongly convex function achieves a minimum value over any closed convex cone at a unique point. Some special cases of symmetric programs are then considered, leading to generalizations of Wolfe's duality as well as generalizations of quadratic and linear programming formulations.