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在最優化理論中的對偶（duality）或對偶性原則（duality principle）是指最佳化問題可以用兩種觀點來看待的理論，兩種觀點分別是「原始問題」（primal problem）及「對偶問題」（dual problem）。
对偶，是我们在最优化里面经常会遇到的一个名词，尤其是在机器学习里面，在优化模型的最后都会使用到一个对偶，比如SVM、最大熵、Fisher 鉴别都会使用到对偶；同时，在很多其他问题，例如线性规划也存在对偶问题。 本人刚刚接触这东西的时候，感觉很奇妙。 妙在对偶问题一般会使得优化问题变得更加容易求解，奇在 为什么对偶形式是那样子写？ KKT条件又是什么鬼···. 本专题就来细细的分析，对偶问题的到底是怎么回事，它其实一点都不神秘。 对偶问题. 从一般问题开始说起. 考虑下面一个一般的带约束的最优化问题： 我们要优化的目标函数是 f 0(x) ,它可以是任意的可导函数，我们的目标是最小化这个函数。 约束条件有两类： m 个不等式约束,以及 p 个 等式约束。
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa).
The dual problem is simply to evaluate $h^{**}(0)$. In other words, the dual problem is: \begin{equation*} \operatorname*{maximize}_{y^* \in Y^*} \quad - h^*(y^*). \end{equation*} We see again the fundamental role that the dual space plays here. It is enlightening to
2021年6月7日 · Every linear programming problem can be thought of as a primal problem which can be converted to its corresponding dual problem. This means solving problem P1 (Primal) is equivalent to solving problem P2 (Dual of P1).
2020年8月3日 · 2 Answers. Sorted by: 17. Here are some uses of the dual problem. Understanding the dual problem leads to specialized algorithms for some important classes of linear programming problems. Examples include the transportation simplex method, the Hungarian algorithm for the assignment problem, and the network simplex method.
The Lagrange dual problem is, min b T ν ∥ ν ∥ ∗ ≤ 1 A T ν = 0 , \begin {array} {ll} \min &b^T \nu \\ & \| \nu \|_* \leq 1 \\ & A^T \nu = 0, \end {array} minbT ν ∥ν ∥∗ ≤1AT ν =0, where we use the fact that the conjugate of a norm is the indicator function of the dual norm unit ball.
