Overview
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two pers…
Linear case
Linear programming problems are optimization problems in which the objective function and the constraints are all linear. In the primal problem, the objecti…
Nonlinear case
In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply.
History
According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the lin…
Applications
In support vector machines (SVMs), the formulating the primal problem of SVMs as the dual problem can be used to implement Kernel trick, but the …
Here's what's really going on with the dual problem. (This is my attempt to answer my own question, over a year after originally asking it.) (A ve...Best answer · 150
I'll take a crack at a couple of these questions (some of them are hard and would require more thought). 1) Here's a nice economic interpretation o...52
Consider the problem
$$
\begin{aligned}
\mbox{min} \quad& f(x) \\
\mbox{subject to} \quad& x\le a\\
\end{aligned}
$$
illustrated below and where $f...52
For #4 and #5, see "The concept of duality in convex analysis, and the characterization of the Legendre transform" by Artstein-Avidan and Milman.12
after reading and learning from this excellent discussion, I summarised an explanation based on my own background. If I made any mistakes, please c...9
A lot of great explanations. The easiest way to get in-depth knowledge, I think, however, is just to study chapter 5 of this book , written by Sta...5
There's a lot of great answers, but they seem to require some understanding of the problem already - so below I write a very quick and basic deduct...5
Here are some counter-examples to help you understand KKT conditions and strong duality. The answer is from my other post: https://math.stackexcha...4