In Boyd's convex optimization book, he gives an example of weak duality where dual optimal is strictly less than primal optimal. He says that the intuition behind this is that we were able to hide the solution inside a dent in G (G is the gray area) duality gap
In this case we clearly have $p* > g(\lambda)$ where $g(\lambda) = \inf_x (f_0(x) + \sum_{i}^{m} \lambda_i . f_i(x))$
What if, instead of taking $f_1(x)$ we build the symetric?
$$g(\lambda) = \inf_x (f_0(x) + \sum_{i}^{m} \lambda_i . (f_i(x) \mathbb{1}_{[-\infty, 0]} (x) + (f_i(-x) \mathbb{1}_{]0, +\infty]} (x) ) )$$
Geometrically that should give us something like this: duality gap with symetric f1. The green line is the new hyperplane $\lambda t + u = g(\lambda)$
Now my questions are:
- Could it be even useful or interesting?
- Would there be any problem with differentiability at $u=0$ ?
- Could you provide an example of $(t,u)=(f_0(x),f_1(x))$ that would give an asymmetric gray area like in the first image so I can play with it?