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I refer to the following paper: (fix the wrong link) https://papers.nips.cc/paper/5723-adaptive-primal-dual-splitting-methods-for-statistical-learning-and-image-processing.pdf

In that paper, we consider the saddle point problem: $$ \min_x \max_y f(x)+y^TAx - g(y) $$

Then, this paper says that variational inequality formulation is that $$ h(u) - h(u^*) + (u-u^*)^TQ(u^*)\geq 0, \quad\forall u\in\Omega \tag{18} $$

where $h(u) = f(x)+g(y)$ and $u=(x,y)$.

However, I am very confused after reading following material in 62page.

http://meeting.xidian.edu.cn/workshop/miis2012/uploads/files/20120409/20120409170826.pdf

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I don't know why $h(u)$, instead of $\nabla h(u)$, appears in (18). Is there anyone who help me?

(added) What I understood is that the solution $x^*$ of the following problem for a convex set $K$, $$ \min_x f(x) \quad \text{s.t. } x\in K $$ satisfy the following: $$ (y-x^*)^T\nabla f(x^*)\geq 0,\quad \forall y\in K $$

So I wonder why $(u-u^*)\nabla h(u^*)$ doesn't appear in (18).

jakeoung
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  • Are you sure about the first link ? The paper doesn't even mention the word "variational" ... – dohmatob May 01 '16 at 18:45
  • I am sorry to write incorrect link. I have fixed the link. Thanks for pointing out this. – jakeoung May 02 '16 at 05:24
  • The necessary inequality (18) has derivatives "hidden" in it. Indeed, the first-order optimality conditions $ Ax^* \in \partial g(y^)$ and $-A^Ty^ \in \partial f(x^),$ for $u^ := (x^, y^)$ to be a saddle-point, are equivalent to inequalities (16) and (17) respectively. – dohmatob May 02 '16 at 06:18
  • I don't understand. Could you explain in more detail about "hidden" derivatives? – jakeoung May 02 '16 at 06:47
  • The inequality (18) involving no derivatives was arrived at by subtracting the first-order inequalities (16) and (17), which involve derivatives (in fact, subdifferentials). Why would you expect $\nabla h$ to appear in (18) ? Maybe I'm missing something in your question ? – dohmatob May 02 '16 at 07:52
  • Thank you for answering. I've added my question in response to your comment. – jakeoung May 02 '16 at 08:23

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