I am studying the course Variational inequality and optimisation in analysis,we have the following definition:
Definition:Let $K$ be a nonempty subset of $\mathbb{R^n}$, $F:K\rightarrow 2^\mathbb{R^n}$ be a set-valued map with nonempty values. The generalized variational inequality problem (GVIP) is to find $\bar{x}$$\in K$ and $\bar{u}\in$ F($\bar{x}$) such that $$⟨\bar{u},y−\bar{x}⟩ ≥ 0$$, for all $y\in K$
My teacher ask the following question:
Question:Describe the existence of a Solution of generalized variational inequality problem by an example.
He provide the hint:
Hint:consider a proper lower semicontinuous convex(non-differentiable) map and then find subdifferential $\partial{f}(x)$,state result by which there exists a solution to generalized variational inequality problem to set valued map $\partial{f}$
I can construct a proper lower semicontinuous convex (non-differentiable) map even can find a solution to generalized variational inequality problem but can't judge which result has to be stated given in hint.
My approach:As suggested in hint, I consider the proper lower semicontinuous convex (non-differentiable) map defined by $$f(x) = \begin{cases} 1 & x>0 \\ -1 & x≤0 . \end{cases}$$,the subdifferential of this map at x=0 I found it to be $(-\infty,0]$,now with x=0 and $\xi=0\in\partial{f}(0)=(-\infty,0]$ we have $$⟨\xi,y-x⟩=⟨0,y⟩=0$$ $\forall y\in \mathbb{R}$ hence variational inequality is satisfied but I haven't use any result as suggested by my teacher,can anybody tell me whether I have gone somewhere wrong or missing some concepts
Your professor's hint is a very good one. However, your function is not convex, e.g. it violates the convexity inequaliy:
$1 = f(0.5) = f(\frac{0 + 1}{2}) > \frac{1}{2}(f(0)+f(1)) = \frac{1}{2}.$
Try your professor's hint with a different function, e.g. absolute value $|\cdot|$ or the hinge loss
$h(x) = \begin{cases} 0 &\text{if }x\leq 0 \\ x &\text{if }x>0 \end{cases}$