convergence of unconstrained convex optimization

Question

I encounter an optimization problem. The simplified version is like following:

Denote function $F(x):\mathbf{R}^n\rightarrow\mathbf{R}$, where $F(x)$ is a smooth lower bounded convex function (i.e. $F(x)>0$). The hessian matrix of $F(x)$ is positive definite. However $F(x)$ is not a strongly convex function, namely we can not assume the hessian matrix $H(x)\succ m I_n$.

Suppose for the sequence $\{\mathbf{x}_k\}_{k=1}^\infty$, the gradients of the $F(x)$ converge to zero: \begin{equation} \lim_{k\rightarrow\infty} \|\mathbf{g}_k\|_2 = 0,\text{ and }F(x_{k+1})\le F(x_{k}), k=0,1,... \end{equation} where $\mathbf{g}_k=\frac{\partial F}{\partial x} |_{x=x_k}$. Do we have the following result \begin{equation} \lim_{k\rightarrow\infty} F(x_{k}) = \inf _{x\in\mathbf{R}^n} F(x)? \end{equation}

I expect $\lim_{k\rightarrow\infty} F(x_k) = \inf_{x\in\mathbf{R}^n} F(x)$. However, I can not find proofs for this result. There are some results about this problem. But unfortunately, either assume the optimum point of $F(x)$ is attainable, namely, there exists $x^*\in\mathbf{R}^n$, such that $F(x^*) = \inf F(x)$ or there exists $x^*\in\mathbf{R}^n$, such that $\lim_{k\rightarrow\infty} x_k=x^*$.

Could any one help me to solve this problem or give me some comments? Thanks for your help.

Answer 1

What about $$ F(x,y) = \frac{x^2}{y} + \frac{\epsilon}{x^2}, \qquad \nabla F(x,y) = \Big( \frac{2x}{y}-\frac{2\epsilon}{x^3}, -\frac{x^2}{y^2} \Big), \qquad x,y>0$$ with the sequence $(x_k,y_k) = (k, k^2)$? Here $\epsilon$ is a small positive real number that makes the Hessian positive definite instead of semidefinite.

Then $F(x_k,y_k) = 1+\epsilon k^{-2}$ does not converge to the global infimum $\inf F(x,y) = 0$, but the sequence of gradients $$\mathbf{g}_k = \Big(\frac2k-\frac{2\epsilon}{k^3}, -\frac{1}{k^2}\Big)$$ converges to zero.

Informally speaking, the usual proof of convergence based on the inequality $f(y)\geq f(x) + \nabla f(x)^T(y-x)$ would not apply here because while the gradients converge as $k^{-1}$ (not fast enough), the distance between $x_k$ and the closest minimum at $x=0$ diverges as $k$.