I have the following problem:

$$\begin{array}{ll} & \boldsymbol{x}^*(t) = \arg \min_{ \boldsymbol{x}}\text{ }g( \boldsymbol{x}) \\ \text{subject to} & \boldsymbol{A}(t)\boldsymbol{x }= \boldsymbol{B}(t)\\ & 0\leq x_i \leq x_{\max} , i=\{1,2,\cdots,N\}\end{array} $$

With $g(\boldsymbol{x})$ being a strictly convex function. $\boldsymbol{x}\in R^{N\times 1}$. $ \boldsymbol{A}(t) \in \mathcal{R}^{M \times N}$ has full row rank with $N>M$. The elements of $ \boldsymbol{A}(t)$ and $ \boldsymbol{B}(t)$ are continuous (possibly smooth if needed) with respect to $t$, and a solution $\boldsymbol{x}^*$ is always feasible for any time t.

The question now is:

Does strict (possibly strong if needed) convexity of $g(\boldsymbol{x})$ this imply that $\boldsymbol{x}^*(t)$ is continuous?

PS: I did ask a similar question previously, where I got a good start. However I am not sure if it is enough yet. ( Is the optimal solution of a convex problem continuous with respect to parameters??)

I have now specified the problem a little more, and my goal is to be able to prove that it is so. Any references along with answers is greatly appreciated.


Though I am not able to formalize it, my thoughts so far is that I somehow can state and use:

1) The solutions of x has a unique minimum

2) The inequalities should restrict the solutions to a convex hull.

3) The equality gives a solution space, that (or so I believe) when changing smoothly also means that the solutions x* change continuously. Possibly involving the implicit function theorem on the KKT conditions.


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