Lower semicontinuity and partial minimization

Question

Assume that $f(\mathbf{x}, \mathbf{y})$ is lower-semicontinuous extended value function on the closed set $\operatorname{dom}(f) \subseteq \mathbb{R}^n \times \mathbb{R}^m$, where $\operatorname{dom}(\cdot)$ is the effective domain of an extended value function. Suppose also that for any $\mathbf{y}$ the function $f(\cdot, \mathbf{y})$ is bounded below.

Is the following function also lower-semicontinuous on its effective domain: $$ g(\mathbf{y}) = \min_{\mathbf{x}} f(\mathbf{x}, \mathbf{y}) $$

If so, can anyone provide a reference? I was not able to find something appropriate.

In addition, can anybody provide a good reference for the properties of lower-semicontinuity, compactness and boundedness?

Answer 1

No, not necessarily. For example, let $$S = \left\lbrace (x, y) : x > 0, y = \frac{1}{x} \right\rbrace,$$ and consider the function $$f(x, y) = \begin{cases} 0 &\text{if } (x, y) \in S \\ 1 &\text{if } (x, y) \notin S \end{cases}.$$ It's not difficult to see that the epigraph of $f$ is closed, hence $f$ is lsc. If we define $g$ as you have, then we get $$g(y) = \begin{cases} 0 &\text{if } y > 0 \\ 1 &\text{if } y \le 0 \end{cases},$$ which is not lsc at $0$.

Answer 2

The function $g$ is lower semicontinuous under the additional assumption that $f$ is coercive, i.e., for all sequences $x_n$, $y_n$ we have $$ \|x_n\| + \|y_n\| \to \infty \quad\Rightarrow\quad f(x_n, y_n) \to \infty.$$

Indeed, let $y_n \to y$ be given and (wlog) let all $y_n$ belong to the effective domain of $g$ and let $g(y_n)$ be bounded. Then, there exists $x_n$ such that $$ g(y_n) \ge f(x_n, y_n) - 1/n.$$ The coercivity assumption ensures that $x_n$ is bounded. W.l.o.g. $x_n \to x$. Now, the lower semicontinuity of $f$ ensures $$ g(y) \le f(x,y) \le \liminf f(x_n, y_n) \le \liminf g(y_n).$$