As we all know, a classic optimization problem can be represented in the following way:
Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers
Sought: an element $x_0 ∈ A$ such that $f(x_0) \le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0) \ge f(x)$ for all $x \in A$ ("maximization").
However, I am now facing an optimization problem, in which the objective function returns a function instead of a real number. The function has two variables, one is the input variable $x$, the other is an unknown variable $z$ within a known interval. The problem can be described as follows:
Given: a function $f: A \rightarrow \mathcal{C}(x, z)$ from some set $A$ to a function space $\mathcal{C}(x, z)$ where $z \in [a, b]$
Sought: a maximum subset $B \subseteq A$, where for each element $x_0 ∈ B$, there exists at least one interval of $z \in [c, d]$ $(a \le c \le d \le b)$, such that $f(x_0, z) \le f(x, z)$ for all $x \in A$ ("minimization") or such that $f(x_0, z) \ge f(x, z)$ for all $x \in A$ ("maximization").
To give a concrete example, say I want to minimize the objective function $f(x, z) = xz^3 - \frac{1}{x} z^2 + z$ where $z \in [-1, 1]$. In the end, I want to get the lowest curve of $f(x, z)$ (scan $z$ from $-1$ to $1$ and connect all lowest parts with different $x$). In my case, I have a finite set of $x$s. The expected result is an optimal subset of $x$s and the corresponding intervals at which $f(x,z)$ is the lowest.
Can anyone suggest how should I approach this kind of problem? I cannot find any literature related to this.
