As we all know, a classic optimization problem can be represented in the following way:
Given a function $f: A \to \mathbb{R}$, find an element $x_0 \in A$ 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 dealing with an optimization problem in which the objective function returns a function instead of a real number. The function has multiple variables, one is the unknown variable $x$ within a known interval, the others are input parameters that belongs to a parameter set $\theta$. The problem can be described as follows:
Given a function $f: A \to \mathcal{F}(x|\theta)$ from some parameter sets $A$ to a function space $\mathcal{F}(x|\theta)$ where $x \in [k, l]$, find a maximum subset of parameter sets $B \subseteq A$, where for each parameter set $\theta_0 \subseteq B$, there exists at least one interval of $x \in [m, n]$ $(k \le m < n \le l)$, such that $f(x|\theta_0) \le f(x|\theta)$ for all $\theta \subseteq A$ ("minimization") or such that $f(x|\theta_0) \ge f(x|\theta)$ for all $\theta \subseteq A$ ("maximization").
I know it might be very confusing from the definition above, so I want to give a concrete example.
Example: suppose I want to minimize an objective function, e.g., $f : [0,1] \to \Bbb R$ defined by
$$f(x) = ax^3 - \frac{x^2}{b} + \sqrt{c}x - \frac{d^2}{x+1} + e\sin{x}$$
where $\theta = \{a, b, c, d, e\}$ is the parameter set. In the end, I want to get the lowest envelope of $f$ over $[0,1]$ and connect all lowest parts with different parameter sets $\theta_0 = \{a_0,b_0,c_0,d_0,e_0\}$. In my case, I have a finite number of parameter sets. I am using an evolutionary algorithm to do a global optimization. The expected result is a subset of optimal parameter sets $\theta_0$ and the corresponding intervals at which $f(x|\theta_0)$ is the lowest envelope. As shown below, curves with different colors represent $f(x|\theta)$ with different parameter sets $\theta$. In each generation of the evolutionary algorithm, the current lowest envelope is highlighted in bold, and the set of all parameter sets $\theta$ that contribute to the lowest envelope is listed in the legend. We can see that the lowest envelope keep going down as the global optimization evolves. Hence this problem can also be seen as minimizing $$\int_0^1 f(x | \theta) \, {\rm d} x$$ i.e., the area enclosed by the lowest envelope and the $x$-axis within the $[0,1]$ interval. The most important thing to notice is: all colored curve segments that contributes to the lowest envelope are equally important.
My questions is: what is the name for this type of optimization problem? I am thinking of terms like "multi-objective optimization", "multimodal optimization", "multivariate optimization", "multidimensional optimization", "convex optimization" or combine, but neither of them feel quite right. Any suggestions?
