There is a lot of literature about numerical methods for finding ( $\bf{x}\in \bf{R}^n$, $\ f$ and $g_1, \ldots,g_k$ sufficiently differentiable):
$ \min f(\bf{x}) $ under $ \cap_{i=1}^k \{g_i({\bf{x}}) \le 0\}$ -AND condition-
but is there anything about
$ \min f(\bf{x}) $ under $ \cup_{i=1}^k \{ g_i({\bf{x}}) \le 0\}$ -OR condition-?
This appears in reliability problems where the first condition describes a parallel system and the second a series system.
I didn't find anything till now.
The AND operator on the constraints makes the set of constraints a system of non-linear equations. When we have an OR operator suddenly we have a no longer such a thing. Typically this is handled using binary variables: a binary variable $b_i$ is used to indicate whether equation $i$ is turned on ($b_i=0$ means ignore this constraint, and $b_i=1$ means keep the constraint): $$ g_i(x) \le (1-b_i)M_i $$ where $M_i$ is a large enough constant to make this equation non-binding (think of it as an upper bound on $g_i$). Finally we add: $$ \sum_i b_i \ge 1 $$ to require that at least one constraint should hold. This makes the model a MINLP (Mixed Integer Non-Linear Program). (For an example related to reliability see here). There are nowadays solvers that can handle those type of problems. You also see that those problems are "solved" using heuristics (including meta-heuristics). These heuristics provide "good" solutions instead of proven optimal solutions. Many reliability models I see try to keep the functions linear using more or less ingenious modeling tricks. Here is an example. This will allow the model to be solved as a straight linear MIP (Mixed Integer Program) for which powerful solvers are available.