Closely related (although not equivalent) to minimax optimization problems is the following:


$$\min\_{x \in \Omega} \min\_{i=1,...,q} f\_i (x).$$ Here, $\Omega \subset \Bbb R^n$ and $f\_i: \Bbb R^n \to \Bbb R$ is continuously differentiable. I am looking for references on algorithms for this kind of problems. Specifically, I am interested in steepest- descent -like methods. Can you suggest a good reference/survey?



On solution methods for min-min optimization problems
This comment is too long so I directly post it as an answer. It is not hard to propose such a method. 


**Assumption**: Every $f\_i(x)$ is Lipschitz differentiable with constant $L$, and define $h(x):=\min\_{i=1,...,q}f\_i(x)$. Suppose further that $h(x)$ is bounded from below. 


First, it is easy to prove:


**Optimality condition**:Assume for every $j$ such that $f\_j(z)=h(z)$ we have $\nabla f\_j(z)=0$, then $h(x)$ is differentiable at $z$ and $\nabla h(z)=0$. 


Second, start from $x\_k$, let $f\_{j\_k}(x\_k)=h(x\_k)$, and $\|\nabla f\_{j\_k}(x\_k)\|$ is the largest one among the index set such that $f\_j(x\_k)=h(x\_k)$. Compute $x\_{k+1}:=x\_k-\frac{1}{L}\nabla f\_{j\_k}(x\_k)$. Then you could prove that $h(x\_k)\leq h(x\_0)-\frac{1}{2L}\sum\_{i=0}^{k-1}\|\nabla f\_{j\_i}(x\_i)\|^2$. Push to the limit, it is easy to prove that any cluster point of $x\_k$ is a stationary point of $h(x)$. 


By the way, in general, I think your problem is a special form of bilevel optimization.


