# Is there any closed-form solution for this MaxMin optimization problem?

Is there any closed-form solution for the following optimization problem? \begin{align} &\max_{\mathbf{u}_1,\ldots,\mathbf{u}_M}\min_{i}\mathbf{1}^{\mathrm{T}}\mathbf{u}_i\\ &\text{s.t.}\quad [\mathbf{A}_1,\ldots,\mathbf{A}_M] \begin{bmatrix} \mathbf{u}_1\\ \vdots\\ \mathbf{u}_M \end{bmatrix} \leq\mathbf{1}_{N\times 1}, \end{align} where $$\mathbf{u}_i$$'s are vectors of size $$Q\times 1$$, $$\mathbf{A}_i$$'s are matrices of size $$N\times Q$$ and $$\mathbf{1}$$ is all-one column vector.

Unfortunately, one can not find a closed form answer if no extra information of $$A_1,...,A_M$$ is provided.
Let $$u_i^*$$ be the optimal solition of problem $$min\ 1^T u_i,\ s.t.\ A_i u\le 1$$. You need to choose from $$u_1^*,...,u_M^*$$ the maximum, which clearly cannot be done without knowing $$A_i,0\le i \le M$$.
Moreover, the dual problem of $$min\ 1^T u_i,\ s.t.\ A_i u\le 1$$ is:
$$\begin{array}{lll} \max & & - 1^T \lambda_i\\ s.t. & & 1 + A_i^T \lambda_i = 0\\ & & \lambda_i \geqslant 0 \end{array}$$
Strong duality holds since the original problem is linear. However the problem may be infeasible, and finding the optimal $$\lambda_i^*$$ also need to know $$A_i$$.