# Min max of quadratic function

$$\min_{x \in \mathbb{R}^n} \max_{1 \leq i \leq k} \frac 1 2 x^T Q_i x + b^T_i x + c_i$$ $$Ax = d$$

$$Q_i$$ are positive definite matrices. Find dual problem.

I found out that this problem called quasiconvex and can be solved by moving "max" part into constraints section. So I did this:

$$\min \gamma$$ $$\frac 1 2 x^T Q_i x + b^T_i x + c_i \leq \gamma$$ $$Ax = d$$

But there are two things I am confused about:

• The constraint is not linear
• I am not sure about how to minimise $$\gamma$$
• The original problem is convex, not just quasiconvex. You minimize $\gamma$ with a QCQP solver. – LinAlg Nov 1 '19 at 12:41