Minimising quadratic objective subject to logarithmic inequality constraints

Question

I need to minimise the following function: $$ (2a_1 + 2a_2 - 1)^2 + (2a_1 + 2a_3 - 1)^2 $$ subject to: $$ \sum a_i \log_2 a_i \geq -1 $$ where all the $i \in \{1,2,3,4\}$ and $a_i \in [0,1]$ and $\sum a_i = 1 $

What would be the best way to do this analytically?

Answer 1

Let us choose $a_1 = a_4 = 1/2$ and $ a_2 = a_3 = 0$

This minimizes $ 0 \leq (2a_1 + 2a_2 - 1)^2 + (2a_1 + 2a_3 - 1)^2 = 0$ and satisfies the constraints:

• $\sum a_i = 1 $
• $ \sum a_i \log_2 a_i = \log_2 (1/2) = -1$, where we have exploited that $ \lim_{x\rightarrow 0} x \log_2 x = 0$.