Projected Gradient Descent with Multiple Constraints for Objective Function

Question

I have the following optimization problem(objective function is non-convex. B and c are unknown variables) $$minimize_{B,C}\sum_{n=1}^N{\Vert\sum_n -\mathbf{B} diag(\mathbf{c}_{n})B^T \Vert_F^2} $$ subject to $$\Vert \mathbf{b}_k \Vert_1 \le \lambda , k=1,...,K $$ $$-1\le\mathbf{b}_k(i)\le1,$$ $$max\vert \mathbf{b}_k(i)\vert =1, i=1,...,P$$ $$\mathbf{c}_n\ge0 , n=1,...,N$$ I want solve it with projected gradient descent and implement with matlab , but it has multiple constraints and I don't know how solve it!!! please help me...

Answer 1

If both $ B $ and $ {c}_{n} $ are to be optimized a simple solution would be to chose $ {c}_{n} = 0 $. Then just chose one element of $ B $ to be 1 or -1.

This will yield cost value of $ 0 $ which for this problem is a lower boundary.