# How to solve this problem by Lagrange duality?

This is a convex problem and although it can be well solved by CVX, I want to know how it can be solved by the Lagrange duality method. The derivations with regard to $$L_k$$ and $$B_k$$ are constants, which means the closed-form solution cannot be presented. I'm confused and I need some insights that help me understand the problem deeply. I will give my honest appreciation to these who clear up the doubts for me.

• Welcome to OR.SE! Please take a look at our help here and try to write your problem using MathJax. – EhsanK Sep 7 '20 at 0:53
• It is extremely unlikely that this has a closed form solution. – ErlingMOSEK Sep 7 '20 at 4:33
• I agree with @ErlingMOSEK (unsurprisingly). If you want to get a closed-form solution you need to get rid of the log. My suggestion would be to write out the KKT conditions and try something there, but it's going to be really tough. If you manage, please do share that here :) – Richard Sep 7 '20 at 7:40
• Even if you replace the log by a linear function it will be hard. Because then you have an LP. That is my argument. – ErlingMOSEK Sep 7 '20 at 8:01
• What is your reason for wanting to solve this by :Lagrange Duality? What are you trying to accomplish which can't be done by numerical optimization, such as with CVX? See the "Edit" portion of my answer or.stackexchange.com/questions/4784/… for reference on how CVXPY can be used to insert (the solution to) this optimization problem as a differentiable modeling layer in PyTorch or TensorFlow models. – Mark L. Stone Sep 7 '20 at 12:10