# how to prove the convergence of piecewise linear function

I have a nonlinear optimization objective function like $$f=\log(\sum_{i\in \mathcal C} k m_i+1) \\ \text{subject to } m_i\leq n_i$$ where both $m_i , n_i$ are binary valued variables. I transform it into a piecewise linear function using tangent line approximation.

I also relax the binary valued constraint sing Lagrange Relaxation technique. The piecewise linear relaxed problem looks like:

Let $r=\frac{m_i}\alpha, \alpha \to 1$ then $$F=\log(r+1)+\log'(r)(m_i-r)-\lambda_i( n_i-m_i)$$

I can solve the above function using any linear solving technique. However, I need to know how I can show that given I solve it, the original nonlinear problem converges?

Thanks

• Why bother with the logarithm? You can just as well optimize the sum, as the logarithm is a simple monotonic transformation. Are you missing a sum outside the logarithm or something? Jul 10, 2017 at 7:15
• There is sum outside the logarithm. In fact, it looks like $\sum_{a\in \mathcal A} \sum_{b\in \mathcal B} \log(\sum_{i\in \mathcal C} k_i m_{i,a,b}+1)$. Moreover, $K$ in $f$ is rate for each $i$ which is also summation function. Jul 11, 2017 at 0:31