I have come across an optimization problem with the following objective function:

$$f(x_0,y_0,z_0,x_1,y_1,z_1,...,x_N,y_N,z_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i, \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i))$$

i.e. the objective function is the sum of the functions $f_i$ that only depend on three variables $x_i,y_i,z_i$ and on a linear combination of the difference to the direct neighbors (for $x_i$ and $y_i$). So far I have tired an NLCG algorithm, but convergence is very slow.

Is there a specialized solver that can exploit the structure of the optimization problem?

I have thought about introducing new variables $s_i = \alpha(x_{i+1}-x_i) + \beta(y_{i+1}-y_i)$ and writing $$f(x_0,y_0,z_0,s_0,...,x_N,y_N,z_N,s_N) = \sum_{i=0}^N f_i(x_i,y_i,z_i,s_i)$$ because then the optimization could be conducted separately for each $f_i$ but I would have to introduce equality constraints to get back the original problem?

$f_i(x_i,y_i,z_i,s_i)$ is nonconvex (there is a $\cos(s_i)$ term in the function); however $f_i(x_i,y_i,z_i)$ is convex.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.