How do you navigate nonlinear programming challenges?

1 What is NLP and why is it hard?

NLP is a generalization of linear programming (LP), where the objective function and/or the constraints can be nonlinear. This allows NLP to model more realistic and complex situations, such as production functions, utility functions, or neural networks. However, this also makes NLP harder to solve, because nonlinear functions can have multiple local optima, non-convex regions, or discontinuities. Moreover, NLP problems can have many variables and constraints, which increases the computational complexity and the difficulty of finding feasible solutions.

2 How to find global optima?

One of the main challenges of NLP is finding the global optimum, which is the best possible solution among all the local optima. However, finding the global optimum is often intractable or very expensive, especially for non-convex problems. Therefore, one has to resort to heuristic methods, such as simulated annealing, genetic algorithms, or tabu search, that explore the search space and try to escape from local optima. Alternatively, one can use convex relaxation techniques, such as Lagrangian relaxation, that transform the original problem into a convex one and provide a lower bound for the global optimum.

4 How to solve large-scale models?

A third challenge of NLP is solving large-scale models, which have a high number of variables and constraints. Large-scale models can arise in many applications, such as portfolio optimization, data mining, or image processing. However, large-scale models can be computationally demanding and hard to store in memory. Therefore, one has to use special techniques to solve them, such as decomposition methods, that split the problem into smaller and easier subproblems and coordinate their solutions. Another technique is to use gradient-based methods, that use the information of the first or second derivatives of the objective function to find the optimal direction and step size.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?