Introduction

Nonlinear programs (NLPs) are a class of optimization problems in which some of the constraints or the objective function are nonlinear:

\[\begin{align} \min_{x \in \mathbb{R}^n} & f_0(x) \\ \;\;\text{s.t.} & l_j \le f_j(x) \le u_j & j = 1 \ldots m \\ & l_i \le x_i \le u_i & i = 1 \ldots n. \end{align}\]

Mixed-integer nonlinear linear programs (MINLPs) are extensions of nonlinear programs in which some (or all) of the decision variables take discrete values.

How to choose a solver

JuMP supports a range of nonlinear solvers; look for "NLP" in the list of Supported solvers. However, very few solvers support mixed-integer nonlinear linear programs. Solvers supporting discrete variables start with "(MI)" in the list of Supported solvers.

If the only nonlinearities in your model are quadratic terms (that is, multiplication between two decision variables), you can also use second-order cone solvers, which are indicated by "SOCP." In most cases, these solvers are restricted to convex quadratic problems and will error if you pass a nonconvex quadratic function; however, Gurobi has the ability to solve nonconvex quadratic terms.

How these tutorials are structured

Having a high-level overview of how this part of the documentation is structured will help you know where to look for certain things.

• The following tutorials are worked examples that present a problem in words, then formulate it in mathematics, and then solve it in JuMP. This usually involves some sort of visualization of the solution. Start here if you are new to JuMP.
  • Example: nonlinear optimal control of a rocket
  • Example: optimal control for a Space Shuttle reentry trajectory
  • Example: portfolio optimization
• The Computing Hessians is an advanced tutorial which explains how to compute the Hessian of the Lagrangian of a nonlinear program. This is useful only in particular cases.
• The remaining tutorials are less verbose and styled in the form of short code examples. These tutorials have less explanation, but may contain useful code snippets, particularly if they are similar to a problem you are trying to solve.