Nonlinear Programming (NLP)
Nonlinear Programming (NLP) is a broad and fundamental class of optimization problems in which the objective function, the constraints, or both are nonlinear. NLP generalizes Linear Programming (LP) by allowing curved objective functions and feasible regions, making it possible to model a much wider range of real-world phenomena.
Nonlinear Programming (NLP) is central to fields such as engineering design, economics, machine learning, energy systems, finance, and control, where relationships between variables are rarely purely linear.
What is Nonlinear Programming (NLP)?
Formally, a Nonlinear Programming (NLP) problem seeks to optimize an objective function subject to constraints, where at least one of these elements is nonlinear. A general NLP can be written as:
- Minimize or maximize a nonlinear objective function
- Subject to nonlinear equality and/or inequality constraints
- With decision variables typically taking continuous values
Unlike Linear Programs (LP), the feasible region of an NLP is not necessarily a polytope and may be nonconvex, disconnected, or curved. This geometric complexity has profound implications for both theory and algorithms.
If all functions involved are linear, the problem reduces to a linear program. If integer variables are added to an NLP, the problem becomes a Mixed-Integer Nonlinear Program (MINLP).
Convex vs. nonconvex NLP
A key distinction within Nonlinear Programming (NLP) is between convex and nonconvex problems.
- In convex NLP, the objective function is convex and the feasible region is a convex set. These problems enjoy strong theoretical properties: any local optimum is also a global optimum, and efficient algorithms with convergence guarantees exist.
- In nonconvex NLP, local optima may not be global, and finding a global solution is generally much harder.
This distinction strongly influences the choice of solution methods and the guarantees that can be provided.
Solving Nonlinear Programs
Nonlinear Programming (NLP) problems are typically solved using continuous optimization algorithms, including:
- Gradient-based methods, such as steepest descent and Newton-type methods,
- Sequential Quadratic Programming (SQP), which solves a sequence of quadratic approximations,
- Interior-Point methods extended from linear and convex optimization,
- Augmented Lagrangian methods, which combine penalty and dual approaches.
For nonconvex problems, these methods often find locally optimal solutions. Global optimization techniques such as spatial branch-and-bound, interval methods, or multi-start strategies are required when global optimality is important.
When derivatives are unavailable or unreliable, derivative-free optimization and surrogate-based methods may be used.
Extensions and related problem classes
Nonlinear programming occupies a central position in optimization:
- It generalizes Linear Programming (LP) by allowing nonlinear relationships.
- It serves as the continuous counterpart to Mixed-Integer Nonlinear Programming (MINLP).
- Convex NLP forms the theoretical foundation of much of modern optimization and underpins many algorithms used in LP, QP, and conic optimization.
Many complex optimization problems are first modeled as NLPs before being approximated, relaxed, or reformulated into other problem classes for tractability.
Further reading and references
Kuhn, H. W., & Tucker, A. W. (1951). Nonlinear Programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability.
A foundational paper that formalized the KKT conditions and established nonlinear programming as a distinct mathematical optimization discipline.
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.
A foundational text for convex nonlinear programming, providing the mathematical framework underlying convex optimization, duality theory, and modern algorithm design.
Luenberger, D. G., & Ye, Y. (2008). Linear and Nonlinear Programming (3rd ed.). Springer.
A classic and widely cited textbook covering the theory of nonlinear programming, including duality, optimality conditions, and convergence analysis, with strong connections to linear and convex optimization.