nonlinear programming

nonlinear programming is a subfield of mathematical optimization that deals with problems where the objective function or the constraints are nonlinear. It extends the principles of linear programming to more complex, real-world scenarios where relationships are not strictly proportional. The field has foundational contributions from pioneers like Harold W. Kuhn and Albert W. Tucker, who developed key optimality conditions. Nonlinear programming is essential in numerous scientific and engineering disciplines, including operations research, control theory, and economics.

Introduction

The study of nonlinear programming emerged prominently in the mid-20th century, building upon the framework established by George Dantzig for linear programming. Early theoretical advances were made by scholars such as William Karush, whose earlier work was later recognized alongside the contributions of Harold W. Kuhn and Albert W. Tucker. The development of the field was further propelled by the establishment of important numerical methods and the increasing computational power available through institutions like IBM and research at universities such as Stanford University. Fundamental texts, like those by David G. Luenberger, helped codify the discipline, which now interfaces with areas like convex optimization and calculus of variations.

Problem formulation

A standard nonlinear programming problem involves minimizing or maximizing an objective function subject to a set of constraints. The general form can be expressed using mathematical notation often associated with the work of Joseph-Louis Lagrange. Problems are frequently categorized by the properties of their functions, such as those studied in convex analysis, a field advanced by Rockafellar. Important special cases include quadratic programming, where the objective is quadratic, and problems with only bound constraints. The formulation often requires specifying domains, which may involve spaces studied in functional analysis.

Solution methods

Numerous algorithmic strategies have been developed for solving nonlinear programming problems. Classical approaches include Newton's method and its variants, such as the BFGS method developed independently by Broyden, Fletcher, Goldfarb, and Shanno. For constrained problems, methods like the sequential quadratic programming technique are widely used. The interior point method, pioneered by Narendra Karmarkar for linear programs, was successfully extended to nonlinear problems. Other important families of algorithms include gradient descent, trust region methods, and simulated annealing, the latter inspired by work in statistical mechanics.

Optimality conditions

The theoretical foundation for identifying solutions is provided by optimality conditions. The most central are the Karush–Kuhn–Tucker conditions, which generalize the method of Lagrange multipliers to problems with inequality constraints. These conditions rely on concepts from differential calculus and require constraint qualifications, such as the Linear Independence Constraint Qualification. For convex problems, these conditions become sufficient, a key result in the theory developed by Rockafellar and Stephen Boyd. The study of duality, related to the Legendre transformation, is also a crucial aspect of the optimality theory.

Applications

Nonlinear programming has vast applications across many fields. In engineering, it is used for optimal control in systems governed by principles from classical mechanics. In finance, it aids in portfolio optimization, a problem studied by Harry Markowitz. The chemical industry employs it for process optimization in plants designed by companies like Dow Chemical. Other applications include machine learning for training neural networks, trajectory optimization for missions by NASA, and resource allocation in projects managed by the World Bank.

Software and solvers

The practical solution of nonlinear programming problems relies on specialized software. Commercial packages include tools from The MathWorks (MATLAB) and the General Algebraic Modeling System. Open-source solvers like IPOPT, developed primarily at Carnegie Mellon University, are widely used in academia and industry. Other notable software includes the CONOPT solver and the KNITRO package. These tools implement algorithms discussed at conferences like the International Symposium on Mathematical Programming and are integral to workflows in companies such as Boeing and Siemens.

Category:Mathematical optimization