In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective function is not a linear function. An optimization problem is one of calculation of the extrema of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.
The general algebraic modeling system (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms. Models are portable from one platform to another.
IPOPT, short for "Interior Point OPTimizer, pronounced I-P-Opt", is a software library for large scale nonlinear optimization of continuous systems.
The Galahad library is a thread-safe library of packages for the solution of mathematical optimization problems. The areas covered by the library are unconstrained and bound-constrained optimization, quadratic programming, nonlinear programming, systems of nonlinear equations and inequalities, and non-linear least squares problems. The library is mostly written in the Fortran 90 programming language.
The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.
Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation. One particular advantage of some algebraic modeling languages like AIMMS, AMPL, GAMS, Gekko, MathProg, Mosel, and OPL is the similarity of their syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization, which is supported by certain language elements like sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of a model does not contain any hints how to process it.
FortMP is a software package for solving large-scale optimization problems. It solves linear programming problems, quadratic programming problems and mixed integer programming problems. Its robustness has been explored and published in the Mathematical Programming journal. FortMP is available as a standalone executable that accepts input in MPS format and as a library with interfaces in C and Fortran. It is also supported in the AMPL modeling system.
nl is a file format for presenting and archiving mathematical programming problems. Initially, this format has been invented for connecting solvers to AMPL. It has also been adopted by other systems such as COIN-OR, FortSP, and Coopr.
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective, but the augmented Lagrangian method adds yet another term designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with, the method of Lagrange multipliers.
WORHP, also referred to as eNLP by ESA, is a mathematical software library for numerically solving large scale continuous nonlinear optimization problems.
APOPT is a software package for solving large-scale optimization problems of any of these forms:
SAMPL, which stands for "Stochastic AMPL", is an algebraic modeling language resulting by expanding the well-known language AMPL with extended syntax and keywords. It is designed specifically for representing stochastic programming problems and, through recent extensions, problems with chance constraints, integrated chance constraints and robust optimization problems. It can generate the deterministic equivalent version of the instances, using all the solvers AMPL connects to, or generate an SMPS representation and use specialized decomposition based solvers, like FortSP.
Convex Over and Under ENvelopes for Nonlinear Estimation (Couenne) is an open-source library for solving global optimization problems, also termed mixed integer nonlinear optimization problems. A global optimization problem requires to minimize a function, called objective function, subject to a set of constraints. Both the objective function and the constraints might be nonlinear and nonconvex. For solving these problems, Couenne uses a reformulation procedure and provides a linear programming approximation of any nonconvex optimization problem.
Deterministic global optimization is a branch of numerical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within some predefined tolerance. The term "deterministic global optimization" typically refers to complete or rigorous optimization methods. Rigorous methods converge to the global optimum in finite time. Deterministic global optimization methods are typically used when locating the global solution is a necessity, when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the best possible solution to a problem.
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.
The NEOS Server is an Internet-based client-server application that provides free access to a library of optimization solvers. Its library of solvers includes more than 60 commercial, free and open source solvers, which can be applied to mathematical optimization problems of more than 12 different types, including linear programming, integer programming and nonlinear optimization.
Algebraic modeling languages like AIMMS, AMPL, GAMS, MPL and others have been developed to facilitate the description of a problem in mathematical terms and to link the abstract formulation with data-management systems on the one hand and appropriate algorithms for solution on the other. Robust algorithms and modeling language interfaces have been developed for a large variety of mathematical programming problems such as linear programs (LPs), nonlinear programs (NPs), mixed integer programs (MIPs), mixed complementarity programs (MCPs) and others. Researchers are constantly updating the types of problems and algorithms that they wish to use to model in specific domain applications.
The GEKKO Python package solves large-scale mixed-integer and differential algebraic equations with nonlinear programming solvers. Modes of operation include machine learning, data reconciliation, real-time optimization, dynamic simulation, and nonlinear model predictive control. In addition, the package solves Linear programming (LP), Quadratic programming (QP), Quadratically constrained quadratic program (QCQP), Nonlinear programming (NLP), Mixed integer programming (MIP), and Mixed integer linear programming (MILP). GEKKO is available in Python and installed with pip from PyPI of the Python Software Foundation.
HiGHS is open-source software to solve linear programming (LP), mixed-integer programming (MIP), and convex quadratic programming (QP) models.
