# What is the difference between linear and nonlinear programming?

The main difference between linear and nonlinear programming is that linear programming helps find the best solution from a set of parameters or requirements that have a linear relationship, while nonlinear programming helps find the best solution from a set of parameters or to find requirements that have a nonlinear relationship.

Linear programming is an important concept in optimization techniques in mathematics as it helps to find the most optimized solution to a given problem. On the other hand, nonlinear programming is the mathematical method for finding the optimized solution taking into account nonlinear constraints or objective functions.

### Key areas covered

1. What is linear programming - definition, functionality 2. What is nonlinear programming - definition, functionality 3. What is the difference between linear and nonlinear programming - comparison of the main differences

### key terms

Linear programming, nonlinear programming

## What is linear programming?

Linear programming is a mathematical method that makes it possible to determine the best possible result or solution from a given set of parameters or a set of requirements. These parameters or requirements are linearly related. In addition, the main advantage of linear programming is that it helps in performing modeling or simulations to find the best solutions according to the available money, energy, resources, time, space and other related factors or variables. In addition, the result of linear programming is maximizing profit and reducing costs.

Figure 1: Example graph for linear programming

The basic components of linear programming are as follows.

Decision variables - quantities to be determined

Objective Function - Describes how each decision variable affects the property to be optimized

Constraints - Represents how any decision variable would consume limited amounts of resources

Data - Explains the relationships between the objective function and the constraints

## What is nonlinear programming?

Nonlinear programming is the process of solving optimization problems that affect some of the nonlinear constraints or nonlinear objective functions. It involves minimizing or maximizing a nonlinear objective function subject to bound constraints, linear constraints, nonlinear constraints, and so on. These constraints can be inequalities or equality. In addition, nonlinear programming helps in the analysis of design compromises, the selection of optimal designs, the calculation of optimal trajectories, as well as portfolio optimization and model calibration in computation finance.

Figure 2: Example graph for nonlinear programming

There are two types of nonlinear programming as follows.

### Unrestricted non-linear programming

Unconstrained nonlinear programming involves finding a vector x that is a local minimum of the nonlinear scalar function f (x). Quasi-Newton, Nelder Mead, and Trust Region are some common unconstrained nonlinear programming algorithms.

### Limited non-linear programming

Bounded nonlinear programming involves finding a vector x that minimizes a nonlinear function f (x) while observing one or more constraints. Inner point, sequential quadratic programming, and confidence region reflection are some common constrained nonlinear programming algorithms.

## Difference between linear and non-linear programming

### definition

Linear programming is a method of getting the best result in a mathematical model whose requirements are represented by linear relationships, while nonlinear programming is a process of solving an optimization problem where the constraints or objective functions are nonlinear. So this is the main difference between linear and non-linear programming.

### purpose of use

In addition, linear programming helps find the best solution to a problem using conditions that are linear, while nonlinear programming helps find the best solution to a problem using nonlinear conditions.

### diploma

The main difference between linear and nonlinear programming is that linear programming helps find the best solution from a set of parameters or requirements that have a linear relationship, while nonlinear programming helps find the best solution from a set of parameters or to find requirements that have a nonlinear relationship.

##### Reference:

1. “What is linear programming (LP)? - Definition of Techopedia. ”Techopedia.com, available here .2. “Nonlinear Programming.” Wikipedia, Wikimedia Foundation, October 26, 2018, available here .

##### Image courtesy:

1. “Example graph for linear programming” By en: User: Jacj - Own work by en: User: Jacj (Public Domain) via Commons Wikimedia 2. “Non-linear programming” By Krishnavedala - Own work (CC0) via Commons Wikimedia