Linear Programming
Linear programming refers to optimizing a linear objective function subject to linear equality and inequality constraints. It has applications across operations research, logistics, scheduling, etc.
The general form is:
Maximize:
c1x1 + c2x2 + … + cn*xn
Subject to:
a11x1 + a12x2 + … + a1n*xn (<=, =, >=) b1
a21x1 + a22x2 + … + a2n*xn (<=, =, >=) b2
…
am1x1 + am2x2 + … + amn*xn (<=, =, >=) bm
Java example:


C++ example:


Python example:


Powerful technique for optimizing allocation problems subject to linear constraints.
Linear Programming (LP) is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. In simpler terms, it’s about finding the best (highest or lowest) value that a linear function can take based on linear equality and inequality constraints.
Algorithm
 Objective Function: Create a linear function that you want to maximize or minimize.
 Constraints: These are the linear equations or inequalities that bound the decision variables.
 Feasible Region: Identify the region where all constraints overlap.
 Optimal Solution: The point(s) in the feasible region where the objective function takes its optimal value.
Java Code
Java does not have builtin support for LP, but you can use thirdparty libraries like Apache Commons Math
.


C++ Code
C++ does not have native support for LP. Libraries like GLPK
can be used.


Python Code
Python has the SciPy
library which provides a convenient function to solve LP problems.


In the Python code, we’re minimizing x  2y
subject to x + y <= 2
and 2x + y <= 3
. The negative sign in result.fun
flips the minimized value to a maximized value, as we intended to maximize x + 2y
.
This covers the basic idea of Linear Programming and how to implement it in different programming languages using thirdparty libraries.