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:
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C++ example:
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Python example:
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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 built-in support for LP, but you can use third-party libraries like Apache Commons Math
.
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C++ Code
C++ does not have native support for LP. Libraries like GLPK
can be used.
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Python Code
Python has the SciPy
library which provides a convenient function to solve LP problems.
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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 third-party libraries.