Linear Programming

Algebra
process

Also known as: LP, optimization with constraints

Grade 9-12

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Linear programming optimizes a linear objective subject to linear inequality or equality constraints. Linear programming is used in scheduling, logistics, budgeting, and resource allocation worldwide.

Definition

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

💡 Intuition

You search the corners of an allowed region for the best score.

🎯 Core Idea

The optimal solution to a linear program always occurs at a vertex (corner point) of the feasible region — never in the interior.

Example

\max z = 3x+2y subject to x+y \le 4,\; x \ge 0,\; y \ge 0 — optimal at a corner.

Formula

max/min;c^Tx; ext{subject to};Axle b

Notation

max z or min z with linear constraints.

🌟 Why It Matters

Linear programming is used in scheduling, logistics, budgeting, and resource allocation worldwide.

💭 Hint When Stuck

First define your variables and write the objective function. Then list all constraints as inequalities. Graph the feasible region, identify corner points, and evaluate the objective function at each corner to find the maximum or minimum.

Formal View

Find xinmathbb{R}^n that optimizes c^Tx over a polyhedral feasible set {xmid Axle b}.

🚧 Common Stuck Point

Students optimize outside the feasible region or forget to include all constraints when finding corner points.

⚠️ Common Mistakes

  • Forgetting to check all corner points of the feasible region for the optimal value
  • Setting up constraints incorrectly — mixing up \leq and \geq based on the problem context
  • Ignoring the non-negativity constraints (x \geq 0, y \geq 0) when they apply

Frequently Asked Questions

What is Linear Programming in Math?

Linear programming optimizes a linear objective subject to linear inequality or equality constraints.

What is the Linear Programming formula?

max/min;c^Tx; ext{subject to};Axle b

When do you use Linear Programming?

First define your variables and write the objective function. Then list all constraints as inequalities. Graph the feasible region, identify corner points, and evaluate the objective function at each corner to find the maximum or minimum.

How Linear Programming Connects to Other Ideas

To understand linear programming, you should first be comfortable with inequalities, systems of equations and constraint system.

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