Linear Programming and Mixed-Integer Linear Programming

Solve linear programming problems with continuous and integer variables

Before you begin to solve an optimization problem, you must choose the appropriate approach: problem-based or solver-based. For details, see First Choose Problem-Based or Solver-Based Approach.

For the problem-based approach, create problem variables, and then represent the objective function and constraints in terms of these symbolic variables. For the problem-based steps to take, see Problem-Based Optimization Workflow. To solve the resulting problem, use solve.

For the solver-based steps to take, including defining the objective function and constraints, and choosing the appropriate solver, see Solver-Based Optimization Problem Setup. To solve the resulting problem, use intlinprog when there are extended integer constraints, or use linprog when there are no integer constraints.

Functions

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Solve and Analyze, Problem-Based

`evaluate`	Evaluate optimization expression or objectives and constraints in problem
`findindex`	Find numeric index equivalents of named index variables
`infeasibility`	Constraint violation at a point
`optimproblem`	Create optimization problem
`optimvar`	Create optimization variables
`prob2struct`	Convert optimization problem or equation problem to solver form
`solve`	Solve optimization problem or equation problem

Import and Solve Problems, Solver-Based

`integerConstraint`	Indices of extended integer variables (Since R2025a)
`intlinprog`	Mixed-integer linear programming (MILP)
`linprog`	Solve linear programming problems
`mpsread`	Read MPS file for LP and MILP optimization data

Objects

integerConstraint Indices of extended integer variables (Since R2025a)

Live Editor Tasks

Optimize

Optimize or solve equations in the Live Editor (Since R2020b)

Topics

Problem-Based Mixed-Integer Linear Programming

Mixed-Integer Linear Programming Basics: Problem-Based
Simple example of mixed-integer linear programming.
Factory, Warehouse, Sales Allocation Model: Problem-Based
This example shows how to set up and solve a mixed-integer linear programming problem.
Traveling Salesman Problem: Problem-Based
This example shows how to use binary integer programming to solve the classic traveling salesman problem.
Optimal Dispatch of Power Generators: Problem-Based
This example shows how to schedule two gas-fired electric generators optimally, meaning to get the most revenue minus cost.
Office Assignments by Binary Integer Programming: Problem-Based
This example shows how to solve an assignment problem by binary integer programming using the optimization problem approach.
Mixed-Integer Quadratic Programming Portfolio Optimization: Problem-Based
This example shows how to solve a Mixed-Integer Quadratic Programming (MIQP) portfolio optimization problem using the problem-based approach.
Cutting Stock Problem: Problem-Based
This example shows how to solve a cutting stock problem using linear programming with an integer linear programming subroutine.
Minimize Makespan in Parallel Processing
Minimize the maximum time for a set of processors to complete a group of tasks.
Solve Sudoku Puzzles via Integer Programming: Problem-Based
This example shows how to solve a Sudoku puzzle using binary integer programming.

Solver-Based Mixed-Integer Linear Programming

Mixed-Integer Linear Programming Basics: Solver-Based
Simple example of mixed-integer linear programming.
Factory, Warehouse, Sales Allocation Model: Solver-Based
Example of optimizing logistics in a small supply chain.
Traveling Salesman Problem: Solver-Based
The classic traveling salesman problem, with setup and solution.
Optimal Dispatch of Power Generators: Solver-Based
Example showing how to schedule power generation when there is a cost for activation.
Office Assignments by Binary Integer Programming: Solver-Based
Solve an assignment problem using binary integer programming.
Mixed-Integer Quadratic Programming Portfolio Optimization: Solver-Based
Example showing how to optimize a portfolio, a quadratic programming problem, with integer and other constraints.
Cutting Stock Problem: Solver-Based
Solve a cutting stock problem using linear programming with an integer programming subroutine.
Solve Sudoku Puzzles via Integer Programming: Solver-Based
Sudoku is a type of puzzle that you can solve using integer linear programming.

Problem-Based Linear Programming

Set Up a Linear Program, Problem-Based
Linear problem formulation using the problem-based approach.
Maximize Long-Term Investments Using Linear Programming: Problem-Based
Optimize a deterministic multiperiod investment problem using linear programming and the problem-based approach.
Optimize Green Hydrogen Production System
Optimize a green hydrogen production system.
Optimal Dispatch of Electric Power
Optimize the power generated or dispatched among renewable and nonrenewable generators in a deterministic system using optimization variables.
Create Multiperiod Inventory Model in Problem-Based Framework
Create an inventory model, where stock is carried between time periods, in the problem-based approach.

Solver-Based Linear Programming

Set Up a Linear Program, Solver-Based
Problem formulation using the solver-based approach.
Typical Linear Programming Problem
This example shows the solution of a typical linear programming problem.
Maximize Long-Term Investments Using Linear Programming: Solver-Based
Optimize a deterministic multiperiod investment problem using linear programming.

Code Generation

Code Generation for linprog Background
Prerequisites to generate C code for quadratic optimization.
Generate Code for linprog
Learn the basics of code generation for the linprog optimization solver.

Model and Analyze Linear and Integer Problems

Integer and Logical Modeling
Techniques for modeling with integer constraints using "Big-M" and other techniques.
Investigate Linear Infeasibilities
Find out which linear constraints cause a problem to be infeasible.

Problem-Based Algorithms

Problem-Based Optimization Algorithms
Learn how the optimization functions and objects solve optimization problems.
Supported Operations for Optimization Variables and Expressions
Explore the supported mathematical and indexing operations for optimization variables and expressions.

Solver-Based Algorithms and Options

Linear Programming Algorithms
Minimizing a linear objective function in n dimensions with only linear and bound constraints.
Mixed-Integer Linear Programming (MILP) Algorithms
The algorithms used for solution of mixed-integer linear programs.
Optimization Options Reference
Explore optimization options.
Tuning Integer Linear Programming
Steps for improving solutions or solution time.
intlinprog Output Function and Plot Function Syntax
How to monitor the progress of the intlinprog solution process.

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