Accelerating the pace of engineering and science

# Optimization Toolbox

## Linear and Quadratic Programming

Optimization Toolbox can solve large-scale linear and quadratic programming problems.

### Linear Programming

Linear programming problems involve minimizing or maximizing a linear objective function subject to bounds, linear equality, and inequality constraints. Linear programming is used in finance, energy, operations research, and other applications where relationships between variables can be expressed linearly.

Optimization Toolbox includes three algorithms used to solve linear programming problems:

• The simplex algorithm is a systematic procedure for generating and testing candidate vertex solutions to a linear program. The simplex algorithm is the most widely used algorithm for linear programming.
• The interior point algorithm is based on a primal-dual predictor-corrector algorithm used for solving linear programming problems. Interior point is especially useful for large-scale problems that have structure or can be defined using sparse matrices.
• The active-set algorithm minimizes the objective at each iteration over the active set (a subset of the constraints that are locally active) until it reaches a solution.
Linear programming used in the design of a plant for generating steam and electrical power.

Quadratic programming problems involve minimizing a multivariate quadratic function subject to bounds, linear equality, and inequality constraints. Quadratic programming is used for portfolio optimization in finance, power generation optimization for electrical utilities, design optimization in engineering, and other applications.

Optimization Toolbox includes three algorithms for solving quadratic programs:

• The interior-point-convex algorithm solves convex problems with any combination of constraints.
• The trust-region-reflective algorithm solves bound constrained problems or linear equality constrained problems.
• The active-set algorithm solves problems with any combination of constraints.

Optimization in MATLAB: An Introduction to Quadratic Programming 36:35
In this webinar, you will learn how MATLAB can be used to solve optimization problems using an example quadratic optimization problem and the symbolic math tools in MATLAB.

Both the interior-point-convex and trust-region-reflective algorithms are large scale, meaning they can handle large, sparse problems. Furthermore, the interior-point-convex algorithm has optimized internal linear algebra routines and a new presolve module that can improve speed, numerical stability, and the detection of infeasibility.

Quadratic programming used to perform a returns-based style analysis for three mutual funds.