## Quadratic Programming |

Quadratic programming (QP) involves minimizing or maximizing an objective function subject to bounds, linear equality, and inequality constraints. Example problems include portfolio optimization in finance, power generation optimization for electrical utilities, and design optimization in engineering.

Quadratic programming is the mathematical problem of finding a vector $x$ that minimizes a quadratic function:

$\underset{x}{\mathrm{min}}\mathrm{}\{\frac{1}{2}x{}^{T}Hx+f{}^{T}x\}$

Subject to the linear constraints:

$Ax\le b$ | (inequality constraint) |

${A}_{\mathrm{eq}}x={b}_{\mathrm{eq}}$ | (equality constraint) |

$\mathrm{lb}\le x\le \mathrm{ub}$ | (bound constraint) |

The following algorithms are commonly used to solve quadratic programming problems:

**Interior-point-convex:**solves convex problems with any combination of constraints**Trust-region-reflective:**solves bound constrained or linear equality constrained problems**Active-set:**solves problems with any combination of constraints

For more information about quadratic programming, see Optimization Toolbox.

- Large-Scale Quadratic Programming (Example)
- Using Quadratic Programming on Portfolio Optimization Problems (Example)
- Quadratic Minimization with Bound Constraints (Example)
- Quadratic Minimization with a Dense but Structured Hessian (Example)
- Large Sparse Quadratic Programming with Interior Point Algorithm (Example)
- Introduction to Optimization Graphical User Interface 6:08 (Video)

- quadprog Function in Optimization Toolbox (Functions)
- Interior Point Convex Algorithm (Documentation)
- Trust Region Reflective Algorithm (Documentation)
- Active Set Algorithm (Documentation)

*See also*: *Optimization Toolbox*, *Global Optimization Toolbox*, *linear programming*, *nonlinear programming*, *multiobjective optimization*, *genetic algorithm*, *simulated annealing*