## Minimize quadratic functions subject to constraints

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:

$\min_{x} \left\{\frac{1}{2}x^{\mathsf{T}}Hx + f^{\mathsf{T}}x\right\}$

Subject to the contraints:

$\begin{eqnarray}Ax \leq b & \quad & \text{(inequality constraint)} \\A_{eq}x = b_{eq} & \quad & \text{(equality constraint)} \\lb \leq x \leq ub & \quad & \text{(bound constraint)}\end{eqnarray}$

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