How does quadratic programming combine a quadratic objective with linear constraints?

Quadratic programming is a specialized class of nonlinear programming where the optimization problem is defined by a quadratic objective function subject to linear constraints.1

In this framework, the objective function includes terms that are multiples of the product of two variables, such as $3(x_1{)}^2+1.4x_1{x}_2+2(x_2{)}^2$. While the objective function is nonlinear, the constraints remain linear, meaning they are expressed as a system of linear equalities or inequalities (often represented in matrix form as $Ax\le b$).123

This combination is particularly significant in fields like finance, where the risk of an investment portfolio is measured by the variance of its returns—a calculation that results in a quadratic function of the amounts invested in each security. Because the constraints are linear, quadratic programming problems are often more computationally manageable than general nonlinear problems, and specific algorithms have been developed to solve them efficiently. Additionally, sequential quadratic programming is a broader algorithmic technique that solves a sequence of these quadratic-objective, linear-constraint subproblems to navigate more complex nonlinear optimization tasks.14