Quadratic optimization is a subfield of mathematical optimization concerned with problems that involve minimizing or maximizing a quadratic objective function. This function is subject to constraints, which are often linear equalities or inequalities, though the field also encompasses problems with more complex constraint sets. The research area investigates the structure of these problems, develops algorithms for computing optimal solutions, and analyzes the theoretical properties and computational complexity of various quadratic optimization formulations, which are fundamental models in numerous applications across science, engineering, and economics.