Complementarity problems is **complementarity problems is** a class of mathematical problems focused on finding vectors that satisfy a system of conditions, typically including inequalities and a 'complementarity' requirement where the product of paired variables is zero. As a research field, it investigates the theoretical properties, such as existence and uniqueness of solutions, and develops computational algorithms for these problems, which are characterized by their structure involving coupled inequalities and an orthogonality condition between the vectors. Complementarity problems are significant due to their capacity to model and unify diverse applications in scientific computing, engineering, economics, and operations research, serving as generalizations of problems like linear and quadratic programming and game theory.