Quadratic programming problems and related linear complementarity problems

This paper investigates the general quadratic programming problem, i.e., the problem of finding the minimum of a quadratic function subject to linear constraints. In the case where, over the set of feasible points, the objective function is bounded from below, this problem can be solved by the minimization of a linear function, subject to the solution set of a linear complementarity problem, representing the Kuhn-Tucker conditions of the quadratic problem.To detect in the quadratic problem the unboundedness from below of the objective function, necessary and sufficient conditions are derived. It is shown that, when these conditions are applied, the general quadratic programming problem becomes equivalent to the investigation of an appropriately formulated linear complementarity problem.This research was supported by the Hungarian Research Foundation, Grant No. OTKA/1044.