SIMPLIFIED COPOSITIVE AND LAGRANGIAN RELAXATIONS FOR LINEARLY CONSTRAINED QUADRATIC OPTIMIZATION PROBLEMS IN CONTINUOUS AND BINARY VARIABLES

For a quadratic optimization problem (QOP) with linear equality constraints in continuous non-negative variables and binary variables, vie propose three relaxations in simplified forms with a parameter lambda: Lagrangian, completely positive, and copositive relaxations. These relaxations are obtained by reducing the QOP to an equivalent QOP with a single quadratic equality constraint in nonnegative variables, and applying the Lagrangian relaxation to the resulting QOP. As a result, an unconstrained QOP with a Lagrangian multiplier lambda in nonnegative variables is obtained. The other two relaxations are a primal-dual pair of a completely positive programming (CPP) relaxation in a variable matrix with the upper-left element set to 1 and a copositive programming (CP) relaxation in a single variable. The CPP relaxation is derived from the unconstrained QOP with the parameter lambda, based on the recent result by Arima, Kim and Kojima. The three relaxations with a same parameter value lambda > 0 work as relaxations of the original QOP. The optimal values zeta(lambda) of the three relaxations coincide, and monotonically converge to the optimal value of the original QOP as lambda tends to infinity under a moderate assumption. The parameter lambda serves as a penalty parameter when it is chosen to be positive. Thus, the standard theory on the penalty function method can be applied to establish the convergence.