Refinement of Lagrangian bounds in optimization problems

Lagrangian constraint relaxation and the corresponding bounds for the optimal value of an original optimization problem are examined. Techniques for the refinement of the classical Lagrangian bounds are investigated in the case where the complementary slackness conditions are not fulfilled because either the original formulation is nonconvex or the Lagrange multipliers are nonoptimal. Examples are given of integer and convex problems for which the modified bounds improve the classical Lagrangian bounds.