Technical Note—Searchability of the Composite and Multiple Surrogate Dual Functions

The values of optimal solutions to the various dual relaxations of integer programs can be viewed as functions of their multipliers. Successful search procedures have been developed for optimal lagrangian (piecewise linear concave) and surrogate (quasi-concave) multipliers. This note demonstrates that the composite and multiple surrogate dual functions lack quasi-concavity and may possess “false optima.” For the composite dual, complementary and alternate optimization over lagrangian and surrogate multipliers are shown to be nonoptimal techniques. However, some promising computational results, based on closing part of the gap between the surrogate dual and the primal, are reported for the alternating approach.