Surrogate Multipliers Research Articles

Surrogate Dual Multiplier Search Procedures in Integer Programming

Search procedures for optimal Lagrange multipliers are highly developed and provide good bounds in branch and bound procedures that have led to the successful application of Lagrangean duality in integer programming. Although the surrogate dual generally provides a better objective bound, there has been little development of surrogate multiplier search procedures. This paper develops and empirically analyzes several surrogate multiplier search procedures. Results indicate that the procedures can produce possibly superior bounds in an amount of time comparable to other techniques. Our discussion also highlights the similarity of the procedures to some well known Lagrangean search techniques.

Surrogate geometric programming estimates of restricted multinomial proportions

Maximum-likelihood estimation of multinomial proportions subject to linear inequality or equality constraints is considered. A demonstration that surrogate constraints yield a zero-degree-of-difficulty geometric programming problem is given. A general solution in terms of surrogate multipliers and the unconstrained estimates is exhibited. A computational algorithm is presented and used to solve an example problem pertaining to coal deliveries received by a power plant.

Surrogate Constraints Algorithm for Reliability Optimization Problems with Two Constraints

This paper presents a surrogate constraints algorithm for solving nonlinear programming, nonlinear integer programming, and nonlinear mixed integer programming problems. The algorithm contains a new technique for generating a succession of vector values of surrogate multiplier (ie, surrogate problems). By using this technique, a computer can keep a polyhedron, which is a vector space of surrogate multipliers to be considered at a certain time, in its memory. Furthermore it can cut the polyhedron by a given hyperplane, and produce the remaining space as the next polyhedron. Simple examples are included.

Surrogate duality in a branch‐and‐bound procedure

AbstractRecent research has led to several surrogate multiplier search procedures for use in a primal branch‐and‐bound procedure. As single constrained integer programming problems, the surrogate subproblems are also solved via branch‐and‐bound. This paper develops the inner play between the surrogate subproblem and the primal branch‐and‐bound trees which can be exploited to produce a number of computational efficiencies. Most important is a restarting procedure which precludes the need to solve numerous surrogate subproblems at each node of a primal branch‐and‐bound tree. Empirical evidence suggests that this procedure greatly reduces total computation time.

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.