This paper proposes a class of surrogate constraint heuristics for obtaining approximate, near optimal solutions to integer programming problems. These heuristics are based on a simple framework that illuminates the character of several earlier heuristic proposals and provides a variety of new alternatives. The paper also proposes additional heuristics that can be used either to supplement the surrogate constraint procedures or to provide independent solution strategies. Preliminary computational results are reported for applying one of these alternatives to a class of nonlinear generalized set covering problems involving approximately 100 constraints and 300–500 integer variables. The solutions obtained by the tested procedure had objective function values twice as good as values obtained by standard approaches (e.g., reducing the best objective function values of other methods from 85 to 40 on the average. Total solution time for the tested procedure ranged from ten to twenty seconds on the CDC 6600.

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Heuristics for integer programming using surrogate constraints

The multidimensional 0–1 knapsack problem: An overview

This paper examines the performance of three heuristic methods (Senju-Toyoda, Kochen-berger et al. and Hillier) when applied to the 0-1 linear programming problem with nonnegative coefficients. Their effectiveness, measured in terms of computing time, error and relative error, is evaluated on a set of problems from the literature and randomly generated 0-1 test problems with nonnegative coefficients. Analysis of variance and stepwise regressions are employed to study the effect of the number of variables, number of constraints and degree of constraint slackness. The methods exhibited some similarities bui also marked differences in their behavior. Interestingly enough, the larger the number of variables the belter the accuracy of each method. Error differences among the three methods were significant (1:0.8:0.2) yet small (less than 2% on the average) for many practical situations. Hillier's algorithm was the most accurate but much slower and more core demanding than the other two, which makes it difficul...

Heuristic 0-1 Linear Programming: An Experimental Comparison of Three Methods

Cutting-plane theory: Algebraic methods

This paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5% of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over ten times faster than the control.

Experimental results on Hillier's linear search

: The paper reports an empirical discovery in integer programming. A version of the branch-and-bound approach is used as a control, and tested against the same algorithm augmented by the use of Hillier's linear search performed at every node of the search tree. It is shown that the imbedded linear search locates solutions within 1%, and solutions within 5%, of the theoretical optimum, which in fact can be seen to have this proximity to the theoretical optimum at the time of termination of computation, over (10) times faster than the control. Evidence is presented, in terms of the number of variables which have to be fixed to locate the 1% and 5% optima, which strongly supports the view that this order-of-magnitude speed-up will be independent of the precise branch-and-bound algorithm that is used. (Author)

Robert G. Jeroslow

Papers

Experimental Results on Hillier's Search Imbedded in a Branch-and-Bound Algorithm.