Abstract
Various theoretical properties of the surrogate dual of a mathematical programming problem are discussed, including some connections with the Lagrangean dual. Two algorithms for solving the surrogate dual, suggested by analogy with Lagrangean optimisation, are described and proofs of their convergence given. A simple example is solved using each method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Banerjee, “Generalized Lagrange multipliers in dynamic programming”, Research report ORC 71-12, Operations Research Center, University of California, Berkeley, CA (1971).
S. Bolwell and M.H. Karwan, “Efficient use of surrogate duality in 0–1 integer programming”, Industrial Engineering report, State University of New York at Buffalo, NY (1979).
S. Bolwell and M.H. Karwan, “A hybrid surrogate dual-cutting plane approach to integer programming”, Industrial Engineering report, State University of New York at Buffalo, NY (1979).
G.H. Bradley, “Transformation of integer programs to knapsack problems”,Discrete Mathematics 1 (1971) 29–45.
B.C. Eaves and W.I. Zangwill, “Generalised cutting plane methods”,SIAM Journal of Control 9 (1971) 529–542.
A.M. Frieze, “Bottleneck linear programming”,Operational Research Quarterly 26 (1975) 871–874.
A.M. Geoffrion, “Duality in nonlinear programming”,SIAM Review 13 (1971) 1–37.
F. Glover, “Surrogate constraints”,Operations Research 16 (1968) 741–749.
F. Glover, “Surrogate constraint duality in mathematical programming”,Operations Research 23 (1975) 434–451.
H.J. Greenberg and W.P. Pierskalla, “Surrogate mathematical programs”,Operations Research 18 (1970) 924–939.
H.J. Greenberg and W. P. Pierskalla, “A review of quasi-convex functions”,Operations Research 19 (1971) 1553–1570.
H.J. Greenberg and W.P. Pierskalla, “Quasi-conjugate functions and surrogate duality”,Cahiers du Center d'Etudes de Recherche Operationelle 15 (1973) 437–448.
H.J. Greenberg, “The generalised penalty function surrogate model”,Operations Research 21 (1973) 162–178.
M. Held, P. Wolfe and H.P. Crowder, “A validation of subgradient optimisation”,Mathematical Programming 6 (1974) 62–88.
M.H. Karwan, “Surrogate constraint duality and extensions in integer programming”, Ph.D. dissertation, Georgia Institute of Technology (1976).
M.H. Karwan and R.L. Rardin, “Surrogate dual multiplier search procedures”, Industrial andSystems Engineering Report Series No. J-76-29, Georgia Institute of Technology, GA (1976).
M.H. Karwan and R.L. Rardin, “Some relationships between Lagrangian and surrogate duality in integer linear programming”,Mathematical Programming 17 (1979) 320–334.
M.H. Karwan and R.L. Rardin, “Surrogate duality in a branch-and-bound procedure”,Naval Research Logistics Quarterly (to appear).
M.H. Karwan and R.L. Rardin, “Searchability of the composite and multiple surrogate dual functions”,Operations Research (to appear).
D.G. Luenberger, “Quasi-convex programming”,SIAM Journal of Applied Mathematics 16 (1968) 1090–1095.
T.L. Magnanti, J.F. Shapiro and M.H. Wagner, “Generalised programming solves the dual”,Management Science 22 (1976) 1195–1203.
T.H. Mattheiss and W.B. Widhelm, “The generalized slack variable linear program”,Management Science 23 (1977) 859–871.
G.L. Nemhauser and W.B. Widhelm, “A modified linear program for columnar methods in mathematical programming”,Operations Research 19 (1971) 1051–1060.
R.P. O'Neill and W.B. Widhelm, “Computational experience with normed and non-normed column generation procedures in nonlinear programming”,Operations Research 23 (1975) 372–382.
B.T. Polyak, “A general method of solving extremal problems”,Doklady Akademii Nauk SSSR 174 (1967) 33–36.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dyer, M.E. Calculating surrogate constraints. Mathematical Programming 19, 255–278 (1980). https://doi.org/10.1007/BF01581647
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01581647