# An outer-approximation algorithm for a class of mixed-integer nonlinear programs

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## Abstract

An outer-approximation algorithm is presented for solving mixed-integer nonlinear programming problems of a particular class. Linearity of the integer (or discrete) variables, and convexity of the nonlinear functions involving continuous variables are the main features in the underlying mathematical structure. Based on principles of decomposition, outer-approximation and relaxation, the proposed algorithm effectively exploits the structure of the problems, and consists of solving an alternating finite sequence of nonlinear programming subproblems and relaxed versions of a mixed-integer linear master program. Convergence and optimality properties of the algorithm are presented, as well as a general discussion on its implementation. Numerical results are reported for several example problems to illustrate the potential of the proposed algorithm for programs in the class addressed in this paper. Finally, a theoretical comparison with generalized Benders decomposition is presented on the lower bounds predicted by the relaxed master programs.

## Key words

Mixed-integer nonlinear programming outer-approximation decomposition cutting planes computer-aided design## Preview

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## References

- [1]S. Albers and K. Brockhoff, “A procedure for new product positioning in an attribute space”,
*European Journal of Operational Research*1 (1977) 230–238.MATHCrossRefGoogle Scholar - [2]E. Balas, “A duality theorem and an algorithm for (mixed-) integer nonlinear programming”,
*Linear Algebra and its Applications*4 (1971) 341–352.MATHCrossRefMathSciNetGoogle Scholar - [3]E. Balas, “Disjunctive programming”, in: P.L. Hammer, E.L. Johnson and B.H. Korte, eds.,
*Annals of Discrete Mathematics 5: Discrete Optimization II*(North-Holland, Amsterdam, 1979) pp. 3–51.Google Scholar - [4]E. Balas and R. Jeroslow, “Canonical Cuts on the Unit Hypercube”,
*SIAM Journal of Applied Mathematics*23 (1972) 61–79.MATHCrossRefMathSciNetGoogle Scholar - [5]J. F. Benders, “Partitioning procedures for solving mixed-variables programming problems”,
*Numerische Mathematik*4 (1962) 238–252.MATHCrossRefMathSciNetGoogle Scholar - [6]D.P. Bertsekas, G.S. Lauer, N.R. Sandell, Jr., and T.A. Posbergh, “Optimal short-term scheduling of large-scale power systems”,
*IEEE Transactions on Automatic Control*AC-28 (1983) 1–11.MATHCrossRefGoogle Scholar - [7]J.A. Bloom, “Mathematical generation planning models using decomposition and probabilistic simulation”, Special Report No. EA-2566-SR, Electric Power Research Institute, Palo Alto, California (August 1982).Google Scholar
- [8]J.A. Bloom, “Solving an electricity generating capacity expansion planning problem by generalized Benders’ decomposition”,
*Operations Research*31 (1983) 84–100.MATHGoogle Scholar - [9]A.L. Brearley, G. Mitra and H.P. Williams, “Analysis of mathematical programming problems prior to applying the simplex algorithm”,
*Mathematical Programming*8 (1975) 54–83.MATHCrossRefMathSciNetGoogle Scholar - [10]H. Crowder, E.L. Johnson and M.W. Padberg, “Solving large-scale zero-one linear programming problems”,
*Operations Research*31 (1983) 803–834.MATHGoogle Scholar - [11]M.A. Duran, “A mixed-integer nonlinear programming approach for the systematic synthesis of engineering systems”, Ph.D. Thesis, Department of Chemical Engineering, Carnegie-Mellon University (Pittsburgh, PA, December 1984).Google Scholar
- [12]M.A. Duran and I.E. Grossmann, “An outer-approximation algorithm for a class of mixed-integer nonlinear programs. The relation with generalized Benders decomposition”, Technical Report DRC-06-68-84, Design Research Center, Carnegie-Mellon University, (Pittsburgh, PA, December 1984).Google Scholar
- [13]M. A. Duran and J. E. Grossmann, “A mixed-integer nonlinear programming algorithm for process systems synthesis”,
*American Institute of Chemical Engineers Journal*, 32 (1986) 592–606.Google Scholar - [14]B.C. Eaves and W.I. Zangwill, “Generalized cutting plane algorithms”,
*SIAM Journal on Control*9 (1971) 529–542.CrossRefMathSciNetGoogle Scholar - [15]J. Elzinga and T.G. Moore, “A central cutting plane algorithm for the convex programming problem”,
*Mathematical Programming*8 (1975) 134–145.MATHCrossRefMathSciNetGoogle Scholar - [16]J.F. Faccenda and S. Debashish, “Computational experiments with a nonlinear 0–1 powerhouse optimization model”, paper presented at the ORSA/TIMS Joint National Meeting, Orlando, FL (November 1983).Google Scholar
- [17]R.S. Garfinkel and G.L. Nemhauser,
*Integer Programming*(John Wiley & Sons, New York, 1972).MATHGoogle Scholar - [18]B. Gavish, D. Horsky and K. Srikanth, “An approach to the optimal positioning of a new product”,
*Management Science*29 (1983) 1277–1297.MATHGoogle Scholar - [19]A.M. Geoffrion, “Elements of large-scale mathematical programming, Parts I and II”,
*Management Science*16 (1970), 652–691.MathSciNetGoogle Scholar - [20]A.M. Geoffrion, “Generalized Benders decomposition”,
*Journal of Optimization Theory and Applications*10 (1972) 237–260.MATHCrossRefMathSciNetGoogle Scholar - [21]F. Glover, “Surrogate constraints”,
*Operations Research*16 (1968), 741–749.MATHMathSciNetGoogle Scholar - [22]R.E. Gomory, “Some polyhedra related to combinatorial problems”,
*Linear Algebra and its Applications*2 (1969) 451–558.MATHCrossRefMathSciNetGoogle Scholar - [23]C. Gonzaga and E. Polak, “On constraint dropping schemes and optimality functions for a class of outer approximations algorithms”,
*SIAM Journal of Control and Optimization*17 (1979) 477–493.MATHCrossRefMathSciNetGoogle Scholar - [24]G.W. Graves, “Water pollution control”, in: A.V. Balakrishnan, ed.,
*Techniques of Optimization*(Academic Press, New York, 1972) pp. 499–509.Google Scholar - [25]I.E. Grossman, “Mixed-integer programming approach for the synthesis of integrated process flow-sheets”,
*Computers and Chemical Engineering*9 (1985) 463–482.CrossRefGoogle Scholar - [26]G. Gunawardane, S. Hoff and L. Schrage, “Identification of special structure constraints in linear programs”,
*Mathematical Programming*21 (1981) 90–97.MATHCrossRefMathSciNetGoogle Scholar - [27]O.K. Gupta, “Branch and bound experiments in nonlinear integer programming”, Ph.D. Thesis, School of Industrial Engineering, Purdue University (West Lafayette, IN, December 1980).Google Scholar
- [28]O.K. Gupta and A. Ravindran, “Nonlinear mixed integer programming and discrete optimization”, in: R.W. Wayne and K.M. Ragsdell, eds.,
*Progress in Engineering Optimization*(ASME, New York, 1981) pp. 27–32.Google Scholar - [29]H.H. Hoang, “Topological optimization of networks: A nonlinear mixed integer model employing generalized Benders decomposition”,
*IEEE Transactions on Automatic Control*AC-27 (1982) 164–169.MATHCrossRefGoogle Scholar - [30]W.W. Hogan, “Optimization and convergence for extremal value functions arising from structured nonlinear programs”, Working Paper No. 180. Western Management Science Institute, University of California, Los Angeles (September 1971).Google Scholar
- [31]J.E. Kelley, Jr.: “The cutting-plane method for solving convex programs”,
*Journal of the Society for Industrial and Applied Mathematics*8 (1960) 703–712.CrossRefMathSciNetGoogle Scholar - [32]G.S. Lauer, D.P. Bertsekas, N.R. Sandell Jr., and T.A. Posbergh, “Solution of large-scale optimal unit commitment problems”,
*IEEE Transactions on Power Apparatus and Systems*, PAS-101 (1982) 79–86.CrossRefGoogle Scholar - [33]L.J. Leblanc, “An algorithm for the discrete network design problems”,
*Transportation Science*9 (1975) 183–199.CrossRefGoogle Scholar - [34]J.H. May, A.D. Shocker and D. Sudharshan, “A simulation comparison of methods for new product location”, Working paper No. 82-126, Graduate School of Business, University of Pittsburgh, Pittsburgh (October 1983).Google Scholar
- [35]D.Q. Mayne, E. Polak and R. Trahan, “An outer approximations algorithm for computer-aided design problems”,
*Journal of Optimization Theory and Applications*28 (1979) 331–352.MATHCrossRefMathSciNetGoogle Scholar - [36]B.A. Murtagh and M.A. Saunders, “A projected lagrangian algorithm and its implementation for sparse nonlinear constraints, and MINOS/AUGMENTED User’s Manual”, Technical Reports SOL 80-1 R and SOL 80-14, Systems Optimization Laboratory, Department of Operations Research, Stanford University, California (February 1981, June 1980).Google Scholar
- [37]R.T. Rockafellar,
*Convex Analysis*(Princeton University Press, Princeton, NJ, 1970).MATHGoogle Scholar - [38]N.R. Sandell, Jr., D.P. Bertsekas, J.J. Shaw, S.W. Gully and R. Gendron, “Optimal scheduling of large scale hydrothermal power systems”, in:
*Proceedings of the 1982 IEEE International Large Scale Systems Symposium*(Virginia Beach, Virginia, October 1982) pp. 141–147.Google Scholar - [39]L. Schrage,
*LP Models with LINDO (Linear Interactive Discrete Optimizer)*(The Scientific Press, Palo Alto, CA, 1981).Google Scholar - [40]J. Stoer and C. Witzgall,
*Convexity and Optimization in Finite Dimensions I*(Springer-Verlag, New York, 1970).Google Scholar - [41]J.A. Tomlin, “Large scale mathematical programming systems”,
*Computers and Chemical Engineering*7 (1983) 575–582.CrossRefGoogle Scholar - [42]D.M. Topkis, “Cutting-plane methods without nested constraint sets,”
*Operations Research*18 (1970) 404–413.MATHMathSciNetGoogle Scholar - [43]D.M. Topkis, “A note on cutting-plane methods without nested constraint sets,”
*Operations Research*18 (1970) 1216–1220.MATHMathSciNetGoogle Scholar - [44]D.M. Topkis, “A cutting-plane algorithm with linear and geometric rates of convergence,”
*Journal of Optimization Theory and Applications*36 (1982) 1–22.CrossRefMathSciNetGoogle Scholar - [45]T.J. Van Roy and L.A. Wolsey, “Valid inequalities for mixed 0–1 programs,” CORE Discussion Paper 8316 (March 1983).Google Scholar
- [46]J.A. Vaselenak, I.E. Grossmann and A.W. Westerberg, “Optimal retrofit design of multi-product batch plants,” paper 50d presented at the American Institute of Chemical Engineers 1986 National Meeting, Houston, TX (April 1986).Google Scholar
- [47]F.S. Zufryden, “ZIPMAP—A zero-one integer programming model for market segmentation and product positioning,”
*Journal of the Operational Research Society*30 (1979) 63–70.MATHCrossRefGoogle Scholar