State of the Art in Global Optimization pp 303-339 | Cite as

# A Finite Algorithm for Global Minimization of Separable Concave Programs

## Abstract

Researchers first examined the problem of separable concave programming more than thirty years ago, making it one of the earliest branches nonlinear programming to be explored. This paper proposes a new finite algorithm for solving this problem. In addition to providing a proof of finiteness, the paper discusses a new way of designing branch-and-bound algorithms for concave programming that ensures finiteness. The algorithm uses domain reduction techniques to accelerate convergence; it solves problems with as many as 100 concave variables, 400 linear variables and 50 constraints in about five minutes on an IBM RS/6000 Power PC.

## Keywords

Global Optimization Global Minimization Linear Complementarity Problem Quadratic Assignment Problem Global Optimization Algorithm## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bazaraa, M.S. and Sherali, H.D. (1982), “On the Use of Exact and Heuristic Cutting Plane Methods for the Quadratic Assignment Problem,”
*Journal Operational Society*,**33**, 991–1003.MathSciNetzbMATHGoogle Scholar - [2]Ben Saad, S. and Jacobsen, S.E. (1990), “A Level Set Algorithm for a Class of Reverse Convex Programs,”
*Annals of Operations Research*,**25**, 19–42.MathSciNetzbMATHCrossRefGoogle Scholar - [3]Benson, H.P. (1985), “A Finite Algorithm for Concave Minimization over a Polyhedron,”
*Naval Research Logistics Quarterly*, 32, 165–177.MathSciNetzbMATHCrossRefGoogle Scholar - [4]Benson, H.P. (1990), “Separable Concave Minimization Via Partial Outer Approximation and Branch and Bound,”
*Operations Research Letters*, 9, 389–394.MathSciNetzbMATHCrossRefGoogle Scholar - [5]Benson, H.P. (1994), “Concave Minimization: Theory, Applications and Algorithms,” in
*Handbook of Global Optimization*, Pardalos, P.M. and Horst, R., eds., Kluwer Academic Publishers, Hingham, Massachusetts.Google Scholar - [6]Benson, H.P. and Sayin, S. (1994), “A Finite Concave Minimization Algorithm Using Branch and Bound and Neighbor Generation,”
*Journal of Global Optimization*,**5**, 1–14.MathSciNetzbMATHCrossRefGoogle Scholar - [7]Bomze, I.M. and Danninger, G. (1992), “A Finite Algorithm for Solving General Quadratic Problems,”
*Journal of Global Optimization*,**4**, 1–16.MathSciNetCrossRefGoogle Scholar - [8]Bomze, I.M. and Danninger, G. (1993), “A Global Optimization Algorithm for Concave Quadratic Programming Problems,”
*SIAM Journal of Optimization*,**3**, 826–842.MathSciNetzbMATHCrossRefGoogle Scholar - [9]Bretthauer, K.M. and Cabot, A.V. (1994), “A Composite Branch and Bound, Cutting Plane Algorithm for Concave Minimization Over a Polyhedron,”
*Computers in Operations Research*,**21**(7), 777–785.MathSciNetzbMATHCrossRefGoogle Scholar - [10]Cabot, A.V. and Francis, R.L. (1970), “Solving Certain Nonconvex Quadratic Minimization Problems by Ranking the Extreme Points,”
*Operations Research*,**18**, 82–86.zbMATHCrossRefGoogle Scholar - [11]Carvajal-Moreno, R. (1972), “Minimization of Concave Functions Subject to Linear Constraints,” Operations Research Center, University of California, Berkeley, ORC 72–3.Google Scholar
- [12]Dorneich, M.C. and Sahinidis, N.V. (1995), “Global Optimization Algorithms for Chip Layout and Compaction,” to appear in
*Engineering Optimization*.Google Scholar - [13]Dyer, M.E. (1983), “The Complexity of Vertex Enumeration Methods,” Mathematics of Operations Research,
**8**, 381–402.MathSciNetzbMATHCrossRefGoogle Scholar - [14]Dyer, M.E. and Proll, L.G. (1977), “An Algorithm for Determining All Extreme Points of a Convex Polytope,”
*Mathematical Programming*,**12**, 81–96.MathSciNetzbMATHCrossRefGoogle Scholar - [15]Falk, J.E. (1973), “A Linear Max-Min Problem,”Mathematical Programming,
**5**, 169–188.MathSciNetzbMATHCrossRefGoogle Scholar - [16]Falk, J.E. and Hoffman, K.R. (1976), “A Successive Underestimation Method for Concave Minimization Problems,”
*Mathematics of Operations Research*,**1**(3), 1976.CrossRefGoogle Scholar - [17]Falk, J.E. and Soland, R.M. (1969), “An Algorithm for Separable Nonconvex Programming Problems,”
*Management Science*,**15**(9), 550–569.MathSciNetzbMATHCrossRefGoogle Scholar - [18]Floudas, C.A. and Pardalos, P.M. (1990), A Collection of Test Problems for
*strained Global Optimization Algorithms*, Lecture Notes in Computer Science,**268**, Springer-Verlag, Berlin- Heidelberg.Google Scholar - [19]Frieze, A.M. (1974), “A Bilinear Programming Formulation of the 3 Dimensional Assignment Problem,”
*Mathematical Programming*,**7**, 376–379.MathSciNetzbMATHCrossRefGoogle Scholar - [20]Gianessi, F. and Niccolucci, F. (1976), “Connections Between Nonlinear and Integer Programming Problems,” in Symposia Mathematica Vol. XIX, Istituto Nazionale Di Alta Math., Academic Press, New York, 161–176.Google Scholar
- [21]Glover, F. (1973), “Convexity Cuts and Cut Search,” Operations Research, 21, 123–134.MathSciNetzbMATHCrossRefGoogle Scholar
- [22]Glover, F. and Klingman, D. (1973), “Concave Programming Applied to a Special Class of 0-1 Integer Programs,”
*Operations Research*, 21, 135–140.MathSciNetzbMATHCrossRefGoogle Scholar - [23]23]Hansen, P, Jaumard, B, and Lu, S-H (1991), “An Analytical Approach to Global Optimization,”
*Mathematical Programming, Series B*, 52, 227–254.MathSciNetzbMATHCrossRefGoogle Scholar - [24]Hoffman, K.L. (1981), “A Method for Globally Minimizing Concave Functions Over Convex Sets,”
*Mathematical Programming*,**22**, 22–32.CrossRefGoogle Scholar - [25]Horst, R. (1976), “An Algorithm for Nonconvex Programming Problems,”
*Mathematical Programming*,**10**, 312–321.MathSciNetzbMATHCrossRefGoogle Scholar - [25]26]Horst, R. (1984), “On the Global Minimization of Concave Functions— Introduction and Survey,”
*OR Spektrum*,**6**, 195–205.MathSciNetzbMATHCrossRefGoogle Scholar - [27]Horst, R., and Tuy, H. (1993),
*Global Optimization: Deterministic Approaches*, Springer-Verlag, 2nd ed., Berlin.Google Scholar - [28]Kalantari, B. and Rosen, J.B. (1987), “An Algorithm for Global Minimization of Linearly Constrained Convex Quadratic Functions,”
*Mathematics of Operations Research*,**12**(3), 544–561.MathSciNetzbMATHCrossRefGoogle Scholar - [29]Krynski, S.L. (1979),
*Minimization of a Concave Function under Linear Constraints (Modification of Tuy’s Method)*, in Survey of Mathematical Programming, Proceedings of the Ninth International Mathematical Programming Symposium, Budapest, 1976, North-Holland, Amsterdam, 1, 479–493.Google Scholar - [30]Lamar, B.W. (1993), “An Improved Branch and Bound Algorithm for Minimum Concave Cost Network Flow Problems,”
*Journal of Global Optimization*,**3**(3), 261–287.MathSciNetzbMATHCrossRefGoogle Scholar - [31]Lawler, E.L. (1963), “The Quadratic Assignment Problem,”
*Management Science*,**9**, 586–699.MathSciNetzbMATHCrossRefGoogle Scholar - [32]Mangasarian, O.L. (1978), “Characterization of Linear Complementarity Problems as Linear Programs,”
*Mathematical Programming Study*,**7**, 74–87.MathSciNetzbMATHGoogle Scholar - [33]Matheiss, T.H. (1973), “An Algorithm for Determining Unrelevant Constraints and All Vertices in Systems of Linear Inequalities,”
*Operations Research*,**21**, 247–260.MathSciNetCrossRefGoogle Scholar - [34]Matheiss, T.H. and Rubin, D.S. (1980), “A Survey and Comparison of Methods for Finding All Vertices of Convex Polyhedral Sets,”
*Mathematics of Operations Research*,**5**, 167–185.MathSciNetzbMATHCrossRefGoogle Scholar - [35]McCormick, G.P. (1972), “Attempts to Calculate Global Solutions of Problems that May Have Local Minima,” in
*Numerical Methods for Non-Linear Optimization*, Lootsma, F.A., Ed. Academic Press, New York, 209–221.Google Scholar - [36]Moshirvaziri, K. (1994), “A Generalization of the Construction of Test Problems for Nonconvex Optimization,”
*Journal of Global Optimization*,**5**, 21–34.MathSciNetzbMATHCrossRefGoogle Scholar - [37]Moshirvaziri, K. (1994), Personal Communication.Google Scholar
- [38]Mukhamediev, B.M. (1982), “Approximate Methods of Solving Concave Programming Problems”,
*USSR Computational Mathematics and Mathematical Physics*,**22**(3), 238–245.MathSciNetzbMATHCrossRefGoogle Scholar - [39]Murty, K.G. and Kabadi, S.N. (1987), “Some A/T-Complete Problems in Quadratic and Nonlinear Programming,”
*Mathematical Programming*,**39**, 117–129.MathSciNetzbMATHCrossRefGoogle Scholar - [40]Nourie, F.J. and Gder, F. (1994), “A Restricted-Entry Method for a Trans-portation Problem with Piecewise-Linear Concave Costs,”
*Computers in Operations Research*,**21**(7), 723–733.zbMATHCrossRefGoogle Scholar - [41]Pardalos, P.M. (1985), “Integer and Separable Programming Techniques for Large-Scale Global Optimization Problems,” Ph.D. Thesis, Computer Science Department, University of Minnesota, Minneapolis.Google Scholar
- [42]Pardalos, P.M. and Rosen, J.B. (1986), “Methods for Global Concave Minimization: A Bibliographic Survey,”
*SIAM Review*,**28**, 367–379.MathSciNetzbMATHCrossRefGoogle Scholar - [43]Pardalos, P.M. and Rosen, J.B. (1987),
*Constrained Global Optimization: Algorithms and Applications*, Lecture Notes in Computer Science,**268**, Springer-Verlag, Berlin- Heidelberg.Google Scholar - [44]Phillips, A.T. (1988), “Parallel Algorithms for Constrained Optimization, Ph.D. Dissertation”, University of Minnesota, Minneapolis, MN.Google Scholar
- [45]Phillips, A.T. and Rosen, J.B. (1987), “A Parallel Algorithm for Constrained Concave Quadratic Global Minimization, Technical Report 87–48,” Computer Science Department, Institute of Technology, University of Minnesota, Minneapolis.Google Scholar
- [46]Phillips, A.T. and Rosen, J.B. (1988), “A Parallel Algorithm for Constrained Concave Quadratic Global Minimization,”
*Mathematical Programming*, 42, 421–448.MathSciNetzbMATHCrossRefGoogle Scholar - [47]Phillips, A.T. and Rosen, J.B. (1990), “A Parallel Algorithm for Partially Separable Non-convex Global Minimization: Linear Constraints,”
*Annals of Operations Research*,**25**, 101–118.MathSciNetzbMATHCrossRefGoogle Scholar - [48]Phillips, A.T. and Rosen, J.B. (1990), “Guaranteed ɛ-Approximate Solution for Indefinite Quadratic Global Minimization,”
*Naval Research Logistics*,**37**, 499–514.MathSciNetzbMATHCrossRefGoogle Scholar - [49]Phillips, A.T. and Rosen, J.B. (1993), “Sufficient Conditions for Solving Linearly Constrained Separable Concave Global Minimization Problems,”
*Journal of Global Optimization*,**3**, 79–94.MathSciNetzbMATHCrossRefGoogle Scholar - [50]Phillips, A.T. and Rosen, J.B. (1994), “Computational Comparison of Two Methods for Constrained Global Optimization,”
*Journal of Global Optimization*,**5**(4), 325–332.MathSciNetzbMATHCrossRefGoogle Scholar - [51]Raghavachari, M. (1969), “On Connections between Zero-One Integer Programming and Concave Programming Under Linear Constraints,”
*Operations Research*,**17**, 680–684.MathSciNetzbMATHCrossRefGoogle Scholar - [52]Rosen, J.B. (1983), “Global Minimization of a Linearly Constrained Con¬cave Function by Partition of Feasible Domain,”
*Mathematics of Operations Research*,**8**(2), 215–230.MathSciNetzbMATHCrossRefGoogle Scholar - [53]Rosen, J.B. and Pardalos, P.M. (1986) “Global Minimization of Large- Scale Constrained Concave Quadratic Problems by Separable Programming,”
*Mathematical Programming*,**34**, 163–174.MathSciNetzbMATHCrossRefGoogle Scholar - [54]Rosen, J.B. and van Vliet, M. (1987), “A Parallel Stochastic Method for the Constrained Concave Global Minimization Problem, Technical Report 87–31,” Computer Science Department, Institute of Technology, University of Minnesota, Minneapolis.Google Scholar
- [55]Ryoo, H.S. (1994), “Range Reduction as a Means of Performance Improvement in Global Optimization: A Branch-and-Reduce Global Optimization Algorithm,” Master’s Thesis, University of Illinois, Urbana.Google Scholar
- [56]Ryoo, H.S. and Sahinidis, N.V. (1994), “A Branch-and-Reduce Approach to Global Optimization,” Submitted to
*Journal of Global Optimization*.Google Scholar - [57]Ryoo, H.S. and Sahinidis, N.V. (1995), “Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design,” Computers & Chemical Engineering, 19 (5), 551–566.Google Scholar
- [58]Sahinidis, N.V. (1992), “Accelerating Branch-and-Bound in Continuous Optimization,” Research Report UILU ENG 92 - 4031, University of Illinois, Urbana.Google Scholar
- [59]Sherali, H.D. and Alameddine, A. (1990) “A New Reformulation- Linearization Technique for Bilinear Programming Problems,” Technical Report, Department of Industrial and Systems Engineering, Virginia Poly-technic Institute and State University, Blacksburg, Virginia.Google Scholar
- [60]Sherali, H.D. and Tuncbilek, C.H. (1994), “Tight Reformulation- Linearization Technique Representations for Solving Nonconvex Quadratic Programming Problems,” Technical Report, Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia.Google Scholar
- [61]Soland, R.M. (1974), “Optimal Facility Location with Concave Costs,
*Operations Research*,”**22**, 373–382.MathSciNetzbMATHCrossRefGoogle Scholar - [62]Suhl, U.H. and Szymanski (1994), “Supemode Processing of Mixed-Integer Models,”
*Computational Optimization and Applications*,**3**, 317–331.MathSciNetzbMATHCrossRefGoogle Scholar - [63]Thakur, N.V. (1990), “Domain Contraction in Nonlinear Programming: Minimizing a Quadratic Concave Function over a Polyhedron,”
*Mathematics of Operations Research*,**16**(2), 390–407.MathSciNetCrossRefGoogle Scholar - [64]Thieu, T.V. (1980), “Relationship Between Bilinear Programming and Concave Programming,” Acta Mathematica Vietnamica, 2, 106–113.Google Scholar
- [65]Thoai, N.V. and Tuy, H. (1980), “Convergent Algorithms for Minimizing a Concave Function,”
*Mathematics of Operations Research*,**5**, 556–566.MathSciNetzbMATHCrossRefGoogle Scholar - [66]Thoai, N.V. and Tuy, H. (1983), “Solving the Linear Complementarity Problem Through Concave Programming,”
*USSR Computational Mathematics and Mathematical Physics*,**23**(3), 55–59.zbMATHCrossRefGoogle Scholar - [67]Tuy, H. (1964), “Concave Programming Under Linear Constraints,” Soviet Mathematics, 5, 1437–1440.Google Scholar
- [68]Tuy, H. (1991), “Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms,”
*Journal of Global 0ptimization*,**1**(1), 23–36.MathSciNetzbMATHCrossRefGoogle Scholar - [69]Tuy, H. and Horst, R. (1988), “Convergence and Restart in Branch-and- Bound Algorithms for Global Optimization. Application to Concave Minimization and D.C. Optimization Problems,”
*Mathematical Programming*,**41**, 161–183.MathSciNetzbMATHCrossRefGoogle Scholar - [70]Tuy, H., Thieu, T.V., and Thai N.Q. (1985), “A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set,”
*Mathematics of Operations Research*,**10**, 498–514.MathSciNetzbMATHCrossRefGoogle Scholar - [71]Visweswaran, V. and Floudas, C.A. (1993), “New Properties and Compu-tational Improvement of the GOP Algorithm for Problems with Quadratic Objective Functions and Constraints,”
*Journal of Global Optimization*,**3**, 439–462.MathSciNetzbMATHCrossRefGoogle Scholar - [72]Zwart, P.B. (1971), “Computational Aspects on the Use of Cutting Planes in Global Optimization,”
*Proceedings of the 1971 Annual Conference of the ACM*, 457–465.Google Scholar - [73]Zwart, P.B. (1973), “Nonlinear Programming: Counterexamples to Global Optimization Algorithms,”
*Operations Research*,**21**, 1260–1266.MathSciNetzbMATHCrossRefGoogle Scholar - [74]Zwart, P.B. (1974), “Global Maximization of a Convex Function with Linear Inequality Constraints,”
*Operations Research*,**22**, 602–609.MathSciNetzbMATHCrossRefGoogle Scholar