Mathematical Programming

, Volume 57, Issue 1–3, pp 49–83 | Cite as

Partially finite convex programming, Part II: Explicit lattice models

  • J. M. Borwein
  • A. S. Lewis


In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L1 approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.

AMS 1985 Subject Classifications

Primary 90C25, 49B27 Secondary 90C48, 52A07, 65K05 

Key words

Convex programming duality constraint qualification semi-infinite programming constrained approximation spectral estimation transportation problem 


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  1. E.J. Anderson and P. Nash,Linear Programming in Infinite-dimensional Spaces (Wiley, Chichester, 1987).Google Scholar
  2. A. Ben-Tal, J.M. Borwein and M. Teboulle, “A dual approach to multidimensionalL p spectral estimation problems,”SIAM Journal on control and Optimization 26 (1988) 985–996.Google Scholar
  3. A. Ben-Tal, J.M. Borwein and M. Teboulle, “Spectral estimation via convex programming,” to appear in:Systems and Management Science by Extremal Methods: Research Honoring Abraham Charnes at Age 70 (Kluwer Academic Publishers, Dordrecht, 1992).Google Scholar
  4. J.M. Borwein, “Convex relations in analysis and optimization,” in: S. Schaible and W.T. Ziemba, eds.,Generalized Concavity in Optimization and Economics (Academic Press, New York, 1981a) pp. 335–377.Google Scholar
  5. J.M. Borwein, “A Lagrange multiplier theorem and a sandwich theorem for convex relations,”Mathematica Scandinavica 48 (1981b) 189–204.Google Scholar
  6. J.M. Borwein, “Automatic continuity and openness of convex relations,”Proceedings of the American Mathematical Society 99 (1987) 49–55.Google Scholar
  7. J.M. Borwein and A.S. Lewis, “Duality relationships for entropy-like minimization problems,”SIAM Journal on Control and Optimization 29 (1991) 325–338.Google Scholar
  8. J.M. Borwein and H. Wolkowicz, “A simple constraint qualification in infinite dimensional programming,”Mathematical Programming 35 (1986) 83–96.Google Scholar
  9. F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).Google Scholar
  10. H.W. Corley Jr. and S.D. Roberts, “A partitioning problem with applications in regional design,”Operations Research 20 (1972) 1010–1019.Google Scholar
  11. C. De Boor, “On ‘best’ interpolation,”Journal of Approximation Theory 16 (1976) 28–42.Google Scholar
  12. A.L. Dontchev and B.D. Kalchev, “Duality and well-posedness in convex interpolation,”Numerical Functional Analysis and Optimization 10 (1989) 673–689.Google Scholar
  13. I. Ekeland and R. Temam,Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).Google Scholar
  14. J. Favard, “Sur l'interpolation,”Journal de Mathematiques Pures et Appliquees 19 (1940) 281–306.Google Scholar
  15. R.L. Francis and G.F. Wright, “Some duality relationships for the generalized Neyman—Pearson problem,”Journal of Optimization 4 (1969) 394–412.Google Scholar
  16. K. Glashoff and S.-A. Gustafson,Linear Optimization and Approximation (Springer, New York, 1983).Google Scholar
  17. R.B. Holmes,Geometric Functional Analysis and Applications (Springer, New York, 1975).Google Scholar
  18. L.D. Irvine, S.P. Marin and P.W. Smith, “Constrained interpolation and smoothing,”Constructive Approximation 2 (1986) 129–151.Google Scholar
  19. S. Karlin and W.J. Studden,Tchebycheff Systems: With Applications in Analysis and Statistics (Wiley, New York, 1966).Google Scholar
  20. K.O. Kortanek and M. Yamasaki, “Semi-infinite transportation problems,”Journal of Mathematical Analysis and Applications 88 (1982) 555–565.Google Scholar
  21. T.J. Lowe and A.P. Hurter Jr., “The generalized market area problem,”Management Science 22 (1976) 1105–1115.Google Scholar
  22. C.A. Micchelli, P.W. Smith, J. Swetits and J.D. Ward, “ConstrainedL p approximation,”Constructive Approximation 1 (1985) 93–102.Google Scholar
  23. R.T. Rockafellar, “Integrals which are convex functionals,”Pacific Journal of Mathematics 24 (1968) 525–539.Google Scholar
  24. R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  25. R.T. Rockafellar,Conjugate Duality and Optimization (SIAM, Philadelphia, PA, 1974).Google Scholar
  26. H.H. Schaefer,Topological Vector Spaces (Springer, New York, 1971).Google Scholar
  27. H.H. Schaefer,Banach Lattices and Positive Operators (Springer, Berlin, 1974).Google Scholar
  28. L.L. Schumaker,Spline Functions: Basic Theory (Wiley, New York, 1981).Google Scholar
  29. I. Singer,Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer, Berlin, 1970).Google Scholar
  30. M.J. Todd, “Solving the generalized market area problem,”Management Science 24 (1978) 1549–1554.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1992

Authors and Affiliations

  • J. M. Borwein
    • 1
  • A. S. Lewis
    • 2
  1. 1.Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifaxCanada
  2. 2.Department of Combinatorics and Optimization, Faculty of MathematicsUniversity of WaterlooWaterlooCanada

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