Partially finite convex programming, Part II: Explicit lattice models
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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,L 1 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.
- E.J. Anderson and P. Nash,Linear Programming in Infinite-dimensional Spaces (Wiley, Chichester, 1987).
- 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.
- 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).
- 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.
- J.M. Borwein, “A Lagrange multiplier theorem and a sandwich theorem for convex relations,”Mathematica Scandinavica 48 (1981b) 189–204.
- J.M. Borwein, “Automatic continuity and openness of convex relations,”Proceedings of the American Mathematical Society 99 (1987) 49–55.
- J.M. Borwein and A.S. Lewis, “Duality relationships for entropy-like minimization problems,”SIAM Journal on Control and Optimization 29 (1991) 325–338.
- J.M. Borwein and H. Wolkowicz, “A simple constraint qualification in infinite dimensional programming,”Mathematical Programming 35 (1986) 83–96.
- F.H. Clarke,Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
- H.W. Corley Jr. and S.D. Roberts, “A partitioning problem with applications in regional design,”Operations Research 20 (1972) 1010–1019.
- C. De Boor, “On ‘best’ interpolation,”Journal of Approximation Theory 16 (1976) 28–42.
- A.L. Dontchev and B.D. Kalchev, “Duality and well-posedness in convex interpolation,”Numerical Functional Analysis and Optimization 10 (1989) 673–689.
- I. Ekeland and R. Temam,Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976).
- J. Favard, “Sur l'interpolation,”Journal de Mathematiques Pures et Appliquees 19 (1940) 281–306.
- R.L. Francis and G.F. Wright, “Some duality relationships for the generalized Neyman—Pearson problem,”Journal of Optimization 4 (1969) 394–412.
- K. Glashoff and S.-A. Gustafson,Linear Optimization and Approximation (Springer, New York, 1983).
- R.B. Holmes,Geometric Functional Analysis and Applications (Springer, New York, 1975).
- L.D. Irvine, S.P. Marin and P.W. Smith, “Constrained interpolation and smoothing,”Constructive Approximation 2 (1986) 129–151.
- S. Karlin and W.J. Studden,Tchebycheff Systems: With Applications in Analysis and Statistics (Wiley, New York, 1966).
- K.O. Kortanek and M. Yamasaki, “Semi-infinite transportation problems,”Journal of Mathematical Analysis and Applications 88 (1982) 555–565.
- T.J. Lowe and A.P. Hurter Jr., “The generalized market area problem,”Management Science 22 (1976) 1105–1115.
- C.A. Micchelli, P.W. Smith, J. Swetits and J.D. Ward, “ConstrainedL p approximation,”Constructive Approximation 1 (1985) 93–102.
- R.T. Rockafellar, “Integrals which are convex functionals,”Pacific Journal of Mathematics 24 (1968) 525–539.
- R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
- R.T. Rockafellar,Conjugate Duality and Optimization (SIAM, Philadelphia, PA, 1974).
- H.H. Schaefer,Topological Vector Spaces (Springer, New York, 1971).
- H.H. Schaefer,Banach Lattices and Positive Operators (Springer, Berlin, 1974).
- L.L. Schumaker,Spline Functions: Basic Theory (Wiley, New York, 1981).
- I. Singer,Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer, Berlin, 1970).
- M.J. Todd, “Solving the generalized market area problem,”Management Science 24 (1978) 1549–1554.
- Partially finite convex programming, Part II: Explicit lattice models
Volume 57, Issue 1-3 , pp 49-83
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Primary 90C25, 49B27
- Secondary 90C48, 52A07, 65K05
- Convex programming
- constraint qualification
- semi-infinite programming
- constrained approximation
- spectral estimation
- transportation problem
- Industry Sectors
- Author Affiliations
- 1. Department of Mathematics, Statistics and Computing Science, Dalhousie University, B3H 3J5, Halifax, N.S., Canada
- 2. Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, N2L 3G1, Waterloo, Ont., Canada