Symmetrized separable convex programming

McLinden, L.

doi:10.1090/S0002-9947-1979-0517685-5

Symmetrized separable convex programming
HTML articles powered by AMS MathViewer

by L. McLinden PDF

Trans. Amer. Math. Soc. 247 (1979), 1-44 Request permission

Abstract:

The duality model for convex programming studied recently by E. L. Peterson is analyzed from the viewpoint of perturbational duality theory. Relationships with the traditional Lagrangian model for ordinary programming are explored in detail, with particular emphasis placed on the respective dual problems, Kuhn-Tucker vectors, and extremality conditions. The case of homogeneous constraints is discussed by way of illustration. The Slater existence criterion for optimal Lagrange multipliers in ordinary programming is sharpened for the case in which some of the functions are polyhedral. The analysis generally covers nonclosed functions on general spaces and includes refinements to exploit polyhedrality in the finite-dimensional case. Underlying the whole development are basic technical facts which are developed concerning the Fenchel conjugate and preconjugate of the indicator function of an epigraph set.

References

N. Bourbaki, Eléments de mathématique. XV. Première partie: Les structures fondamentales de l’analyse. Livre V: Espaces vectoriels topologiques. Chapitre I: Espaces vectoriels topologiques sur un corps valué. Chapitre II: Ensembles convexes et espaces localement convexes, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1189, Hermann & Cie, Paris, 1953 (French). MR 0054161
C. Roger Glassey, Explicit duality for convex homogeneous programs, Math. Programming 10 (1976), no. 2, 176–191. MR 416590, DOI 10.1007/BF01580666
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4
L. McLinden, Symmetric duality for structured convex programs, Trans. Amer. Math. Soc. 245 (1978), 147–181. MR 511404, DOI 10.1090/S0002-9947-1978-0511404-3
L. McLinden, Affine minorants minimizing the sum of convex functions, J. Optim. Theory Appl. 24 (1978), no. 4, 569–583. MR 503769, DOI 10.1007/BF00935300

Fonctionnelles convexes

Generalization and symmetrization of duality in geometric programming

Elmor L. Peterson, Geometric programming and some of its extensions, Optimization and design (Internat. Summer School, Louvain, 1971) Prentice-Hall, Englewood Cliffs, N.J., 1973, pp. 228–289. MR 0381702
Elmor L. Peterson, Geometric programming, SIAM Rev. 18 (1976), no. 1, 1–51. MR 395849, DOI 10.1137/1018001
E. L. Peterson, Constrained duality via unconstrained duality in generalized geometric programming, J. Optim. Theory Appl. 26 (1978), no. 1, 43–50. MR 514095, DOI 10.1007/BF00933269
Elmor L. Peterson, The duality between suboptimization and parameter deletion, Math. Oper. Res. 2 (1977), no. 4, 311–319 (1978). MR 475884, DOI 10.1287/moor.2.4.311
E. L. Peterson, Saddle points and duality in generalized geometric programming, J. Optim. Theory Appl. 26 (1978), no. 1, 15–41. MR 514094, DOI 10.1007/BF00933268
E. L. Peterson, Optimality conditions in generalized geometric programming, J. Optim. Theory Appl. 26 (1978), no. 1, 3–13. MR 514093, DOI 10.1007/BF00933267

Ordinary duality vis-a-vis geometric duality

Geometric duality via Rockafellar duality

R. T. Rockafellar, Duality theorems for convex functions, Bull. Amer. Math. Soc. 70 (1964), 189–192. MR 165429, DOI 10.1090/S0002-9904-1964-11074-0
R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966), 46–63. MR 192318, DOI 10.1090/S0002-9947-1966-0192318-X
R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966), 81–89. MR 187062, DOI 10.1215/S0012-7094-66-03312-6

Convex analysis

R. Tyrrell Rockafellar, Some convex programs whose duals are linearly constrained, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 293–322. MR 0281500
R. Tyrrell Rockafellar, Conjugate duality and optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. MR 0373611, DOI 10.1137/1.9781611970524
Josef Stoer and Christoph Witzgall, Convexity and optimization in finite dimensions. I, Die Grundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York-Berlin, 1970. MR 0286498, DOI 10.1007/978-3-642-46216-0