Symmetric duality for structured convex programs

McLinden, L.

doi:10.1090/S0002-9947-1978-0511404-3

Symmetric duality for structured convex programs
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Trans. Amer. Math. Soc. 245 (1978), 147-181 Request permission

Abstract:

A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar’s general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.

References

N. Bourbaki, Éléments de mathématique. Fasc. XV. 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, Paris, 1966 (French). Deuxième édition revue et corrigée. MR 0203425
Richard W. Cottle, Symmetric dual quadratic programs, Quart. Appl. Math. 21 (1963/64), 237–243. MR 156707, DOI 10.1090/S0033-569X-1963-0156707-4
R. J. Duffin, Infinite programs, Linear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, pp. 157–170. MR 0087573
E. Eisenberg, Duality in homogeneous programming, Proc. Amer. Math. Soc. 12 (1961), 783–787. MR 129021, DOI 10.1090/S0002-9939-1961-0129021-9

Convex cones, sets and functions

David Gale, Harold W. Kuhn, and Albert W. Tucker, Linear programming and the theory of games, Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, Inc., New York, N.Y.; Chapman & Hall, Ltd., London, 1951, pp. 317–329. MR 0046018
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

A decomposition principle for minimax problems

Operators obeying Hölder’s inequality

Extremum problems involving a convex process

L. McLinden, Symmetrized separable convex programming, Trans. Amer. Math. Soc. 247 (1979), 1–44. MR 517685, DOI 10.1090/S0002-9947-1979-0517685-5
Jean-Jacques Moreau, Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France 93 (1965), 273–299 (French). MR 201952, DOI 10.24033/bsmf.1625

Fonctionnelles convexes

Generalization and symmetrization of duality in geometric programming

Elmor L. Peterson, Geometric programming, SIAM Rev. 18 (1976), no. 1, 1–51. MR 395849, DOI 10.1137/1018001
Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130–143. MR 430181, DOI 10.1287/moor.1.2.130
R. Tyrrell Rockafellar, Duality and stability in extremum problems involving convex functions, Pacific J. Math. 21 (1967), 167–187. MR 211759, DOI 10.2140/pjm.1967.21.167

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
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