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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by L. McLinden

Trans. Amer. Math. Soc. 245 (1978), 147-181

DOI: https://doi.org/10.1090/S0002-9947-1978-0511404-3

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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.

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