Mathematical Programming

, Volume 57, Issue 1–3, pp 15–48 | Cite as

Partially finite convex programming, Part I: Quasi relative interiors and duality theory

  • J. M. Borwein
  • A. S. Lewis
Article

Abstract

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable.

AMS 1985 Subject Classifications

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

Key words

Convex programming duality constraint qualification Fenchel duality semi-infinite programming 

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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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