Genericity Results in Linear Conic Programming—A Tour d’Horizon

This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all proble...

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Bibliographic Details
Published in:Mathematics of operations research 2017-02, Vol.42 (1), p.77-94
Main Authors: Dür, Mirjam, Jargalsaikhan, Bolor, Still, Georg
Format: Article
Language:English
Subjects:
Analysis
conic optimization
generic properties
Geometry
Linear programming
Operations research
Set theory
Slater’s condition
stability
strict complementarity
Studies
Uniqueness
uniqueness and nondegeneracy of optimal solutions
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Summary:This paper is concerned with so-called generic properties of general linear conic programs. Many results have been obtained on this subject during the last two decades. For example, it is known that uniqueness, strict complementarity, and nondegeneracy of optimal solutions hold for almost all problem instances. Strong duality holds generically in a stronger sense, i.e., it holds for a generic subset of problem instances. In this paper, we survey known results and present new ones. In particular we give an easy proof of the fact that Slater’s condition holds generically in linear conic programming. We further discuss the problem of stability of uniqueness, nondegeneracy, and strict complementarity. We also comment on the fact that in general, a conic program cannot be treated as a smooth problem and that techniques from nonsmooth geometric measure theory are needed.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.2016.0793