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Degeneracy Graphs and Simplex Cycling

'Lecture Notes in Economics and Mathematical Systems'. Softcover reprint of the original 1st ed. 1991. Book.…
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Many problems in economics can be formulated as linearly
constrained mathematical optimization problems, where the
feasible solution set X represents a convex polyhedral set.
In practice, the set X freque … weiterlesen
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Degeneracy Graphs and Simplex Cycling als Buch

Produktdetails

Titel: Degeneracy Graphs and Simplex Cycling
Autor/en: Peter Zörnig

ISBN: 354054593X
EAN: 9783540545934
'Lecture Notes in Economics and Mathematical Systems'.
Softcover reprint of the original 1st ed. 1991.
Book.
Sprache: Englisch.
Springer Berlin Heidelberg

13. November 1991 - kartoniert - 216 Seiten

Beschreibung

Many problems in economics can be formulated as linearly
constrained mathematical optimization problems, where the
feasible solution set X represents a convex polyhedral set.
In practice, the set X frequently contains degenerate verti-
ces, yielding diverse problems in the determination of an
optimal solution as well as in postoptimal analysis.The so-
called degeneracy graphs represent a useful tool for des-
cribing and solving degeneracy problems. The study of dege-
neracy graphs opens a new field of research with many theo-
retical aspects and practical applications. The present pu-
blication pursues two aims. On the one hand the theory of
degeneracy graphs is developed generally, which will serve
as a basis for further applications. On the other hand dege-
neracy graphs will be used to explain simplex cycling, i.e.
necessary and sufficient conditions for cycling will be de-
rived.

Inhaltsverzeichnis

1. Introduction.- 2. Degeneracy problems in mathematical optimization.- 2.1. Convergence problems in the case of degeneracy.- 2.1.1 Cycling in linear complementarity problems.- 2.1.2 Cycling in network problems.- 2.1.3 Cycling in bottleneck linear programming.- 2.1.4 Cycling in integer programming.- 2.2 Efficiency problems in the case of degeneracy.- 2.2.1 Efficiency loss by weak redundancy.- 2.2.2 Efficiency problems from the perspective of the theory of computational complexity.- 2.3 Degeneracy problems within the framework of postoptimal analysis.- 2.4. On the practical meaning of degeneracy.- Summary of Chapter 2.- 3. Theory of degeneracy graphs.- 3.1. Fundamentals.- 3.1.1 The concept of degeneracy.- 3.1.2 The graphs of a polytope.- 3.1.3 Degeneracy graphs.- 3.2 Theory of ? × n-degeneracy graphs.- 3.2.1 Foundations of the theory of finite sets.- 3.2.2 Characterization of ? × n-degeneracy graphs.- 3.2.3 Properties of ? × n-degeneracy graphs.- 3.3. Theory of 2 × n-degeneracy graphs.- 3.3.1 Characterization of 2 × n-degeneracy graphs.- 3.3.2 Properties of 2 × n-degeneracy graphs.- Summary of Chapter 3.- 4. Concepts to explain simplex cycling.- 4.1. Specification of the question.- 4.2 A pure graph theoretical approach.- 4.2.1 The concept of the LP-degeneracy graph.- 4.2.2 Characterization of simplex cycles by means of the LP-degeneracy graph.- 4.3 Geometrically motivated approaches.- 4.3.1 Fundamentals.- 4.3.2 Characterization of simplex cycles by means of the induced point set.- 4.3.3 Properties of the induced point set.- 4.3.4 Characterization of simplex cycles by means of the induced cone.- 4.4 A determinant approach.- 4.4.1 Terms and foundations.- 4.4.2 Characterization of simplex cycles by means of determinant inequality systems.- Summary of Chapter 4.- 5. Procedures for constructing cycling examples.- 5.1 On the practical use of constructed cycling examples.- 5.2 Successive procedures for constructing cycling examples.- 5.2.1 Modification of a row in the initial tableau.- 5.2.2 Modification of a column in the initial tableau.- 5.2.3 Addition of a column to the initial tableau.- 5.2.4 Addition of a row to the initial tableau.- 5.2.5 Combination of construction steps.- 5.2.5.1 Successive modification of rows.- 5.2.5.2 Successive addition of columns.- 5.2.6 Open questions in connection with the practical performance of the procedures.- 5.3 On the construction of general cycling examples.- Summary of Chapter 5.- A. Foundations of linear algebra and the theory of convex polytopes.- B. Foundations of graph theory.- C. Problems in the solution of determinant inequality systems.- References.
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