Degeneracy Graphs and Simplex Cycling

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 vertices, 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 describing and solving degeneracy problems. The study of degeneracy graphs opens a new field of research with many theoretical aspects and practical applications. The present publication 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 degeneracy graphs will be used to explain simplex cycling, i. e. necessary and sufficient conditions for cycling will be derived.

1. Introduction. - 2. Degeneracy problems in mathematical optimization. - 2. 1. Convergence problems in the case of degeneracy. - 2. 2 Efficiency problems in the case of degeneracy. - 2. 3 Degeneracy problems within the framework of postoptimal analysis. - 2. 4. On the practical meaning of degeneracy. - 3. Theory of degeneracy graphs. - 3. 1. Fundamentals. - 3. 2 Theory of ? × n-degeneracy graphs. - 3. 3. Theory of 2 × n-degeneracy graphs. - 4. Concepts to explain simplex cycling. - 4. 1. Specification of the question. - 4. 2 A pure graph theoretical approach. - 4. 3 Geometrically motivated approaches. - 4. 4 A determinant approach. - 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. 3 On the construction of general cycling examples. - 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.