Presentation on theme: "Forty Years of Corner Polyhedra. Two Types of I.P. All Variables (x,t) and data (B,N) integer. Example: Traveling Salesman Some Variables (x,t) Integer,"— Presentation transcript:
Two Types of I.P. All Variables (x,t) and data (B,N) integer. Example: Traveling Salesman Some Variables (x,t) Integer, some continuous, data continuous. Example: Scheduling,Economies of scale. Corner Polyhedra relevant to both
Corner Polyhedra Origins Stock Cutting Computing Lots of Knapsacks Periodicity observed Gomory-Gilmore 1966 "The Theory and Computation of Knapsack Functions
Comparing Integer Programs and Corner Polyhedron General Integer Programs – Complex, no obvious structure Corner Polyhedra – Highly structured, but complexity increases rapidly with group size. Next Step: Making this supply of cutting planes available for non-integer data and continuous variables. Gomory-Johnson 1970
Cutting Planes for Type Two Example: Gomory Mixed Integer Cut Variables t i Integer Variables t +, t - Non-Integer Valid subadditive function
Summary Corner Polyhedra are very structured The structure can be exploited to create the 2D facets analogous to the Gomory Mixed Integer Cut There is much more to learn about Corner Polyhedra and it is learnable
Challenges Generalize cuts from 2D to n dimensions Work with families of cutting planes (like stock cutting) Introduce data fuzziness to exploit large facets and ignore small ones Clarify issues about functions that are not piecewise linear.
References Some Polyhedra Related to Combinatorial Problems, Journal of Linear Algebra and Its Applications, Vol. 2, No. 4, October 1969, pp.451-558 Some Continuous Functions Related to Corner Polyhedra, Part I with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 1, North-Holland, August, 1972, pp. 23-85. Some Continuous Functions Related to Corner Polyhedra, Part II with Ellis L. Johnson, Mathematical Programming, Vol. 3, No. 3, North-Holland, December 1972, pp. 359- 389. T-space and Cutting Planes Paper, with Ellis L. Johnson, Mathematical Programming, Ser. B 96: Springer-Verlag, pp 341-375 (2003).