MATHEMATICAL PROGRAMMING (CS 856/450)
Assignment 1 :
- Devise a linear program formulation
for the ATM Cash Problem discussed in the class.
- Show that the General, Canonical and Standard Formulations of
Linear Programs are equivalent.
- Show the equivalence of the Canonical Form to a polyhedron
formed by the intersection of m halfspaces.
- Prove that degenerate basic feasible solutions
which are same in non-zero components correspond to the same vertex.
- Show: If no adjacent BFS of a BFS B has cost less than that of B
then B is optimal.
- Supoose b1 and b2 are two BFS's which are not adjacent to
each other and C1 and C2 are the columns corresponding to
the BFS's b1 and b2 then there exists another
basis in .
- Prove that if e is an edge of a polyhedron and , i.e
,then z1,z2 belong to e.
- Resolve the assumption b>=0 in the derivation of Simplex Method.
- Prove that Bland's Rule which is
Use lowest numbered column for basis change
and use lowest numbered row for determining .
prevents cyclicity in the Simplex Method.
- Design Linear Programs which demonstrate the relationship possible
between primal & dual from the set of properties of bounded feasiblity,
unbounded feasibility and infeasibility.
- Analyze the worst case behaviour of the Simplex. This should include
the time for pivoting also.