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Reconnect ‘04 A Couple of General Classes of Cutting Planes. Cynthia Phillips Sandia National Laboratories. Knapsack Cover (KC) Inequalities. A. C. Moving Away from Graphs. The cuts apply to more general For this discussion, assume Let I be a set of variable indices such that.

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Reconnect 04 a couple of general classes of cutting planes

Reconnect ‘04A Couple of General Classes of Cutting Planes

Cynthia Phillips

Sandia National Laboratories



Moving away from graphs
Moving Away from Graphs

The cuts apply to more general

For this discussion, assume

Let I be a set of variable indices such that


Cover cuts
Cover Cuts

We can remove the assumption that

Consider a general inequality

Set

Apply a regular cover cut to and substitute


Review linear programming basis

B

L

U

Review: Linear Programming Basis

What does a corner look like algebraically?

Ax=b

Partition A matrix into three parts

where B is nonsingular (invertible, square).

Reorder x: (xB, xL, xU)

We have BxB + LxL + UxU = b

xB

xL

xU


A basic solution
A Basic Solution

We have BxB + LxL + UxU = b

Set all members of xL to their lower bound.

Set all members of xU to their upper bound.

Let (this is a constant because bounds and u are)

Thus we have

Set

So we can express each basic variable in the current optimal LP solution x* as a function of the nonbasic variables.


Gomory cuts
Gomory Cuts

Assume we have a pure integer program (not necessarily binary)

Express each basic variable in the current optimal LP solution x* as a function of the nonbasic variables (tableau):

fr(gj) is the fractional part of gj

Split gj into integral and fractional pieces:



Gomory cuts2
Gomory Cuts

In a feasible solution xi is integral (pure integer program), so the whole left side is integral. Thus the right side must be as well:

This is (one type of) Gomory Cut.


Global validity
Global Validity

Cuts like the TSP subtour elimination cuts are globally valid (apply to all subproblems).

  • Can be shared

    Recall the key step for Gomory cuts:


Global validity1
Global Validity

We require for the and uj in effect at the subproblem where the Gomory cut was generated.

  • Gomory cuts are globally valid for binary variables

    • Need fixed at 1 to be fixed at upper and fixed at 0 to be at lower

  • Gomory cuts are not generally valid for general integer variables


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