Cutting Plane Methods for Global Optimization

In solving global and combinatorial optimization problems cuts are used as a device to discard portions of the feasible set where it is known that no optimal solution can be found. Specifically, given the optimization problem

(1)

if x ⁰ is an unfit solution and there exists a function l(x) satisfying l(x ⁰)> 0, while l(x)≤ 0 for every optimal solution x, then by adding the inequality l(x)≤ 0 to the constraint set we exclude x ⁰ without excluding any optimal solution. The inequality l(x)≤ 0 is called a valid cut , or briefly, a cut. Most often the function l(x) is affine: the cut is then said to be linear, and the hyperplane l(x) = 0 is called a cutting plane. However, nonlinear cuts have proved to be useful, too, for a wide class of problems.

Cuts may be employed in different contexts: outer and inner approximation (conjunctive cuts), branch and bound (disjunctive cuts), or in combined form.

Outer Approximation

Let Ω⊂ R ⁿ be the set of optimal solutions of problem (2). Suppose there...