Cutting-plane method

rdf:langString Cutting-plane method

rdf:langString May 2022

rdf:langString November 2019

rdf:langString First: this is rather technical detail of the simplex algorithm, which cannot be presumed known to readers of this article. Second: this appears to presume an equality Ax=b formulation of the linear program rather than the inequality Ax≤b form used above, and that transformation will introduce extra variables. Do we know those have to be integer-valued? Third: do we even need to distinguish basic and nonbasic variables??

rdf:langString An explanation would be interesting, as it would make the statement feel less opinionated.

rdf:langString In mathematical optimization, the cutting-plane method is any of a variety of optimization methods that iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts. Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily differentiable convex optimization problems. The use of cutting planes to solve MILP was introduced by Ralph E. Gomory. Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a corner point that is optimal. The obtained optimum is tested for being an integer solution. If it is not, there is guaranteed to exist a linear inequality that separates the optimum from the convex hull of the true feasible set. Finding such an inequality is the separation problem, and such an inequality is a cut. A cut can be added to the relaxed linear program. Then, the current non-integer solution is no longer feasible to the relaxation. This process is repeated until an optimal integer solution is found. Cutting-plane methods for general convex continuous optimization and variants are known under various names: Kelley's method, Kelley–Cheney–Goldstein method, and bundle methods. They are popularly used for non-differentiable convex minimization, where a convex objective function and its subgradient can be evaluated efficiently but usual gradient methods for differentiable optimization can not be used. This situation is most typical for the concave maximization of Lagrangian dual functions. Another common situation is the application of the Dantzig–Wolfe decomposition to a structured optimization problem in which formulations with an exponential number of variables are obtained. Generating these variables on demand by means of delayed column generation is identical to performing a cutting plane on the respective dual problem.