From a practical standpoint, we don't need a full representation of conv(x), only an approximation around the optimal solution Assume that that feasible set is x = p \ zn Typically, f may contain a large number of elements (exponentially many) Thus we cannot introduce all inequalities a priori. The cutting plane method is very useful for solving integer programming problems, but there is a di culty lies in the choice of inequalities which represent the cut of only a very small piece of the feasible set of the linear programming relaxation. By using a cutting plane method, we can (hopefully) nd the optimal point without computing the entire convex hull
One famous method for creating valid cuts is called the gomory cut, discovered by american mathematician ralph gomory (1950). “new and interesting” greedy strategies for maxcut and mvc “which intuitively make sense but have not been analyzed before,” thus could be a “good assistive tool for discovering new algorithms.” Cutting plane is the first algorithm developed for integer programming that could be proved to converge in a finite number of steps Even though the algorithm is considered not efficient, it has provided insights into integer programming that have led to other, more efficient, algorithms. The algorithm development process uses great deluge heuristic method. Gomory proposed a nite cutting plane algorithm for pure ips (1958)
In practice, { these algorithms are hopeless except some very easy cases. Linear inequalities called cutting planes Numerous variants of this basic idea are among standard tools used in convex nonsmooth ptimization and integer linear programing Optimization methods which are based on the idea of iteratively refining the objective function or set of feasible constraints of a problem through linear inequalities
OPEN
