The Kelley cutting plane method is one of the methods commonly used to optimize the dual function in the Lagrangian relaxation scheme. Usually the Kelley cutting plane method uses the simplex method as the optimization engine. It is well known that the simplex method leaves the current vertex, follows an ascending edge and stops at the nearest vertex. What would happen if one would continue the...

ci = 1, and all Pi in Σ. For example, the line segment between two points P and Q is the convex hull of those two points. This is clearly convex. An extremal point of a convex region is a point that does not lie on the interior of a line segment in the region. Any convex region is the convex hull of its extremal points. As was proved in [Weyl:1935], the convex hull of any finite set of points a...

We study the parallelization of the steepest-edge version of the dual sim-plex algorithm. Three diierent parallel implementations are examined, each of which is derived from the CPLEX dual simplex implementation. One alternative uses PVM, one general-purpose System V shared-memory constructs, and one the PowerC extension of C on a Silicon Graphics multi-processor. These versions were tested on ...

In this paper, we describe a method to enhance the FTRAN and BTRAN operations in the revised simplex algorithm by using a reduced basis matrix defined by basic columns and nonbasic rows. This submatrix of the standard basis matrix is potentially much smaller, but may change its dimension dynamically from iteration to iteration. For the classical product form update (“eta update”), the idea has ...

We describe a cutting plane algorithm for solving linear ordering problems. The algorithm uses a primal-dual interior point method to solve the rst few relaxations and then switches to a simplex method to solve the last few relaxations. The simplex method uses CPLEX 4.0. We compare the algorithm with one that uses only an interior point method and with one that uses only a simplex method. We so...

There are two interesting methods, in the literature, for solving fuzzy linear programming problems in which the elements of coefficient matrix of the constraints are represented by real numbers and rest of the parameters are represented by symmetric trapezoidal fuzzy numbers. The first method, named as fuzzy primal simplex method, assumes an initial primal basic feasible solution is at hand. T...

We propose a new simplex-based direct search method for unconstrained minimization of a realvalued function f of n variables. As in other methods of this kind, the intent is to iteratively improve an n-dimensional simplex through certain reflection/expansion/contraction steps. The method has three novel features. First, a user-chosen integer m̄k specifies the number of “good” vertices to be reta...