We show that the simplex method can be interpreted as a cutting-plane method, assumed that a special pricing rule is used. This approach is motivated by the recent success of the cutting-plane method in the solution of special stochastic programming problems. We compare the classic Dantzig pricing rule and the rule that derives from the … Read more
In this paper, we propose a simple variant of the Mizuno-Todd-Ye predictor-corrector algorithm for linear programming problem (LP). Our variant executes a natural finite termination procedure at each iteration and it is easy to implement the algorithm. Our algorithm admits an objective-function free polynomial-time complexity when it is applied to LPs whose dual feasible region … Read more
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the diameters of polyhedral graphs. One particular variant has a diameter which satisfies the best known upper bound on the diameters of … Read more
We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different … Read more
The paper proposes a polynomial algorithm for solving systems of linear inequalities. The algorithm uses a polynomial relaxation-type procedure which either finds a solution for Ax = b, 0
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the dual simplex method with the most negative pivoting rule for LP. The bound is comparable with the bound given by Kitahara and Mizuno (2010) for the primal simplex method. We apply the result to the … Read more
We give an upper bound for the number of different basic feasible solutions generated by Dantzig’s simplex method (the simplex method with the most negative pivoting rule) for LP with bounded variables by extending the result of Kitahara and Mizuno (2010). We refine the analysis by them and improve an upper bound for a standard … Read more
The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, … Read more
In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the … Read more
Kitahara and Mizuno (2010) get two upper bounds for the number of different basic feasible solutions generated by Dantzig’s simplex method. The size of the bounds highly depends on the ratio between the maximum and minimum values of all the positive elements of basic feasible solutions. In this paper, we show some relations between the … Read more