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Due to recent advances in the development of linear programming solvers, some of the formerly considered difficult multicommodity problems can today be solved in few minutes, even faster than with specialized methods. However, for other kind of multicommodity instances, general linear solvers can still be quite inefficient. In this paper we will give an overview of the current state-of-the-art in solving large-scale multicommodity problems, comparing an specialized interior-point algorithm with CPLEX 6.5 in the solution of difficult multicommodity problems of up lo I million of variables and 300,000 constraints. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0–1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables. [ABSTRACT FROM AUTHOR]

Semi-infinite linear programs often arise as the limit of a sequence of approximating linear programs. Hence, studying the behavior of extensions of linear programming algorithms to semi-infinite problems can yield valuable insight into the behavior of the underlying linear programming algorithm when the number of constraints or the number of variables is very large In this paper, we study the behavior of the affine-scaling algorithm on a particular semi-infinite linear programming problem. We show that the continuous trajectories converge to the optimal solution but that, for any strictly positive step, there are starting points for which the discrete algorithm converges to nonoptimal boundary points. [ABSTRACT FROM AUTHOR]

It is shown that the problem of calculating the importance factors of objects on the basis of their interval paired comparisons, which is set up as a multicriteria problem of the minimization of equally important criteria that manifest themselves as multiplicative residuals of the system of inequalities for the required factors, comes to the sequence of a few problems of linear programming. [ABSTRACT FROM AUTHOR]

We present a family of variants of the Simplex method, which are based on a Constraint-By-Constraint procedure: the solution to a linear program is obtained by solving a sequence of subproblems with an increasing number of constraints. We discuss several probabilistic models for generating linear programs. In all of them the underlying distribution is assumed to be invariant under changing the signs of rows or columns in the problem data. A weak regularity condition is also assumed. Under these models, for linear programs with d variables and m + d inequality constraints, the expected number of pivots required by these algorithms is bounded by a function of min(m, d) only. In particular this means that, for a fixed number of variables, the expected number of pivots is bounded by a constant when the number of constraints tends to infinity. Since Smale's original model [S1] satisfies our probabilistic assumptions, the same results apply to his model, although not to the particular algorithm he analyzes. We also present some results for models generating only feasible linear programs, and for Bland's pivoting rule. We conclude with a discussion of our probabilistic models, and show why they are inadequate for obtaining meaningful results unless d and m are of the same order of magnitude. [ABSTRACT FROM AUTHOR]

The article reports on the computational solution of linear programming equations involving almost-block-diagonal matrices. The author focuses on dynamic programming methods that can be used to solve weakly coupled and multi-stage economic systems. He suggests that most economic and industrial problems can be reduced to the problem of maximizing the linear form. It is suggested that finding the mathematical solutions to multi-dimensional problems is important because of the wide range of values of the solution's parameters. Dantzig's simplex algorithm enables programming problems of this type to be solved using digital computers.