A. I. Golikov

Parallel implementation of Newton’s method for solving large-scale linear programs

Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.

New perspective on the theorems of alternative

New general theorems of the alternative are presented. The constructive proofs based on the duality theory are given. From these results many well-known theorems of the alternative are obtained by simple substitutions. Computational applications of theorems of the alternative to solving linear systems, LP and NLP problems are given. A linear systems of possibly unsolvable equalities and inequalities are considered. With original linear system an alternative system is associated such that one and only one of these systems is consistent. If the original system is solvable then numerical method for solving this system consists of minimization of the residual of the alternative inconsistent system. From the results of this minimization the normal solution of the original system is determined.

Finding the Projection of a Given Point on the Set of Solutions of a Linear Programming Problem

The problem of finding the projections of points on the sets of solutions of primal and dual problems of linear programming is considered. This problem is reduced to a single solution of the problem of minimizing a new auxiliary function, starting from some threshold value of the penalty coefficient. Estimates of the threshold value are obtained. A software implementation of the proposed method is compared with some known commercial and research software packages for solving linear programming problems.

Generalized Newton method for linear optimization problems with inequality constraints

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.

Sensitivity function: Properties and applications

The sensitivity function induced by a convex programming problem is examined. Its monotonicity, subdifferentiability, and closure properties are analyzed. A relation to the Pareto optimal solution set of the multicriteria convex optimization problem is established. The role of the sensitivity function in systems describing optimization problems is clarified. It is shown that the solution of these systems can often be reduced to the minimization of the sensitivity function on a convex set. Numerical methods for solving such problems are proposed, and their convergence is proved.

On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.

Parallel Implementation of Generalized Newton Method for Solving Large-Scale LP Problems

The augmented Lagrangian and Generalized Newton methods are used to simultaneously solve the primal and dual linear programming (LP) problems. We propose parallel implementation of the method to solve the primal linear programming problem with very large number (≈ 2 ·106) of nonnegative variables and a large (≈ 2 ·105) number of equality type constraints.

Projective-Dual Method for Solving Systems of Linear Equations with Nonnegative Variables

In order to solve an underdetermined system of linear equations with nonnegative variables, the projection of a given point onto its solutions set is sought. The dual of this problem—the problem of unconstrained maximization of a piecewise-quadratic function—is solved by Newton’s method. The problem of unconstrained optimization dual of the regularized problem of finding the projection onto the solution set of the system is considered. A connection of duality theory and Newton’s method with some known algorithms of projecting onto a standard simplex is shown. On the example of taking into account the specifics of the constraints of the transport linear programming problem, the possibility to increase the efficiency of calculating the generalized Hessian matrix is demonstrated. Some examples of numerical calculations using MATLAB are presented.

A New Class of Theorems of the Alternative

The connection is established between theorems of the alternative for linear systems of equations and/or inequalities and duality theorems in linear programming. We give new versions of theorems of the alternative in which the alternative systems have different matrices of various sizes.

Regularization and normal solutions of systems of linear equations and inequalities

The paper provides some examples of mutually dual unconstrained optimization problems originating from regularization problems for systems of linear equations and/or inequalities. The solution of each of these mutually dual problems can be found from the solution of the other problem by means of simple formulas. Since mutually dual problems have different dimensions, it is natural to solve the unconstrained optimization problem of the smaller dimension.