An improving procedure of the interior projective method for linear programming (original) (raw)

On the Complexity of a Class of Projective Interior Point Methods

Mathematics of Operations Research, 1995

We present a generic projective interior point algorithm for linear programming which includes modified versions of Karmarkar's original algorithm and most other projective algorithms that have been proposed as special cases. We show that this class of algorithms has a worst case iteration bound of O(√nL) and is closely related to the class of potential reduction interior point methods.

A New Method Combining Interior and Exterior Approaches for Linear Programming

Journal of Mathematics and System Science, 2015

In this paper we present a new method combining interior and exterior approaches to solve linear programming problems. With the assumption that a feasible interior solution to the input system is known, this algorithm uses it and appropriate constraints of the system to construct a sequence of the so called station cones whose vertices tend very fast to the solution to be found. The computational experiments show that the number of iterations of the new algorithm is significantly smaller than that of the second phase of the simplex method. Additionally, when the number of variables and constraints of the problem increase, the number of iterations of the new algorithm increase in a slower manner than that of the simplex method.

An interior point method for linear programming

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1990

Design of an interior point method for linear programming is discussed, and results of a simulation study reported. Emphasis is put on guessing the optimal vertex at as early a stage as possible.

On Interior-Point Methods and Simplex Method in Linear Programming

Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica

In this paper we treat numerical computation methods for linear programming. Started from the analysis of the efficiency and defficiency of the simplex procedure, we present new possibilities offered by the interior-point methods, which appears from practical necessity, from the need of efficient means of solving large-scale problems. We realise the implementation in Java of the Karmarkar's method.

The projective method for solving linear matrix inequalities

Mathematical Programming, 1997

Numerous problems in control and systems theory can be formulated in terms of linear matrix inequalities (LMI). Since solving an LMI amounts to a convex optimization problem, such formulations are known to be numerically tractable. However, the interest in LMI-based design techniques has really surged with the introduction of efficient interior-point methods for solving LMIs with a polynomial-time complexity.

A modification of Karmarkar's linear programming algorithm

Algorithmica, 1986

We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviating a priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.

Projective geometry and the outer approximation algorithm for multiobjective linear programming

2010

A key problem in multiobjective linear programming is to find the set of all efficient extreme points in objective space. In this paper we introduce oriented projective geometry as an efficient and effective framework for solving this problem. The key advantage of oriented projective geometry is that we can work with an "optimally simple" but unbounded efficiency-equivalent polyhedron, yet apply to it the familiar theory and algorithms that are traditionally restricted to bounded polytopes. We apply these techniques to Benson's outer approximation algorithm, using oriented projective geometry to remove an exponentially large complexity from the algorithm and thereby remove a significant burden from the running time.

A primal - dual exterior point algorithm for linear programming problems

Yugoslav Journal of Operations Research, 2009

The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal-Dual method is a Simplex-type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal-Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.