Algorithm For Linear Programming Problems Research Articles

Generalized affine scaling algorithms for linear programming problems

Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal Affine Scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we integrate Nesterov’s restarting strategy in the primal Affine Scaling method with an extra parameter, which in turn generalizes the original primal Affine Scaling method. We provide the proof of convergence for the proposed generalized algorithm considering long step size. We also provide the proof of convergence for the primal and dual sequence without the degeneracy assumption. This convergence result generalizes the original convergence result for the Affine Scaling methods and it gives us hints about the existence of a new family of methods. Then, we introduce a second algorithm to accelerate the convergence rate of the generalized algorithm by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal Affine Scaling method.

A Simple Projection Algorithm for Linear Programming Problems

Fujishige et al. propose the LP-Newton method, a new algorithm for linear programming problem (LP). They address LPs which have a lower and an upper bound for each variable, and reformulate the problem by introducing a related zonotope. The LP-Newton method repeats projections onto the zonotope by Wolfe’s algorithm. For the LP-Newton method, Fujishige et al. show that the algorithm terminates in a finite number of iterations. Furthermore, they show that if all the inputs are rational numbers, then the number of projections is bounded by a polynomial in L, where L is the input length of the problem. In this paper, we propose a modification to their algorithm using a binary search. In addition to its finiteness, if all the inputs are rational numbers and the optimal value is an integer, then the number of projections is bounded by $$L+1$$ , that is, a linear bound.

Search for feasible solutions by interior point algorithms

A family of interior point algorithms for linear programming problems is considered. In these algorithms, entering the feasible solution region of the original problem is presented as an optimization process of an extended problem. This extension is performed by adding a new variable. The main objective of the paper is a theoretical justification of the procedure of entering the feasible region of the original problem under the assumption of non-degeneracy of the extended problem. Specifically, it is proved that under consistent constraints of the original problemthe procedure leads to a relative interior point of the feasible region.

Some fundamental issues of basic line search algorithm for linear programming problems

In this article we present the fundamental idea, concepts and theorems of a basic line search algorithm for solving linear programming problems which can be regarded as an extension of the simplex method. However, unlike the iteration of the simplex method from a basic point to an improved adjacent basic point via pivot operation, the basic line search algorithm, also by pivot operation, moves from a basic line which contains two basic feasible points to an improved basic line which also contains two basic feasible points whose objective values are no worse than that of the two basic feasible points on the previous basic line. The basic line search algorithm may skip some adjacent vertices so that it converges to an optimal solution faster than the simplex method. For example, for a 2-dimensional problem, the basic line search algorithm can find an optimal solution with only one iteration.

An $${O(\sqrt{n}L)}$$ iteration primal-dual second-order corrector algorithm for linear programming

In this paper, we propose a primal-dual second-order corrector interior point algorithm for linear programming problems. At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction [Ai and Zhang in SIAM J Optim 16:400–417 (2005)], in an attempt to improve performance. The corrector is multiplied by the square of the step-size in the expression of the new iterate. We prove that the use of the corrector step does not cause any loss in the worst-case complexity of the algorithm. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm enjoyed the low iteration bound of \({O(\sqrt{n}L)}\), the same as the best known complexity results for interior point methods.

Average case complexity results for a centering algorithm for linear programming problems under Gaussian distributions

Abstract To solve linear programming problems by interior point methods an approximately centered interior point has to be known. Such a point can be found by an algorithmic approach – a so-called phase 1 algorithm or centering algorithm. For random linear programming problems distributed according to the rotation symmetry model, especially with normal distribution, we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm.

A primal-dual exterior point algorithm for linear programming problems

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.

An improving procedure of the interior projective method for linear programming

We propose in this study, a practical modification of Karmarkar’s projective algorithm for linear programming problems. This modification leads to a considerable reduction of the cost and the number of iterations. This claim is confirmed by many interesting numerical experimentations.

Performance evaluation of a family of criss–cross algorithms for linear programming

In this article we evaluate a family of criss–cross algorithms for linear programming problems, comparing the results obtained by these algorithms over a set of test problems with those obtained by the simplex algorithms implemented in the XPRESS commercial package. We describe the known criss–cross variants existing in the literature and introduce new versions obtained with a slight modification based on the original criss–cross algorithm. Moreover, we consider some computational details of our implementation and describe the set of test problems used.

A new efficient primal dual simplex algorithm

The purpose of this paper is to present a revised primal dual simplex algorithm (RPDSA) for linear programming problems. RPDSA has interesting theoretical properties. The advantages of the new algorithm are the simplicity of implementation, low computational overhead and surprisingly good computational performance. The algorithm can be combined with interior point methods to move from an interior point to a basic optimal solution. The new algorithm always proved to be more efficient than the classical simplex algorithm on our test problems. Numerical experiments on randomly generated sparse linear problems are presented to verify the practical value of RPDSA. The results are very promising. In particular, they reveal that RPDSA is up to 146 times faster in terms of number of iterations and 94 times faster in terms of CPU time than the original simplex algorithm (SA) on randomly generated problems of size 1200×1200 and density 2.5%. Scope and purpose In this paper, we present a new simplex-type algorithm for linear programming problems. This algorithm can be viewed as a modification of dual simplex algorithm and it performs sufficiently well in practice, particularly on linear problems of small or medium size. A revised version of the algorithm and solution of general LPs using a modified big- M method are also presented. A computational study on randomly generated sparse linear problems reveals that the new algorithm is surprisingly faster than the original simplex algorithm. In our computational study, we use three different density cases of LPs: 2.5%, 5% and 10%. Compared to the original simplex algorithm, our algorithm's found to be up to 146 times faster in terms of number of iterations and up to 94 times faster in terms of CPU time. Our algorithm can also be seen as a procedure to move from any interior point to an optimal basic solution.

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path.

Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems

This note studies A , a condition number used in the linear programming algorithm of Vavasis and Ye [14] whose running time depends only on the constraint matrix A∈ℝ m×n , and (A), a variant of another condition number due to Ye [17] that also arises in complexity analyses of linear programming problems. We provide a new characterization of A and relate A and (A). Furthermore, we show that if A is a standard Gaussian matrix, then E(ln A )=O(min{mlnn,n}). Thus, the expected running time of the Vavasis-Ye algorithm for linear programming problems is bounded by a polynomial in m and n for any right-hand side and objective coefficient vectors when A is randomly generated in this way. As a corollary of the close relation between A and (A), we show that the same bound holds for E(ln (A)).

A global-filtering algorithm for linear programming problems with stochastic elements

In this paper, an interior-point based global filtering algorithm is proposed to solve linear programming problems with the right-hand-side and cost vectors being stochastic. Previous results on the limiting properties of the Kalman filtering process have been extended to handle some non-stationary situations. A global Kalman filter, across all iterations of the interior-point method, is designed to filter out noises while improving the objective value and reducing the primal and dual infeasibilities. Under appropriate assumptions, the proposed algorithm is shown to be globally convergent to an optimal solution of the underlying “true value” system.

An Affine Scaling Algorithm for Linear Programming Problem and its Application to Utility Plant Optimization

Since Karmarkar's projective algorithm, many interior point methods for linear programming problem have been proposed. This paper presents a primal affine scaling algorithm for linear programming problem with bounded variables. This algorithm is derived from the standard affine scaling algorithm, separating the equality constraints about the bounded variables with the added slack variables. An implementation detail of this algorithm is also described. A program code based on this algorithm is developed using sparse matrix data structures, and applied to utility plant optimization problem. Numerical examples show that the proposed algorithm is practical for industrial plant optimization problem, and much superior in computing time to the standard primal affine scaling algorithm.

An exterior-point polytope sliding and deformation algorithm for linear programming problems

In this research, we develop an algorithm for linear programming problems based on a new interpretation of Karmarkar's representation for this problem. Accordingly, we examine a suitable polytope for which the origin is an exterior point, and in order to determine an optimal solution, we need to ascertain the minimum extent by which this polytope needs to be slid along a one-dimensional axis so that the origin belongs to it. To accomplish this, we employ strongly separating hyperplanes between the origin and the polytope using a closest point routine. The algorithm is further enhanced by the generation of dual solutions which enable us to deform the polytope so that it is favorably positioned with respect to the origin and the axis of sliding motion. The overall scheme is easy to implement, requires a minimal amount of storage, and produces quick good quality lower bounds for the problem in its infinite convergence process. A switchover to the simplex method or an interior point method is also possible, using the current available solution as an advanced start. Preliminary computational results are provided along with implementation guidelines.

An affine scaling algorithm for linear programming problems with inequality constraints

A primal, interior point method is developed for linear programming problems for which the linear objective function is to be maximised over polyhedra that are not necessarily in standard form. This algorithm concurs with the affine scaling method of Dikin when the polyhedron is in standard form, and satisfies the usual conditions imposed for using that method. If the search direction is regarded as a function of the current iterate, then it is shown that this function has a unique, continuous extension to the boundary. In fact, on any given face, this extension is just the value the search direction would have for the problem of maximising the objective function over that face. This extension is exploited to prove convergence. The algorithm presented here can be used to exploit such special constraint structure as bounds, ranges, and free variables without increasing the size of the linear programming problem.

A quadratically convergent $$O((\kappa + 1)\sqrt n L)$$ -iteration algorithm for theP *(κ)-matrix linear complementarity problem

An interior-point predictor-corrector algorithm for theP *(κ)-matrix linear complementarity problem is proposed. The algorithm is an extension of Mizuno—Todd—Ye's predictor—corrector algorithm for linear programming problem. The extended algorithm is quadratically convergent with iteration complexity $$O((\kappa + 1)\sqrt n L)$$ . It is the first polynomially and quadratically convergent algorithm for a class of LCPs that are not necessarily monotone.

On affine scaling and semi-infinite programming

We consider an extension of the affine scaling algorithm for linear programming problems with free variables to problems having infinitely many constraints, and explore the relationship between this algorithm and the finite affine scaling method applied to a discretization of the problem.

A new variant of the primal affine scaling algorithm for linear programs

In this paper, a new variant of the primal affine scaling algorithm for linear programming problem is developed. We show that under the assumptions of bounded feasible region and primal and dual nondegeneracy, the algorithm generates two sequences, {xk } and {π k }, of points that converge to the primal and the dual optimal solutions, respectively. Compared to the primal affine scaling algorithm proposed by Barnes, Cavalier-Soyster, and Vanderbei-Meketon-Freedman, the new algorithm shows a better asymptotic convergent rate. For a standard linear program with n variables and m constraints, the limiting convergent rate of the new algorithm is favored by a factor of . This result is obtained by expending “ball” to “circular cone” in the step of finding a direction vector.

Packing, Covering and Partitioning Problems with Strongly Unimodular Constraint Matrices

A 0-1 matrix A is called strongly unimodular if all its nonsingular square submatrices are triangular. We present an efficient algorithm for linear programming problems in binary variables, when all constraints are of the packing, covering or partitioning type, and the constraint matrix is strongly unimodular. The algorithm uses a certain decomposition of strongly unimodular matrices into their so-called “restricted unimodular” components, and an efficient optimization algorithm for linear programs with restricted unimodular constraints.