A Primal-Dual Interior Point Algorithm for Linear Programming

Mathematics of Operations Research, 1997

In the adaptive step primal-dual interior point method for linear programming, polynomial algorithms are obtained by computing Newton directions towards targets on the central path, and restricting the iterates to a neighborhood of this central path. In this paper, the adaptive step methodology is extended, by considering targets in a certain central region, which contains the usual central path, and subsequently generating iterates in a neighborhood of this region. The size of the central region can vary from the central path to the whole feasible region by choosing a certain parameter. An 𝒪(√nL) iteration bound is obtained under mild conditions on the choice of the target points. In particular, we leave plenty of room for experimentation with search directions. The practical performance of the new primal-dual interior point method is measured on part of the Netlib test set for various sizes of the central region.

SIAM Journal on Optimization, 1992

This paper presents a convergence rate analysis for interior point primal-dual linear programming algorithms. Conditions that guarantee Q-superlinear convergence are identified in two distinct theories. Both state that, under appropriate assumptions, Qsuperlinear convergence is achieved by asymptotically taking the step to the boundary of the positive orthant and letting the barrier parameter approach zero at a rate that is superlinearly faster than the convergence of the duality gap to zero. The first theory makes no nondegeneracy assumption and explains why in recent numerical experimentation Q-superlinear convergence was always observed. The second theory requires the

Modern convex optimization, has been one of the most exciting and active research areas in optimization starting from 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interior-point algorithms for solving convex programing problem(CPP) and the depth and elegance of the underlying optimization theory. This article includes mathematical formation and application of the primal-dual interior-point method to solve CPP.

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.

In this paper, we describe a new primal-dual interior point method for finding search directions for linear optimization. The simplex method can be interpreted as a systematic procedure for approaching an optimal extreme point satisfying the Karush-Kuhn-Tucker (KKT) conditions. At each iteration, primal feasibility is sat-isfied. Also complementary slackness is always satisfied. Condition dual feasibility is partially violated during the iterations of the simplex method. The dual feasibility is violated, until of course, the optimal solution is reached. Therefore, we com-pute optimal solution by interior point method that was satisfied in KKT condition. KKT conditions are the nonlinear equations system. So, we reduce the premen-tioned system to the linear matrix system and change the coefficient matrix to the symmetric and positive definite matrix for applying the Cholesky factorization. We improve the (x, y, s) by computing the search-directions in different cases. Finally, the met...

Kybernetika, 2024

In this paper, we first present a polynomial-time primal-dual interior-point method (IPM) for solving linear programming (LP) problems, based on a new kernel function (KF) with a hyperbolic-logarithmic barrier term. To improve the iteration bound, we propose a parameterized version of this function. We show that the complexity result meets the currently best iteration bound for large-update methods by choosing a special value of the parameter. Numerical experiments reveal that the new KFs have better results comparing with the existing KFs including log t in their barrier term. To the best of our knowledge, this is the first IPM based on a parameterized hyperboliclogarithmic KF. Moreover, it contains the first hyperbolic-logarithmic KF (Touil and Chikouche in Filomat 34:3957-3969, 2020) as a special case up to a multiplicative constant, and improves significantly both its theoretical and practical results.

SIAM Journal on Optimization, 1993

The choice of the centering parameter and the step-length parameter are the fundamental issues in primal-dual interior-point algorithms for linear programming. Various choices for these two parameters have been proposed that lead to polynomial algorithms. Recently, Zhang, Tapia and Dennis derived conditions for the choices of the two parameters that were sufficient for superlinear or quadratic convergence. However, prior to this work it had not been shown that these conditions for fast convergence are compatible with the choices that lead to polynomiality; none of the existing polynomial primal-dual interior-point algorithms satisfies these fast convergence requirements. This paper gives an affirmative answer to the question: can a primal-dual algorithm be both polynomial and superlinearly convergent for general problems? We construct and analyze a "large step" algorithm that possesses both polynomiality and, under the assumption of the convergence of the iteration sequence, Q-superlinear convergence. For nondegenerate problems, the convergence is actually Q-quadratic.

Operations Research Letters, 1994

We present a predictor ~corrector algorithm for solving a primal dual pair of linear programming problems, The algorithm starts from an infeasible interior point and it solves the pair in O(nL) iterations, where n is the number of variables and L is the size of the problems. At each iteration of the algorithm, the predictor step decreases the infeasibility and the corrector step decreases the duality gap. The main feature of the algorithm is the simplicity of the predictor step, which performs a line search along a fixed search direction computed at the beginning of the algorithm. The corrector step uses a procedure employed in a feasible-interior-point algorithm. The proof of polynomiality is also sirnple.

Journal of Computational and Applied Mathematics, 2017

The purpose of this paper is to improve the complexity of a large-update primal-dual interior point method for a semidefinite programming problem. We define a proximity function for the semidefinite programming problem based on a new parametric kernel function and prove that the worst-case iteration bound for the new correspondent algorithm is O √ n (ln n) pq+1 pq ln n ε , where p, q ≥ 1.

SIAM Journal on Optimization, 2004

Recently, so-called self-regular barrier functions for primal-dual interior-point methods (IPMs) for linear optimization were introduced. Each such barrier function is determined by its (univariate) self-regular kernel function. We introduce a new class of kernel functions. The class is defined by some simple conditions on the kernel function and its derivatives. These properties enable us to derive many new and tight estimates that greatly simplify the analysis of IPMs based on these kernel functions. In both the algorithm and its analysis we use a single neighborhood of the central path; the neighborhood naturally depends on the kernel function. An important conclusion is that inverse functions of suitable restrictions of the kernel function and its first derivative more or less determine the behavior of the corresponding IPMs. Based on the new estimates we present a simple and unified computational scheme for the complexity analysis of kernel function in the new class. We apply this scheme to seven specific kernel functions. Some of these functions are self-regular, and others are not. One of the functions differs from the others, and from all self-regular functions, in the sense that its growth term is linear. Iteration bounds for both large-and small-update methods are derived. It is shown that small-update methods based on the new kernel functions all have the same complexity as the classical primal-dual IPM, namely, O( √ n log n ε ). For large-update methods the best obtained bound is O( √ n(log n) log n ε ), which until now has been the best known bound for such methods.