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About 6,096 papers for “Interior point method”

Interior-point method flowchart.

Du Chigan (22320279), Jianbei Liu (22320282), Yang Zhao (39052) et al. · Figshare · 2025

The process terminates when convergence criteria are satisfied; otherwise, constraint parameters are updated and iterative solving proceeds.

Cited by 0Open access

Affine-scaling interior point method

Lauschke, Andreas · Acervo Digital da Universidade Estadual Paulista (Universidade Estadual Paulista) · 2011

Interior point methods have taken the world of linear optimization by storm. Explore the path taken by an affine-scaling interior point method (a variant of Karmarkar's original 1984 primal projection method) for a simple linear optimization problem, defined by constraints, objective function, and initial point

Cited by 0

An Inexact Interior Point method

BELLAVIA, STEFANIA · Florence Research (University of Florence) · 1998

In this paper, we introduce an inexact interior-point algorithm for a constrained system of equations. The formulation of the problem is quite general and includes nonlinear complementarity problems of various kinds. In our convergence theory, we interpret the inexact interior-point method as an inexact Newton method. This enables us to establish a global convergence theory for the proposed algorithm. Under the additional assumption of the invertibility of the Jacobian at the solution, the superlinear convergence of the iteration sequence is proved.

Cited by 0

Affine-Scaling Interior Point Method

Wolfram Demonstrations Project · 2008

Interior point methods have taken the world of linear optimization by storm. Explore the path taken by an affine-scaling interior point method (a variant of Karmarkars original 1984 primal projection method) for a simple linear optimization problem defined by constraints objective function and initial point.;

Cited by 1Open access

Algorithms of the interior point method

Valery Zorkaltsev · 2017

The results of development, theoretical justification and experimental research of interior point algorithms for solving linear programming problems are presented in the article.

Cited by 3

Interior Point Method: History and Prospects

V. I. Zorkaltsev · Computational Mathematics and Mathematical Physics · 2019

Two mutually dual families of interior point algorithms are considered. The history of creating the algorithms, the main theoretical results on their justification, the experience of practical use, possible directions of development, and methods for counteracting calculation errors are presented. Subsets of algorithms with various special properties are distinguished, including those that necessarily lead to relatively interior points of optimal solutions. An algorithm for finding the Chebyshev projection onto a linear manifold is presented, in which the properties of relatively interior points of optimal solutions are efficiently employed. This algorithm always elaborates a unique projection and allows one to dispense with the hard-to-verify and sometimes violated Haar condition.

Cited by 6

An interior point method for constrained saddle point problems

Alfredo N. Iusem, Markku Kallio · Computational and Applied Mathematics · 2004

We present an algorithm for the constrained saddle point problem with a convexconcave function L and convex sets with nonempty interior.The method consists of moving away from the current iterate by choosing certain perturbed vectors.The values of gradients of L at these vectors provide an appropriate direction.Bregman functions allow us to define a curve which starts at the current iterate with this direction, and is fully contained in the interior of the feasible set.The next iterate is obtained by moving along such a curve with a certain step size.We establish convergence to a solution with minimal conditions upon the function L, weaker than Lipschitz continuity of the gradient of L, for instance, and including cases where the solution needs not be unique.We also consider the case in which the perturbed vectors are on certain specific curves starting at the current iterate, in which case another convergence proof is provided.In the case of linear programming, we obtain a family of interior point methods where all the iterates and perturbed vectors are computed with very simple formulae, without factorization of matrices or solution of linear systems, which makes the method attractive for very large and sparse matrices.The method may be of interest for massively parallel computing.Numerical examples for the linear programming case are given.

Cited by 4

The interior-point method for linear programming

Greg Astfalk, Irvin J. Lustig, Roy E. Marsten et al. · IEEE Software · 1992

A robust, reliable, and efficient implementation of the primal-dual interior-point method for linear programs, which is based on three well-established optimization algorithms, is presented. The authors discuss the theoretical foundation for interior-point methods which consists of three crucial building blocks: Newton's method for solving nonlinear equations, Joseph Lagrange's methods for optimization with equality constraints, and Fiacco and McCormick's barrier method for optimization with inequality constraints. The construction of the primal-dual interior-point method using these methods is described. An implementation of the primal-dual interior-point method, its performance, and a comparison to other interior-point methods are also presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Cited by 49

Interior point method in tensor optimal transport

Shmuel Friedland · arXiv (Cornell University) · 2023

We study a tensor optimal transport (TOT) problem for $d\ge 2$ discrete measures. This is a linear programming problem on $d$-tensors. We introduces an interior point method (ipm) for $d$-TOT with a corresponding barrier function. Using a "short-step" ipm following central path within $\varepsilon$ precision we estimate the number of iterations.

Cited by 0Open access

Interior Point Method for Utility Plant Optimization

Masahiko Murai, Yutaka Iino, Takashi Shigemasa · Transactions of the Society of Instrument and Control Engineers · 1997

This paper focuses on the re-scheduling problem for utility plant operation. The optimal scheduling problem is described as LP model, and solved by the primal affine scaling algorithm. We try to re-optimize the perturbed LP model using warm-start. Numerical example shows that the warm-start solve the problem about half the time which the cold-start does.

Cited by 0Open access

Parallel inexact Newton and interior point method

L. Bergamaschi, G. Zilli · 2000

In this paper we present the results obtained in the solution of sparse and large systems of non-linear equations by inexact Newton methods combined with an block iterative row-projection linear solver of Cimmino-type. Moreover, we propose a suitable partitioning of the Jacobian matrix A. In view of the sparsity, we obtain a mutually orthogonal row-partition of A that allows a simple solution of the linear least squares subproblems. We present numerical results obtained on a CRAY-T3E when this method is used to solve both non linear problems arising from dicretization of PDEs. Preliminary sequential results are also shown in the solution of nonlinear mixed complementary problems solved with interior point methods. 1. -- The Inexact Newton and Quasi-Newton Cimmino method We are concerned with the numerical solution of a system of nonlinear equations F (x) = 0 F = (f 1 ; :::; fn ) T (1) ( )Indirizzo dell&amp;apos;autore: Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Universit&amp;apos;a di Padova, Via Belzoni 7, 35131 Padova, ITALY 2 l. bergamaschi and g. zilli where F : R n ! R n is a nonlinear C 1 function, and its Jacobian matrix J(x) is sparse and n is large. Moreover, in Section 5 we solve the nonlinear systems which arise from the application of the primal-dual interior point method (see [7]) to mixed complementary problems. Let As = b be the linear system to be solved. Here below we give the general lines of the block iterative Cimmino method. Let us partition A into p rowblocks: A i ; i = 1; : : : ; p; i.e. A T = [A 1 ; A 2 ; : : : ; A p ] and partition the vector b conformally. Then the original system is premultiplied (preconditioning) by H p = [A + 1 ; : : : ; A + i ; : : : ; A + p ] (2) where A + i = A T i (A i A T i ) \\Ga...

Cited by 0