A new infeasible interior-point algorithm for linear programming

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.

Mathematical Programming, 1989

We consider the generalization of a variant of Karmarkar's algorithm to semi-infinite programming. The extension of interior point methods to infinite-dimensional linear programming is discussed and an algorithm is derived. An implementation of the algorithm for a class of semi-infinite linear programs is described and the results of a number of test problems are given. We pay particular attention to the problem of Chebyshev approximation. Some further results are given for an implementation of the algorithm applied to a discretization of the semi-infinite linear program, and a convergence proof is given in this case.

Optimization Methods and Software, 1995

This paper describes two algorithms for the problem of minimizing a linear function over the intersection of an a ne set and a convex set which is required to be the closure of the domain of a strongly self-concordant barrier function. One algorithm is a path-following method, while the other is a primal potential-reduction method. We give bounds on the number of iterations necessary to attain a given accuracy.

Mathematical Programming, 1993

This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primal-dual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special case of Rule G, ensures the O(nL) iteration polynomial-time computational complexity. Both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions. They rely neither on neighborhoods of the central trajectory nor on potential function. These rules allow large steps without performing any line search. Rule G is especially exible enough for implementation in practically e cient primal-dual interior point algorithms.

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.

SIAM Journal on Optimization, 1998

A primal-dual infeasible-interior-point path-following algorithm is proposed for solving semide nite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms nds an optimal solution in at most O( p nL) iterations, where n is the size of the problem and L is the logarithm of the ratio of the initial error and the tolerance. If the starting point is large enough then the algorithm terminates in at most O(nL) steps either by nding a solution or by determining that the primal-dual problem has no solution of norm less than a given number. Moreover, we propose a su cient condition for the superlinear convergence of the algorithm. In addition, we give two special cases of SDP for which the algorithm is quadratically convergent.

1991

The notion of the central path plays an important role in the convergence analysis of interior-point methods. Many interior-point algorithms have been developed based on the principle of following the central path, either closely or otherwise. However, whether such algorithms actually converge to the center of the solution set has remained an open question. In this paper, we demonstrate that under mild conditions, when the iteration sequence generated by a primal-dual interior-point method converges, it converges to the center of the solution set.

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.

1999

The paper studies numerical stability problems arising in the application of interior-point methods to primal degenerate linear programs. A stabilization procedure based on Gaussian elimination is proposed and it is shown that it stabilizes all path following methods, original and modified Dikin's method, Karmarkar's method, etc.

In this paper we use the interior point methodology to cover the main issues in linear programming: duality theory, parametric and sensitivity analysis, and algorithmic and computational aspects. The aim is to provide a global view on the subject matter.

1996

This note derives bounds on the length of the primal-dual affine scaling directions associatedwith a linearly constrained convex program satisfying the following conditions: 1) the problemhas a solution satisfying strict complementarity, 2) the Hessian of the objective function satisfiesa certain invariance property. We illustrate the usefulness of these bounds by establishing thesuperlinear convergence of the algorithm presented in Wright and

Mathematical Programming, 1998

The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrix A. The main drawback of the algorithm is the use of an unknown big constant ZA in computing the search direction and to initiate the algorithm. We propose a modified layered-step interior-point algorithm which does not use the big constant in computing the search direction. The constant is required only for initialization when a well-centered feasible solution is not available, and it is not required if an upper bound on the norm of a primal dual optimal solution is known in advance. The complexity of the simplified algorithm is the same as that of Vavasis and Ye.

RAIRO - Operations Research, 2007

This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorithm to an associate problem allows to obtain one. Finally, we present some numerical experiments which show that the algorithm works properly.

Mathematics of Operations Research, 1999

We consider an infeasible-interior-point algorithm, endowed with a nite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)).

Computational Optimization and Applications, 2019

Numerical experiments using the Netlib test set are made, which show that this approach is competitive when compared to well established solvers, such as PCx.