An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations

The main result of this paper is an O(i-n L ) iteration interior-point algorithm for linear programming, that does not need to follow the central trajectory to retain the complexity bound and so can be augmented by a standard line search of the potential function. This where q = n + fii. We show that this algorithm is optimal with respect to the choice of q in the following sense. Suppose that q = n + nt where t > 0. Then the algorithm will solve the linear program in O( nr L ) iterations, where r = max ( t, 1 -t }. Thus the value of t that minimizes the complexity bound is t = 1/2.

mentioning

confidence: 99%

Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function

Freund

1991

“…solving linear programming problems, see Renegar [19], Gonzaga [13], and Monteiro and Adler [17,18], among others.…”

Section: IImentioning

confidence: 99%

“…Karmarkar Note that P is a generalization of the analytic center problem first analyzed by Sonnevend [22], [23]. This problem has had numerous applications in mathematical programming, see Renegar [19], Gonzaga [13], and Monteiro and Adler [17,18], among others. Also note the P is defined for the most general polyhedral representation, namely inequality as well as equality constraints of arbitrary form.…”

Section: Introductionmentioning

confidence: 99%

Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

Freund

1993

The purpose of this study is to broaden the scope of projective transformation methods in mathematical programming, both in terms of theory and algorithms. We start by generalizing the concept of the analytic center of a polyhedral system of constraints to the w-center of a polyhedral system, which stands for weighted center, where there are positive weights on the logarithmic barrier terms for reach inequality constraint defining the polyhedron X . We prove basic results regarding contained and containing ellipsoids centered at the w-center of the system X . We next shift attention to projective transformations, and we exhibit an elementary projective transformation that transforms the polyhedron X to another polyhedron Z , and that transforms the current interior point to the w-center of the transformed polyhedron Z . We work throughout with a polyhedral system of the most general form, namely both inequality and equality costraints.This theory is then applied to the problem of finding the w-center of a polyhedral system X . We present a projective transformation algorithm, which is an extension of Karmarkar's algorithm, for finding the w-center of the system X . At each iteration, the algorithm exhibits either a fixed constant objective function improvement, or converges superlinearly to the optimal solution. The algorithm produces upper bounds on the optimal value at each iteration. The direction chosen at each iteration is shown to be a positively scaled Newton direction. This broadens a result of Bayer and Lagarias regarding the connection between projective transformation methods and Newton's method. Furthermore, the algorithm specializes to Vaidya's algorithm when used with a line-search, and so shows that Vaidya's algorithm is superlinearly convergent as well. Finally, we show how the algorithm can be used to construct well-scaled containing and contained ellipsoids at near-optimal solutions to the w-center problem.

“…Moreover, he used rank one updates to obtain an O(n3'SL) algorithm for linear programming. Since then, there have been several path following algorithms for linear programming that need O(v/'~L) iterations (Renegar [12], Gonzaga [4]) with time complexity O(n3L). The drawback of the latter approaches is that, although they achieve the best known worst case bound, they are restricted to make small steps by following the central trajectory.…”

Section: Introductionmentioning

confidence: 99%

On the worst case complexity of potential reduction algorithms for linear programming

Bertsimas¹,

Luo

1997

There are several classes of interior point algorithms that solve linear programming problems in O(x/ffL) iterations. Among them, several potential reduction algorithms combine both theoretical (o(vr~L) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(v/'ffL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least ll(v/ffL) iterations to find an optimal solution. (~) 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.