A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

Generalized linear fractional programming under interval uncertainty

Data in many real-life engineering and economical problems suffer from inexactness. Herein we assume that we are given some intervals in which the data can simultaneously and independently perturb. We consider a generalized linear fractional programming problem with interval data and present an efficient method for computing the range of optimal values. The method reduces the problem to solving from two to four real-valued generalized linear fractional programs, which can be computed in polynomial time using an appropriate interior point method solver.     

求解一类非单调线性互补问题的宽邻域内点方法及其计算复杂性

对于一类非单调线性互补问题给出了一种新的算法——宽邻域内点算法，并讨论了其计算复杂性。     

Finding a strict feasible solution of a linear semidefinite program

This study deals with the performance of projective interior point methods for linear semidefinite program. We propose a modification in the initialization phases of the method in order to reduce the computation time.This purpose is confirmed by numerical experiments showing the efficiency which are presented in the last section of the paper.     

On the block upper-triangularity of undiscounted multi-chain markov decision problems

We show that the LP formulation for an undiscounted multi-chain Markov decision problem can be put in a block upper-triangular form by a polynomial time procedure. Each minimal block (after an appropriate dynamic revision) gives rise to a single-chain Markov decision problem which can be treated independently. An optimal solution to each single-chain problem can be connected by auxiliary dual programs to obtain an optimal solution to a multi-chain problem.     

Deepest point of a polyhedron and linear programming

Abstract

The problem of finding a deepest point (a ball centre) of a polyhedron is studied. A finite combinatorial interior point method is presented for this problem which yields an algorithm for linear programming. We conjecture that this is a strongly polynomial algorithm. Meanwhile developing the algorithm, several auxiliary results were found; among others, Gorokh and Werner’s algorithm for linear inequalities is slightly extended. Our numerical experiments with the problem detected bugs in a linear interior point solver used in MATLAB 6 Optimization Toolbox.     

New trajectory-following polynomial-time algorithm for linear programming problems

A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomial time bound. The method is based on the use of the trajectory of the problem, which makes it conceptually very simple. It has the advantage above related methods that it requires no problem transformation (either affine or projective) and that the feasible region may be unbounded. More importantly, the method generates at each stage solutions of both the primal and the dual problem. This implies that, contrary to the simplex method, the quality of the present solution is known at each stage. The paper also contains a practical (i.e., deepstep) version of the algorithm.The author is indebted to J. Bisschop, P. C. J. M. Geven, J. H. Van Lint, J. Ponstein, and J. P. Vial for their remarks on an earlier version of this paper.     

一类线性约束凸规划的内椭球算法

1引言自从1984年Karmarkar的著名算法——梯度投影算法发表以来,由其理论上的多项式收敛性及实际计算的有效性,使得内点算法成为近十几年来优化界研究的热点（［1］）．通过中外学者的深入研究,线性规划与凸二次规划的内点算法研究已取得了不少成果（［2」、［3〕）．这些算法大致可分为四种类型：梯度投影算法、仿射尺度算法、路径跟踪法和势函数减少法吸3］、〔9〕）．近来,人们开始着手将这些方法推广到非线性规划中的凸规划问题、线性互补问题和非线性互补问题（【6」、［7」、〔sj、［10」、Ill〕）．例如：文［8」对一类凸可分规…     

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n ² L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.     

一类框式凸规划的原始-对偶内点算法

本文为框式约束的一类凸规划提出了一个新的内点算法,原始－对偶路径跟踪法,并了政算法的迭代复杂性为多项式时间性。     

Some properties of generalized proximal point methods for quadratic and linear programming

The proximal point method for convex optimization has been extended recently through the use of generalized distances (e.g., Bregman distances) instead of the Euclidean one. One advantage of these extensions is the possibility of eliminating certain constraints (mainly positivity) from the subproblems, transforming an inequality constrained problem into a sequence of unconstrained or equality constrained problems. We consider here methods obtained using a certain class of Bregman functions applied to convex quadratic (including linear) programming, which are of the interior-point type. We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance. These results do not hold for the standard proximal point method, i.e., when the square of the Euclidean norm is used as the Bregman distance.The research leading to this paper was partially supported by CNPq Grant 301280/86.We thank two anonymous referees whose comments and suggestions allowed us to improve our results in a significant way.     

On the use of consistent approximations in the solution of semi-infinite optimization and optimal control problems

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 is a positive weight on the logarithmic barrier term for each inequality constraint defining the polyhedronX. We prove basic results regarding contained and containing ellipsoids centered at the w-center of the systemX. We next shift attention to projective transformations, and we exhibit an elementary projective transformation that transforms the polyhedronX to another polyhedronZ, and that transforms the current interior point to the w-center of the transformed polyhedronZ. 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 systemX. We present a projective transformation algorithm, which is an extension of Karmarkar's algorithm, for finding the w-center of the systemX. 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.This paper is a revision of the two papers Projective transformations for interior point methods, part I: Basic theory and linear programming, O.R. working paper 179-88 and Projective transformations for interior point methods, part II: Analysis of an algorithm for finding the weighted center of a polyhedral system, O.R. working paper 180-88, M.I.T.     

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln ³) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.     

Complexity of the Gravitational Method for Linear Programming

In the gravitational method for linear programming, a particle is dropped from an interior point of the polyhedron and is allowed to move under the influence of a gravitational field parallel to the objective function direction. Once the particle falls onto the boundary of the polyhedron, its subsequent motion is constrained to be on the surface of the polyhedron with the particle moving along the steepest-descent feasible direction at any instant. Since an optimal vertex minimizes the gravitational potential, computing the trajectory of the particle yields an optimal solution to the linear program.Since the particle is not constrained to move along the edges of the polyhedron, as the simplex method does, the gravitational method seemed to have the promise of being theoretically more efficient than the simplex method. In this paper, we first show that, if the particle has zero diameter, then the worst-case time complexity of the gravitational method is exponential in the size of the input linear program. As a simple corollary of the preceding result, it follows that, even when the particle has a fixed nonzero diameter, the gravitational method has exponential time complexity. The complexity of the version of the gravitational method in which the particle diameter decreases as the algorithm progresses remains an open question.     

Interior point methods for optimal control of discrete time systems

We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discrete-time optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete-time linear-quadratic regulator problem with mixed state/control constraints and show how they can be efficiently-incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interior-point method is the narrow-banded structure of the coefficient matrix which is factorized at each iteration.This research was supported by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38.     

A Class of Infeasible Interior Point Algorithms for Convex Quadratic Programming

This article proposes a class of infeasible interior point algorithms for convex quadratic programming, and analyzes its complexity. It is shown that this algorithm has the polynomial complexity. Its best complexity is O(nL).     

On lower bound updates in primal potential reduction methods for linear programming

We present a procedure for computing lower bounds for the optimal cost in a linear programming problem. Whenever the procedure succeeds, it finds a dual feasible slack and the associated duality gap. Although no projective transformations or problem restatements are used, the method coincides with the procedures by Todd and Burrell and by de Ghellinck and Vial when these procedures are applicable. The procedure applies directly to affine potential reduction algorithms, and improves on existent techniques for finding lower bounds.     

New developments in the primal–dual column generation technique

The optimal solutions of the restricted master problems typically leads to an unstable behavior of the standard column generation technique and, consequently, originates an unnecessarily large number of iterations of the method. To overcome this drawback, variations of the standard approach use interior points of the dual feasible set instead of optimal solutions. In this paper, we focus on a variation known as the primal–dual column generation technique which uses a primal–dual interior point method to obtain well-centered non-optimal solutions of the restricted master problems. We show that the method converges to an optimal solution of the master problem even though non-optimal solutions are used in the course of the procedure. Also, computational experiments are presented using linear-relaxed reformulations of three classical integer programming problems: the cutting stock problem, the vehicle routing problem with time windows, and the capacitated lot sizing problem with setup times. The numerical results indicate that the appropriate use of a primal–dual interior point method within the column generation technique contributes to a reduction of the number of iterations as well as the running times, on average. Furthermore, the results show that the larger the instance, the better the relative performance of the primal–dual column generation technique.