一类凸规划的多项式预估校正内点法

1、引言 1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法．1993年Mizuno,Todd和Y．Ye．给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步．1994年F．A．Potra给出了不可行预估校正内点法,使得可以从一个不可行的初始点开始算法的迭代,并证明了其为二次收敛．     

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interior-point method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.     

A predictor-corrector interior-point algorithm for P ∗ ( κ ) -horizontal linear complementarity problem

We present a predictor-corrector path-following interior-point algorithm for \(P_*(\kappa )\) horizontal linear complementarity problem based on new search directions. In each iteration, the algorithm performs two kinds of steps: a predictor (damped Newton) step and a corrector (full Newton) step. The full Newton-step is generated from an algebraic reformulation of the centering equation, which defines the central path and seeks directions in a small neighborhood of the central path. While the damped Newton step is used to move in the direction of optimal solution and reduce the duality gap. We derive the complexity for the algorithm, which coincides with the best known iteration bound for \(P_*(\kappa )\) -horizontal linear complementarity problems.     

线性互补问题的宽邻域预估校正算法

对线性互补问题提出了一种新的宽邻域预估校正算法,算法是基于经典线性规划路径跟踪算法的思想,将Maziar Salahi关于线性规划预估校正算法推广到线性互补问题中,给出了算法的具体迭代步骤并讨论了算法迭代复杂性,最后证明了算法具有多项式复杂性为O(ηlog(X~0)~Ts~0/ε)。     

A modified predictor-corrector method for linear programming

In the predictor-corrector method of Mizuno, Todd and Ye [1], the duality gap is reduced only at the predictor step and is kept unchanged during the corrector step. In this paper, we modify the corrector step so that the duality gap is reduced by a constant fraction, while the predictor step remains unchanged. It is shown that this modified predictor-corrector method retains the iteration complexity as well as the local quadratic convergence property.     

Predictor-Corrector Algorithm for Convex Quadratic Programming with Upper Bounds

1.IntroductionThepredictor-correctormethodforlinearprogrammingisawellknownillteriorpointmethoddevelopedbyMizunoetal.[1])duetoitsquadraticaJlyconvergentanalysis.ThiskindofanalysisusuaJlycontainstwosteps,i.e.,predictorstepandcorrectorstepasoneiteration.ThecorrectorstepisusedonlytoensurethattheiteratesstayclosetothecelltraJpathsothatlargestepcanbetakenduringthepredictorstep.Thedualitygapremainsunchangedatcorrectorstepforlinearprogramming,butincaseofconvexquadraticprogramming,asshownlaterofthisp…     

On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms

Recently, Mehrotra [3] proposed a predictor—corrector primal—dual interior-point algorithm for linear programming. At each iteration, this algorithm utilizes a combination of three search directions: the predictor, the corrector and the centering directions, and requires only one matrix factorization. At present, Mehrotra's algorithmic framework is widely regarded as the most practically efficient one and has been implemented in the highly successful interior-point code OB1 [2]. In this paper, we study the theoretical convergence properties of Mehrotra's interior-point algorithmic framework. For generality, we carry out our analysis on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under the monotonicity assumption, we establish polynomial complexity bounds for two variants of the Mehrotra-type predictor—corrector interior-point algorithms. These results are summarized in the last section in a table.Research supported in part by NSF DMS-9102761, DOE DE-FG05-91ER25100 and DOE DE-FG02-93ER25171.Corresponding author.     

二阶锥规划两个新的预估-校正算法

基于不可行内点法和预估-校正算法的思想,提出两个新的求解二阶锥规划的内点预估-校正算法．其预估方向分别是Newton方向和Euler方向,校正方向属于Alizadeh-Haeberly-Overton(AHO)方向的范畴．算法对于迭代点可行或不可行的情形都适用．主要构造了一个更简单的中心路径的邻域,这是有别于其它内点预估-校正算法的关键．在一些假设条件下,算法具有全局收敛性、线性和二次收敛速度,并获得了O(rln(ε0/ε))的迭代复杂性界,其中r表示二阶锥规划问题所包含的二阶锥约束的个数．数值实验结果表明提出的两个算法是有效的．     

A Predictor-corrector algorithm with multiple corrections for convex quadratic programming

Recently an infeasible interior-point algorithm for linear programming (LP) was presented by Liu and Sun. By using similar predictor steps, we give a (feasible) predictor-corrector algorithm for convex quadratic programming (QP). We introduce a (scaled) proximity measure and a dynamical forcing factor (centering parameter). The latter is used to force the duality gap to decrease. The algorithm can decrease the duality gap monotonically. Polynomial complexity can be proved and the result coincides with the best one for LP, namely, $O(\sqrt{n}\log n\mu^{0}/\varepsilon)$ .     

On quadratic andO\left( {\sqrt {nL} } \right) convergence of a predictor—corrector algorithm for LCP

Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor—corrector algorithm for linear programming (LP) exhibits asymptotic quadratic convergence of the primal—dual gap to zero, without any assumptions concerning nondegeneracy, or the convergence of the iteration sequence. In this paper we prove a similar result for the monotone linear complementarity problem (LCP), assuming only that a strictly complementary solution exists. We also show by example that the existence of a strictly complementarity solution appears to be necessary to achieve superlinear convergence for the algorithm.Research supported in part by NSF Grants DDM-8922636 and DDM-9207347, and an Interdisciplinary Research Grant of the University of Iowa, Iowa Center for Advanced Studies.     

A Primal-dual Interior-point Algorithm for Symmetric Cone Convex Quadratic Programming Based on the Commutative Class Directions

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we present a neighborhood following primal-dual interior-point algorithm for solving symmetric cone convex quadratic programming...     

P*(κ)阵线性互补问题一种新的宽邻域预估-校正内点算法

基于邻近度量函数的最小值,对P*(κ)阵线性互补问题提出了一种新的宽邻域预估-校正算法,在较一般的条件下,证明了算法的迭代复杂性为O(κ+1)23n log(x0ε)Ts0.算法既可视为Miao的P*(κ)阵线性互补问题Mizuno-Todd-Ye预估-校正内点算法的一种变形,也可以视为最近Zhao提出的线性规划基于邻近度量函数最小值的宽邻域内点算法的推广.     

A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is O(?nL){O(\sqrt{n}L)} for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.     

A predictor-corrector algorithm for linear optimization based on a modified Newton direction

We present a predictor-corrector algorithm for linear optimization based on a modified Newton direction. In each main iteration, the algorithm operates two kinds of steps: a modified Newton step and a damped predictor step. The modified Newton step is generated from an equivalent reformulation of the centering equation from the system, which defines the central path, and move in the direction of a small neighborhood of the central path. While the damped predictor step is used to move in the direction of optimal solution and reduce the duality gap. The procedure is repeated until an ?-approximate solution is found. We derive the complexity for the algorithm, and obtain the best-known result for linear optimization.     

A quadratically convergent O(\sqrt n L)-iteration algorithm for linear programming

Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.Supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.Supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Supported in part by NSF Grant DMS-9102761 and DOE Grant DE-FG05-91ER25100.     

框式线性规划的多项式预估校正内点算法

本文研究带线性约束的框式线性规划问题,给出了一个预估校正内点算法,分析了该算法的多项式计算复杂性,并证明其迭代复杂度为Ο(nL).     

Adaptive constraint reduction for convex quadratic programming

We propose an adaptive, constraint-reduced, primal-dual interior-point algorithm for convex quadratic programming with many more inequality constraints than variables. We reduce the computational effort by assembling, instead of the exact normal-equation matrix, an approximate matrix from a well chosen index set which includes indices of constraints that seem to be most critical. Starting with a large portion of the constraints, our proposed scheme excludes more unnecessary constraints at later iterations. We provide proofs for the global convergence and the quadratic local convergence rate of an affine-scaling variant. Numerical experiments on random problems, on a data-fitting problem, and on a problem in array pattern synthesis show the effectiveness of the constraint reduction in decreasing the time per iteration without significantly affecting the number of iterations. We note that a similar constraint-reduction approach can be applied to algorithms of Mehrotra’s predictor-corrector type, although no convergence theory is supplied.     

Parallel Interior Point Schemes for Solving Multistage Convex Programming

The predictor–corrector interior-point path-following algorithm is promising in solving multistage convex programming problems. Among many other general good features of this algorithm, especially attractive is that the algorithm allows possibility to parallelise the major computations. The dynamic structure of the multistage problems specifies a block-tridiagonal system at each Newton step of the algorithm. A wrap-around permutation is then used to implement the parallel computation for this step.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.