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ABS Methods for Continuous and Integer Linear Equations and Optimization

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ABS Methods for Continuous and Integer Linear Equations and Optimization CEJOR DOI 10.1007/s10100-009-0128-9 ORIGINAL PAPER ABS methods for continuous and integer linear equations and optimization Spedicato Emilio · Bodon Elena · Xia Zunquan · Mahdavi-Amiri Nezam © Springer-Verlag 2009 Abstract ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egerváry and can be considered as the most fruitful extension of such technique. They have led to unification of classes of methods for several problems. Moreover they have produced some special algorithms with better complexity than the standard methods. For the linear integer case they have provided the most general polynomial time class of algorithms so far known; such algorithms have been extended to other integer problems, as linear inequalities and LP problems, in over a dozen papers written by Iranian mathematicians led by Nezam Mahdavi-Amiri. ABS methods can be implemented generally in a stable way, techniques existing to enhance their accuracy. Extensive numerical experiments have There are now probably over 400 papers on ABS methods. A partial list up to 1999 of about 350 papers was published by Spedicato et al. (2000) in the journal Ricerca Operativa. Here we give only the handful of references most related to the content of this review paper. S. Emilio (B) · B. Elena University of Bergamo, Bergamo, Italy e-mail: [email protected] X. Zunquan University of Dalian, Dalian, China M.-A. Nezam Sharif University, Tehran, Iran 123 S. Emilio et al. shown that they can outperform standard methods in several problems. Here we pro- vide a review of their main properties, for linear systems and optimization. We also give the results of numerical experiments on some linear systems. This paper is dedi- cated to Professor Egerváry, developer of the rank reducing matrix update, that led to ABS methods. Keywords Egerváry rank reduction matrix update · ABS methods · Linear systems · Diophantine Linear systems · Quasi-Newton equation · Feasible direction methods for linearly constrained optimization · Simplex method · Primal-dual interior point methods · ABSPACK 1 ABS methods 1981–2008, main results The work on ABS methods started in 1981 after a seminar given by Jozsef Abaffy at the University of Bergamo, and developed via an international collaboration involving mathematicians mainly from Hungary, China, Iran, led by prof. Emilio Spedicato of Bergamo University. It is available in: – two monographs, see the references, one published in 1989 by Abaffy and Spedicato (1989), one in 1999 by Xia and Zhang (1999), in Chinese, English updated version planned – the Proceedings of three international conferences held in China (one in Luoyang and two in Beijing) – over 400 papers, of which more than 300 are listed in Spedicato et al. (2000) The main results, obtained in almost thirty years, can be summarized as follows: – general unification of methods for linear algebraic equations and for linearly con- strained optimization, including the LP problem; the unification is provided by the construction of a multiparameter class which, for special choices of the parame- ters, provides all possible methods of a general type for solving certain problems; in practice essentially all methods found insofar in the literature can be shown to correspond to some choices in this class; moreover some open problems have been solved within the provided general frame – the class is based upon the rank-reducing matrix update of Egerváry, that was exploited by him in just a few papers. It has shown the immense potential of the Egerváry’s discovery – the class contains the so called implicit LX method, due essentially to Zunquan Xia, hence the acronym; this is the most efficient general solver of a linear system of the classic type, improving over Gaussian elimination because of lower storage, no need of pivoting and different roundoff properties that seem to provide better accuracy; the overhead is the same as in Gaussian elimination – via the implicit LX algorithm we get an implementation of the simplex algorithm for the LP problem that saves a factor 8 in memory, and up to one order operations over the Forrest-Goldfarb method, the choice classical method – nonlinear ABS methods, see for a partial review Galántai and Spedicato (2007), are in many cases more efficient than Newton method, requiring less storage and 123 ABS methods for continuous and integer linear equations and optimization keeping a quadratic rate of convergence even when the Jacobian is rank deficient in the solution – a new class of algorithms for solving a linear Diophantine system, which is Hilbert 10-th problem in the linear case; this class can be extended to linear integer inequalities and integer LP problems – reduction of condition number in linear systems arising in important problems, as in the primal-dual interior point method – the package ABSPACK1 developed in FORTRAN 77 is usually better than LAPACK on problems that are difficult due to ill conditioning or low numeri- cal rank. 2 The Egerváry-ABS class of algorithms for systems of linear equations This is a class of methods using the rank reducing matrix recursion of Egerváry (1955–1956), that was first published by Abaffy et al. (1984) in the unscaled form, with three parameters, while Abaffy and Spedicato (1989, 1990) gave the scaled form, with four parameters (this class also appeared with a different derivation in Abaffy (1984)). Preliminary restricted versions of the unscaled class were given by Abaffy (1979), with three scalar parameters, and by Abaffy and Spedicato (1984), with two parameters. The relation with the Egerváry’s technique was noticed only later, thanks to Aurel Galantai. It deals with the following problem: Given the linear system = ( T = ) = ,..., Ax b ai x bi i 1 m with T n m A = (a1,...,am) , ai , x ∈ R , b ∈ R , m ≤ n, rank(A) = q ≤ m find x1,...,xm+1 such that for x1 arbitrary, xm+1 solves Ax = b via the iteration xi+1 = xi − αi pi so that the Petrov-Galerkin criterion is satisfied, namely vT ( ) = ( ≤ , ( ) = − ) , j r xi+1 0 j i r x Ax b m with v1,...,vi linearly independent vectors in R . The scaled ABS class can be shown to provide a complete realization of the above process. Equivalent processes are found in the literature via alternative algebra: – Steward 1973—conjugate direction method, where the search directions are com- puted via standard linear solvers – Broyden (1985)—general finite iteration, where some properties are investigated, but no procedure is given to compute the search directions 123 S. Emilio et al. – Dennis-Turner 1987—generalized conjugate directions, under conditions that are more restrictive than in the ABS class. 3 The scaled ABS class (Abaffy 1984, Abaffy and Spedicato, 1989, 1990) The class extends the unscaled or basic class introduced by Abaffy et al. (1984), that was derived via the idea of generating a sequence of approximations xi to the solution having the property that the approximation xi+1 is a particular solution of the first i equations. From the approximation xi one moves to the next one by taking a step along T a search direction pi built by multiplying a matrix Hi by a certain parameter vec- tor. Then the matrix is updated by the Egerváry process. In the 1984 paper by Abaffy et al. (1984) the matrix Hi was called Abaffian, a name that we apply to the case V = I in the class below. For the general case we prefer to call it Egervarian to recognize the original derivation by Egerváry. The scaled class can be obtained nicely by applying the unscaled algorithm to the scaled system V T Ax = V T b, whose set of solutions is the same as the one of the original system Ax = b. The class is characterized by the parameters H1, zi ,wi ,vi . They define four matri- ces of parameters, but one can show that only two of such matrices can generate all possible sets of approximations xi to the solution. We define the class by the following process (Scaled ABS class): n m n,n (A) Let x1 ∈ R ,v1 ∈ R be arbitrary nonzero and H1 ∈ R be arbitrary non- singular; set i = 1 (B) Here we check for redundant equations or incompatible ones. Compute the resid- ual vector ri = Axi − bi .If ri = 0 stop, xi is a solution, otherwise compute = T v ,τ= vT si Hi A i i i ri . If si = 0 goto(C). If si = 0 and τi = 0,set xi+1 = xi , Hi+1 = Hi and go to (D),thei-th equation is redundant. Otherwise stop, the system is unsolvable. (C) Basic recursions Compute a search vector by = T T v = pi Hi zi zi Hi A i 0 = − α α = vT / Update the estimate of the solution by xi+1 xi i pi where i i ri T T v pi A i . Update the Egervarian by T v wT Hi A i i Hi T T H + = H − w H A v = 0 i 1 i wT T v i i i i Hi A i (D) Stop if i = m. Otherwise give vi+1 not dependent on v1,...,vi , increment i by one, go to (B). 123 ABS methods for continuous and integer linear equations and optimization 3.1 Basic properties of the above iterates For proofs in the statements here and below see the monograph of Abaffy and Spedicato (1989), or its updated versions in the Chinese (1991) or Russian (1996) translations; see also Galántai (2003) for a very general approach to projection matrices and pro- jections methods.
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