Khachiyan, Leonid
Overview
Works:  23 works in 24 publications in 1 language and 67 library holdings 

Roles:  Author, Honoree 
Publication Timeline
.
Most widely held works by
Leonid Khachiyan
On the complexity of nonnegative matrix scaling by
Bahman Kalantari(
Book
)
1 edition published in 1990 in English and held by 5 WorldCat member libraries worldwide
Abstract: "An n x n nonnegative matrix is said to be (doubly stochastic) scalable if there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. We derive an upper bound on the norms of the scaling factors X and Y and give a polynomial time complexity bound on the problem of computing the scaling factors with prescribed accuracies."
1 edition published in 1990 in English and held by 5 WorldCat member libraries worldwide
Abstract: "An n x n nonnegative matrix is said to be (doubly stochastic) scalable if there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. We derive an upper bound on the norms of the scaling factors X and Y and give a polynomial time complexity bound on the problem of computing the scaling factors with prescribed accuracies."
Diagnonal [i.e. diagonal] matrix scaling and linear programming by Leonid G Khachiyan(
Book
)
2 editions published in 1990 in English and held by 5 WorldCat member libraries worldwide
Abstract: "A positive semidefinite symmetric matrix either has a nontrivial nonnegative zero, or, can be scaled by a positive diagonal matrix into a doubly quasi stochastic matrix. In this paper we describe a simple path following Newton algorithm [sic] of the complexity [formula] iterations to either scale an n by n matrix or give a nontrivial nonnegative zero. The latter problem is wellknown to be equivalent to linear programming."
2 editions published in 1990 in English and held by 5 WorldCat member libraries worldwide
Abstract: "A positive semidefinite symmetric matrix either has a nontrivial nonnegative zero, or, can be scaled by a positive diagonal matrix into a doubly quasi stochastic matrix. In this paper we describe a simple path following Newton algorithm [sic] of the complexity [formula] iterations to either scale an n by n matrix or give a nontrivial nonnegative zero. The latter problem is wellknown to be equivalent to linear programming."
An exponentialfunction reduction method for blockangular convex programs by Michael D Grigoriadis(
Book
)
1 edition published in 1993 in English and held by 5 WorldCat member libraries worldwide
Abstract: "An exponential potentialfunction reduction algorithm for convex blockangular optimization problems is described. These problems are characterized by K disjoint convex compact sets called blocks and M nonnegativevalued convex blockseparable coupling inequalities with a nonempty interior. A given convex blockseparable function is to be minimized. The method reduces the optimization problem to two resource sharing problems. The first of these problems is solved to obtain a feasible solution interior to the coupling constraints. Starting from this solution, the algorithm proceeds to solve the second problem on the original constraints, but with a modified exponential potential function
1 edition published in 1993 in English and held by 5 WorldCat member libraries worldwide
Abstract: "An exponential potentialfunction reduction algorithm for convex blockangular optimization problems is described. These problems are characterized by K disjoint convex compact sets called blocks and M nonnegativevalued convex blockseparable coupling inequalities with a nonempty interior. A given convex blockseparable function is to be minimized. The method reduces the optimization problem to two resource sharing problems. The first of these problems is solved to obtain a feasible solution interior to the coupling constraints. Starting from this solution, the algorithm proceeds to solve the second problem on the original constraints, but with a modified exponential potential function
Coordination complexity of parallel pricedirective decomposition by Michael D Grigoriadis(
Book
)
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "Parallel price directive decomposition (PPD) methods for the approximate solution of blockangular convex resource sharing problems are considered. This general model in structured optimization consists of K nonempty disjoint compact sets called 'blocks' and M nonnegativevalued convex 'coupling' inequalities. It has a number of applications in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of PDD approximation methods, i.e., the number of iterations required to solve the general resource sharing problem to a fixed relative accuracy, as a function of K and M. First, a simple PDD method based on the classical logarithmic potential function is presented and analyzed. For a fixed accuracy, its coordination complexity is shown to be O(M ln M), which is within a logarithmic factor of the best possible bound for any PDD method that works with the original blocks. An important property of the logarithmicpotential PDD method is that its coordination complexity depends neither on the 'width' nor on the dimension of the blocks. Second, polylogarithmicallymatching upper and lower coordination complexity bounds are presented for a wider class of PDD methods in which each block can be partially restricted by the coupling constraints. The upper bound for this class of PDD is obtained by a combined method, which uses either the logarithmic or the exponential potential function, depending on the number of coupling constraints per block."
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "Parallel price directive decomposition (PPD) methods for the approximate solution of blockangular convex resource sharing problems are considered. This general model in structured optimization consists of K nonempty disjoint compact sets called 'blocks' and M nonnegativevalued convex 'coupling' inequalities. It has a number of applications in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of PDD approximation methods, i.e., the number of iterations required to solve the general resource sharing problem to a fixed relative accuracy, as a function of K and M. First, a simple PDD method based on the classical logarithmic potential function is presented and analyzed. For a fixed accuracy, its coordination complexity is shown to be O(M ln M), which is within a logarithmic factor of the best possible bound for any PDD method that works with the original blocks. An important property of the logarithmicpotential PDD method is that its coordination complexity depends neither on the 'width' nor on the dimension of the blocks. Second, polylogarithmicallymatching upper and lower coordination complexity bounds are presented for a wider class of PDD methods in which each block can be partially restricted by the coupling constraints. The upper bound for this class of PDD is obtained by a combined method, which uses either the logarithmic or the exponential potential function, depending on the number of coupling constraints per block."
On the complexity of approximating extremal determinants in matrices by Leonid Khachiyan(
Book
)
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "For a d x n matrix A, let B = B(A) be the set of all nondegenerate d x d submatrices (bases) of A, and let [delta](A) = min[[absolute value of det B] : B [element of] B]. We show that for any polynomial p = poly(d, n) in the dimension of A, the problem of approximating [delta](A) within a factor of 2[superscript p] is NPhard. We also show that it is NPhard to determine whether a set of n rational points in n dimensions is affinely or linearly degenerate. On the other hand, we give an algorithm for approximating [delta](A) = max[[absolute value of det B] : B [element of] B] within a factor of [(1+[epsilon])d][superscript (d1)/2] in O(nd²([epsilon][superscript 1] + log d + log log n)) arithmetic operations and comparisons over the reals."
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "For a d x n matrix A, let B = B(A) be the set of all nondegenerate d x d submatrices (bases) of A, and let [delta](A) = min[[absolute value of det B] : B [element of] B]. We show that for any polynomial p = poly(d, n) in the dimension of A, the problem of approximating [delta](A) within a factor of 2[superscript p] is NPhard. We also show that it is NPhard to determine whether a set of n rational points in n dimensions is affinely or linearly degenerate. On the other hand, we give an algorithm for approximating [delta](A) = max[[absolute value of det B] : B [element of] B] within a factor of [(1+[epsilon])d][superscript (d1)/2] in O(nd²([epsilon][superscript 1] + log d + log log n)) arithmetic operations and comparisons over the reals."
On the complexity of dualization of monotone disjunctive normal forms by Michael L Fredman(
Book
)
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "We show that the duality of a pair of monotone disjunctive normal forms of size n can be tested in n[superscript o(log n)] time."
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "We show that the duality of a pair of monotone disjunctive normal forms of size n can be tested in n[superscript o(log n)] time."
Fast approximation schemes for convex programs with many blocks and coupling constraints by Michael D Grigoriadis(
Book
)
1 edition published in 1991 in English and held by 4 WorldCat member libraries worldwide
1 edition published in 1991 in English and held by 4 WorldCat member libraries worldwide
On the rate of convergence of the RAS method by
Bahman Kalantari(
Book
)
1 edition published in 1992 in English and held by 4 WorldCat member libraries worldwide
Abstract: "It is wellknown that given an n x n matrix with positive entries, there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. One of the best known algorithms for computing the scaling factors X and Y is the so called RAS algorithm which alternatively normalizes rows and columns of the matrix. In this paper we prove that the RAS is a fullypolynomial time approximation scheme and give a bound of O((1/[epsilon] + ln n) [squareroot n]ln 1/v), on the number of iterations of the RAS for scaling A to an accuracy of [epsilon], where v is the ratio of the least entry of A to its largest."
1 edition published in 1992 in English and held by 4 WorldCat member libraries worldwide
Abstract: "It is wellknown that given an n x n matrix with positive entries, there exists two positive diagonal matrices X and Y such that XAY is doubly stochastic. One of the best known algorithms for computing the scaling factors X and Y is the so called RAS algorithm which alternatively normalizes rows and columns of the matrix. In this paper we prove that the RAS is a fullypolynomial time approximation scheme and give a bound of O((1/[epsilon] + ln n) [squareroot n]ln 1/v), on the number of iterations of the RAS for scaling A to an accuracy of [epsilon], where v is the ratio of the least entry of A to its largest."
A sublineartime randomized approximation algorithm for matrix games by Michael D Grigoriadis(
Book
)
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "This paper presents a parallel randomized algorithm which computes a pair of [epsilon]optimal strategies for a given (m, n) matrix game A = [a[subscript i, subscript j]] [is in the set of] [1,1] in O([epsilon][superscript2] log² (n + m)) expected time on an (n + m) / log (n + m)processor EREW PRAM. This algorithm is a natural extension of the classical method of fictitious play by Brown and Robinson. For any fixed accuracy [epsilon]> 0, the expected sequential running time of the suggested algorithm is O((n + m) log (n + m)), which is sublinear in mn, the number of input elements of A. On the other hand, simple arguments are given to show that for [epsilon] <1/2, any sequential deterministic algorithm for computing a pair of [epsilon]optimal strategies of an (m, n) matrix game A with [plus or minus] 1 elements examines [omega] (m, n) of its elements. In particular, for m = n the randomized algorithm achieves an almost quadratic expected speedup relative to any deterministic method."
1 edition published in 1994 in English and held by 4 WorldCat member libraries worldwide
Abstract: "This paper presents a parallel randomized algorithm which computes a pair of [epsilon]optimal strategies for a given (m, n) matrix game A = [a[subscript i, subscript j]] [is in the set of] [1,1] in O([epsilon][superscript2] log² (n + m)) expected time on an (n + m) / log (n + m)processor EREW PRAM. This algorithm is a natural extension of the classical method of fictitious play by Brown and Robinson. For any fixed accuracy [epsilon]> 0, the expected sequential running time of the suggested algorithm is O((n + m) log (n + m)), which is sublinear in mn, the number of input elements of A. On the other hand, simple arguments are given to show that for [epsilon] <1/2, any sequential deterministic algorithm for computing a pair of [epsilon]optimal strategies of an (m, n) matrix game A with [plus or minus] 1 elements examines [omega] (m, n) of its elements. In particular, for m = n the randomized algorithm achieves an almost quadratic expected speedup relative to any deterministic method."
Diagonal matrix scaling is NPhard by Leonid G Khachiyan(
Book
)
1 edition published in 1992 in English and held by 4 WorldCat member libraries worldwide
Abstract: "A symmetric matrix A is said to be scalable if there exists a positive diagonal matrix X such that the row and column sums of XAX are all ones. We show that testing the scalability of arbitrary matrices is NPhard. Equivalently, it is NPhard to check for a given symmetric matrix A whether the logarithmic barrier function [formula] has a stationary point in the positive orthant x> 0."
1 edition published in 1992 in English and held by 4 WorldCat member libraries worldwide
Abstract: "A symmetric matrix A is said to be scalable if there exists a positive diagonal matrix X such that the row and column sums of XAX are all ones. We show that testing the scalability of arbitrary matrices is NPhard. Equivalently, it is NPhard to check for a given symmetric matrix A whether the logarithmic barrier function [formula] has a stationary point in the positive orthant x> 0."
Complexity of polytope volume computation by Leonid G Khachiyan(
Book
)
1 edition published in 1990 in English and held by 4 WorldCat member libraries worldwide
Abstract: "We survey some recent results on the complexity of computing the volume of convex ndimensional polytopes."
1 edition published in 1990 in English and held by 4 WorldCat member libraries worldwide
Abstract: "We survey some recent results on the complexity of computing the volume of convex ndimensional polytopes."
Approximate solution of matrix games in parallel by Michael D Grigoriadis(
Book
)
1 edition published in 1991 in English and held by 3 WorldCat member libraries worldwide
Abstract: "We consider the problem of solving matrix games to a fixed relative accuracy and present a parallel algorithm which runs in polylogarithmic time on a quadratic number of processors."
1 edition published in 1991 in English and held by 3 WorldCat member libraries worldwide
Abstract: "We consider the problem of solving matrix games to a fixed relative accuracy and present a parallel algorithm which runs in polylogarithmic time on a quadratic number of processors."
Rounding of polytopes in R[superscript n] by Leonid Khachiyan(
Book
)
1 edition published in 1993 in English and held by 3 WorldCat member libraries worldwide
Abstract: "Let A be a set of m points in R[superscript n]. We show that the problem of (1 + [epsilon])nrouding of A, i.e. the problem of computing an ellipsoid E [subset of] R[superscript n] such that E [subset of] conv.hull(A) [subset of] (1 + [epsilon])nE, can be solved in O(mn²([epsilon]⁻¹ + log n + log log m)) arithmetic operations and comparisons. This result implies that the problem of approximating the minimum volume ellipsoid circumscribed about A can be solved in O(m[superscript 3.5] ln(m[epsilon]⁻¹)) operations to a relative accuracy of [epsilon] in the volume. The latter bound also applies to the (1 + [epsilon])nrounding problem. Our bounds hold for the real number model of computation."
1 edition published in 1993 in English and held by 3 WorldCat member libraries worldwide
Abstract: "Let A be a set of m points in R[superscript n]. We show that the problem of (1 + [epsilon])nrouding of A, i.e. the problem of computing an ellipsoid E [subset of] R[superscript n] such that E [subset of] conv.hull(A) [subset of] (1 + [epsilon])nE, can be solved in O(mn²([epsilon]⁻¹ + log n + log log m)) arithmetic operations and comparisons. This result implies that the problem of approximating the minimum volume ellipsoid circumscribed about A can be solved in O(m[superscript 3.5] ln(m[epsilon]⁻¹)) operations to a relative accuracy of [epsilon] in the volume. The latter bound also applies to the (1 + [epsilon])nrounding problem. Our bounds hold for the real number model of computation."
On the conductance of order Markov chains by
A Karzanov(
Book
)
1 edition published in 1990 in English and held by 3 WorldCat member libraries worldwide
Abstract: "Let Q be a convex solid in R[superscript n], partitioned into two volumes u and v by an area s. We show that s [greater than or equal to] min(u, v)/diam Q, and use this inequality to obtain the lower bound [formula] on the conductance of order Markov chains, which describe nearly uniform generators of linear extensions for posets of size n. We also discuss an application of the above results to the problem of sorting of posets."
1 edition published in 1990 in English and held by 3 WorldCat member libraries worldwide
Abstract: "Let Q be a convex solid in R[superscript n], partitioned into two volumes u and v by an area s. We show that s [greater than or equal to] min(u, v)/diam Q, and use this inequality to obtain the lower bound [formula] on the conductance of order Markov chains, which describe nearly uniform generators of linear extensions for posets of size n. We also discuss an application of the above results to the problem of sorting of posets."
An interior point method for bordered block diagonal linear programs by
Rutgers University(
Book
)
1 edition published in 1994 in English and held by 3 WorldCat member libraries worldwide
Abstract: "This paper presents an interior point method for solving a bordered block diagonal linear program which consists of a number of disjoint blocks, coupled by a total of p variables and constraints. This structure includes the wellknown blockangular and dual blockangular structures, as well as their special cases, such as staircase problems, generalized bounds and multicommodity flows. When p is small relative to the total dimension the problem [sic], the method achieves a substantial speedup relative to other generalpurpose methods."
1 edition published in 1994 in English and held by 3 WorldCat member libraries worldwide
Abstract: "This paper presents an interior point method for solving a bordered block diagonal linear program which consists of a number of disjoint blocks, coupled by a total of p variables and constraints. This structure includes the wellknown blockangular and dual blockangular structures, as well as their special cases, such as staircase problems, generalized bounds and multicommodity flows. When p is small relative to the total dimension the problem [sic], the method achieves a substantial speedup relative to other generalpurpose methods."
Generating cut conjunctions in graphs and related problems(
Book
)
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
Linear scheduling is nearly optimal by
Alain Darte(
Book
)
1 edition published in 1991 in English and held by 1 WorldCat member library worldwide
1 edition published in 1991 in English and held by 1 WorldCat member library worldwide
Dualbounded generating problems: efficient and inefficient points for discrete probability distributions and sparse boxes
for multidimensional data(
Book
)
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
Enumerating spanning and connected subsets in graphs and matroids(
Book
)
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
1 edition published in 2007 in English and held by 1 WorldCat member library worldwide
Computing integral points in convex semialgebraic sets by Leonid Khachiyan(
Book
)
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
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Related Identities
 Grigoriadis, Michael D. Author
 Kalantari, Bahman Author
 Fredman, Michael Author
 Karzanov, Alexander Author
 Gurvich, V. Author
 Darte, Alain Author
 Porkolab, Lorant
 Boros, Endre
 Robert, Yves
Associated Subjects
Approximation theory Combinatorial optimization Computational complexity Convex bodies Convex programming Isoperimetric inequalities Linear operators Linear programming Linear programmingData processing Markov operators Markov processes Matrices Nonlinear programming Operations research Parallel processing (Electronic computers) Polytopes
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