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17
PrimalDual TargetFollowing Algorithms for Linear Programming
 ANNALS OF OPERATIONS RESEARCH
, 1993
"... In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Al ..."
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Cited by 26 (1 self)
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In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Along with the convergence analysis we provide a general framework which enables us to analyze various primaldual algorithms in the literature in a short and uniform way.
Polynomiality of PrimalDual Affine Scaling Algorithms for Nonlinear Complementarity Problems
, 1995
"... This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to ..."
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Cited by 11 (4 self)
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This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primaldual affine scaling algorithms generates an approximate solution (given a precision ffl) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial of n, ln(1=ffl) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in [13].
A Family of Polynomial Affine Scaling Algorithms for Positive SemiDefinite Linear Complementarity Problems
 SIAM JOURNAL ON OPTIMIZATION
, 1993
"... In this paper the new polynomial affine scaling algorithm of Jansen, Roos and Terlaky for LP is extended to PSD linear complementarity problems. The algorithm is immediately further generalized to allow higher order scaling. These algorithms are also new for the LP case. The analysis is based on Lin ..."
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Cited by 11 (5 self)
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In this paper the new polynomial affine scaling algorithm of Jansen, Roos and Terlaky for LP is extended to PSD linear complementarity problems. The algorithm is immediately further generalized to allow higher order scaling. These algorithms are also new for the LP case. The analysis is based on Ling's proof for the LP case, hence allows an arbitrary interior feasible pair to start with. With the scaling of Jansen et al. the complexity of the algorithm is O( n ae 2 (1\Gammaae 2 ) ln (x (0) ) T s (0) ffl ), where ae 2 is a uniform bound for the ratio of the smallest and largest coordinate of the iterates in the primaldual space. Finally we show that Monteiro, Adler and Resende's polynomial complexity result for the classical primaldual affine scaling algorithm can easily be derived from our analysis. In addition our result is valid for arbitrary not necessarily centered, initial points.
LongStep PrimalDual TargetFollowing Algorithms for Linear Programming
, 1995
"... In this paper we propose a long{step target{following methodology for linear programming. This is a general framework, that enables us to analyze various long{step primal{dual algorithms in the literature in a short and uniform way. Among these are long{step central and weighted path{following metho ..."
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Cited by 8 (0 self)
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In this paper we propose a long{step target{following methodology for linear programming. This is a general framework, that enables us to analyze various long{step primal{dual algorithms in the literature in a short and uniform way. Among these are long{step central and weighted path{following methods and algorithms to compute a central point or a weighted center. Moreover, we use it to analyze a method with the property that it, starting from an initial non{ central point, generates iterates that simultaneously get closer to optimality and closer to centrality. Key words: interiorpoint method, ane scaling method, primal{dual method, long{step method. Running title: Target{Following Methods for LP. This work is completed with the support of a research grant from SHELL. The rst author is supported by the Dutch Organization for Scientic Research (NWO), grant 611304028. The fourth author is supported by the Swiss National Foundation for Scientic Research, grant 1234002.92. ii 1...
A constraintreduced variant of Mehrotra’s predictorcorrector algorithm. submitted for publication
 In Preparation
, 2007
"... Consider linear programs in dual standard form with n constraints and m variables. When typical interiorpoint algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires O(nm 2) ..."
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Cited by 6 (5 self)
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Consider linear programs in dual standard form with n constraints and m variables. When typical interiorpoint algorithms are used for the solution of such problems, updating the iterates, using direct methods for solving the linear systems and assuming a dense constraint matrix A, requires O(nm 2) operations. When n ≫ m it is often the case that at each iteration most of the constraints are not very relevant for the construction of a good update and could be ignored to achieve computational savings. This idea has been considered in the 1990s by Dantzig and Ye, Tone, Kaliski and Ye, den Hertog et al. and others. More recently, Tits et al. proposed a simple “constraintreduction ” scheme and proved global and local quadratic convergence for a dualfeasible primaldual affinescaling method modified according to that scheme. In the present work, similar convergence results are proved for a dualfeasible constraintreduced variant of Mehrotra’s predictorcorrector algorithm. Some promising numerical results are reported. 1
Primaldual affine scaling interior point methods for linear complementarity problems
 Siam Journal on Optimization
"... Abstract. A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL2 (log nL2)(log log nL2)) ite ..."
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Cited by 5 (4 self)
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Abstract. A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL2 (log nL2)(log log nL2)) iteration complexity. If the LCP admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with Qorder two. If m = Ω(log ( √ nL)), then both higher order methods have O ( √ n)L iteration complexity. The Qorder of convergence of one of the methods is (m + 1) for problems that admit a strict complementarity solution while the Qorder of convergence of the other method is (m + 1)/2 for general monotone LCPs.
PrimalDual AffineScaling Algorithms Fail For Semidefinite Programming
, 1998
"... In this paper, we give an example of a semidefinite programming problem in which primaldual affinescaling algorithms using the HRVW/KSH/M, MT, and AHO directions fail. We prove that each of these algorithm can generate a sequence converging to a nonoptimal solution, and that, for the AHO directio ..."
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Cited by 4 (0 self)
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In this paper, we give an example of a semidefinite programming problem in which primaldual affinescaling algorithms using the HRVW/KSH/M, MT, and AHO directions fail. We prove that each of these algorithm can generate a sequence converging to a nonoptimal solution, and that, for the AHO direction, even its associated continuous trajectory can converge to a nonoptimal point. In contrast with these directions, we show that the primaldual affinescaling algorithm using the NT direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution. Both primal and dual problems have interior feasible solutions, unique optimal solutions which satisfy strict complementarity, and are nondegenerate everywhere.
Improved complexity using higherorder correctors for primaldual Dikin affine scaling
, 1994
"... In this paper we show that the primaldual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O( p nL) complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semidefinite linear ..."
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Cited by 3 (1 self)
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In this paper we show that the primaldual Dikin affine scaling algorithm for linear programming of Jansen, Roos and Terlaky enhances an asymptotical O( p nL) complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semidefinite linear complementarity problems.
Polynomial primaldual cone affine scaling for semidefinite programming
, 1996
"... In this paper we generalize the primaldual cone affine scaling algorithm of Sturm and Zhang to semidefinite programming. We show in this paper that the underlying ideas of the cone affine scaling algorithm can be naturely applied to semidefinite programming, resulting in a new algorithm. Compared t ..."
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Cited by 2 (0 self)
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In this paper we generalize the primaldual cone affine scaling algorithm of Sturm and Zhang to semidefinite programming. We show in this paper that the underlying ideas of the cone affine scaling algorithm can be naturely applied to semidefinite programming, resulting in a new algorithm. Compared to other primaldual affine scaling algorithms for semidefinite programming (see, De Klerk, Roos and Terlaky [3]), our algorithm enjoys the lowest computational complexity.
1An O(√nL) Iteration Bound PrimalDual Cone Affine Scaling Algorithm for Linear Programming
, 1995
"... In this paper we introduce a primaldual affine scaling method. The method uses a searchdirection obtained by minimizing the duality gap over a linearly transformed conic section. This direction neither coincides with known primaldual affine scaling directions [12, 21], nor does it fit in the gene ..."
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In this paper we introduce a primaldual affine scaling method. The method uses a searchdirection obtained by minimizing the duality gap over a linearly transformed conic section. This direction neither coincides with known primaldual affine scaling directions [12, 21], nor does it fit in the generic primaldual method [15]. The new method requires O(√nL) main iterations. It is shown that the iterates follow the primaldual central path in a neighbourhood larger than the conventional N2 neighbourhood. The proximity to the primaldual central path is measured by trigonometric functions. Key words: Linear programming, primaldual interior point algorithm, affine scaling, polynomiality.