An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems

Kojima, M.; Mizuno, Shinji; Yoshise, Akiko

doi:10.1007/bf01594942

Cited by 148 publications

(67 citation statements)

References 14 publications

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“…Subsequently, Kojima, Megiddo, Yoshise [13] and Kojima, Mizuno, Ye [14] developed polynomial-time algorithms for solving (LCP), using a different notion of potential reduction. [Some of these papers treated only the special case of (LCP) where A = 0, b = 0.…”

Section: Introductionmentioning

confidence: 99%

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

Tseng

1992

Mathematical Programming

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We consider a path following algorithm for solving linear complementarity problems with positive semi-definite matrices. This algorithm can start from any interior solution and attain a linear rate of convergence. Moreover, if the starting solution is appropriately chosen, this algorithm achieves a complexity of O(j--mL) iterations, where m is the number of variables and L is the size of the problem encoding in binary. We present a simple complexity analysis for this algorithm, which is based on a new Lyapunov function for measuring the nearness to optimality. This Lyapunov function has itself interesting properties that can be used in a line search to accelerate convergence.We also develop an inexact line search procedure in which the line search stepsize is obtainable in a closed form. Finally, we extended this algorithm to handle directly variables which are unconstrained in sign and whose corresponding matrix is positive definite. The rate of convergence of this extended algorithm is shown to be independent of the number of such variables.

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Section: Introductionmentioning

confidence: 99%

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

Tseng

1992

Mathematical Programming

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“…Much of the modern work in numerical algorithms has focused on interior-point methods [166,37,1631. Initially such work was limited to LPs [88,133,73,89,109,105,621, but was soon extended to encompass other CPs as well 1117, 118,7,87,119,162,15,1201. Now a number of excellent solvers are readily available, both commercial and freely distributed.…”

Section: Numerical Algorithmsmentioning

confidence: 99%

“…Michael Grant, Stephen Boyd, and Yinyu Ye function exp( x ) convex, increasing, >= 0; f (x) = ex (89) As with the monotonicity operations, the range must indeed be specified in the extended-valued sense, so it will inevitably be one-sided: that is, all convex functions are unbounded above, and all concave functions are unbounded below.…”

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confidence: 99%

Disciplined Convex Programming

Grant

Boyd

Nonconvex Optimization and Its Applications

3,684

4,391

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“…However, our example may not establish the worst case lower bound for the potential reduction algorithms proposed in [8,10,17,19].…”

Section: +~ 2+wmentioning

confidence: 96%

“…By direct substitution it is easy to check that the solution 2+7 w*=7 and or*---(1 +7) 2 satisfies the system of equations (9) and (13), which is equivalent to the original system (7), (8 Proof. We first show by induction on k that for all primal steps, a k = ol* and rk/t k = t k+l/r *+l .…”

Section: +~ 2+wmentioning

confidence: 99%

On the worst case complexity of potential reduction algorithms for linear programming

Bertsimas¹,

Luo

1997

Mathematical Programming

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There are several classes of interior point algorithms that solve linear programming problems in O(x/ffL) iterations. Among them, several potential reduction algorithms combine both theoretical (o(vr~L) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(v/'ffL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least ll(v/ffL) iterations to find an optimal solution. (~) 1997 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

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An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems

Cited by 148 publications

References 14 publications

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

Complexity analysis of a linear complementarity algorithm based on a Lyapunov function

Disciplined Convex Programming

On the worst case complexity of potential reduction algorithms for linear programming

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