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On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and compleme ..."
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Cited by 24 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Convergence of Interior Point Algorithms for the Monotone Linear Complementarity Problem
, 1994
"... The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence ..."
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Cited by 23 (4 self)
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The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primaldual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases (whenever the convergence is not too fast in a certain sense), the sequences converge to the analytic center of the optimal face.
Superlinear Convergence Of An Algorithm For Monotone Linear Complementarity Problems, When No Strictly Complementary Solution Exists
 Mathematics of Operations Research
, 1996
"... A new predictorcorrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast ..."
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Cited by 12 (2 self)
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A new predictorcorrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast local convergence is attained without any need for estimating the optimal partition. In the special case that a strictly complementary solution does exist, the order of convergence becomes quadratic. The proof relies on an investigation of the asymptotic behavior of first and second order derivatives that are associated with trajectories of weighted centers for LCP. AMS 1991 subject classification: 90C33. Key words. monotone linear complementarity problem, primaldual interior point method, superlinear convergence, central path. 1 1. Introduction Given n \Theta n real matrices Q and R and a real vector b of order n, the horizontal linear complementarity problem (LCP) is the problem of fin...
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 mappi ..."
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Cited by 10 (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].
On Dual Convergence of the Generalized Proximal Point Method with Bregman Distances
, 1997
"... The use of generalized distances (e.g. Bregman distances), instead of the Euclidean one, in the proximal point method for convex optimization, allows for elimination of the inequality constraints from the subproblems. In this paper we consider the proximal point method with Bregman distances applied ..."
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Cited by 1 (1 self)
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The use of generalized distances (e.g. Bregman distances), instead of the Euclidean one, in the proximal point method for convex optimization, allows for elimination of the inequality constraints from the subproblems. In this paper we consider the proximal point method with Bregman distances applied to linearly constrained convex optimization problems, and study the behavior of the dual sequence obtained from the optimal multipliers of the linear constraints of each subproblem. Under rather general assumptions, which cover most Bregman distances of interest, we obtain an ergodic convergence result, namely that a sequence of weighted averages of the dual sequence converges to the centroid of the dual optimal set. As an intermediate result, we prove under the same assumptions that the dual central path generated by a large class of barriers, including the generalized Bregman distances, converges to the same point. Keywords: generalized proximal point methods, barrier function, centroid o...
Digital Object Identifier (DOI) 10.1007/s1010700304600
, 2004
"... Abstract. This paper presents some examples of illbehaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. Key words ..."
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Abstract. This paper presents some examples of illbehaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. Key words. Central path – Convex optimization – Interior point algorithm – Nonlinear programming – Penalty function methods 1.