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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].
An interiorpoint method for a class of saddlepoint problems
 Journal of Optimization Theory and Applications
, 2003
"... Abstract. We present a polynomialtime interiorpoint algorithm for a class of nonlinear saddlepoint problems that involve semidefiniteness constraints on matrix variables. These problems originate from robust optimization formulations of convex quadratic programming problems with uncertain input p ..."
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Cited by 7 (1 self)
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Abstract. We present a polynomialtime interiorpoint algorithm for a class of nonlinear saddlepoint problems that involve semidefiniteness constraints on matrix variables. These problems originate from robust optimization formulations of convex quadratic programming problems with uncertain input parameters. As an application of our approach, we discuss a robust formulation of the Markowitz portfolio selection model. Key Words. Interiorpoint methods, robust optimization, portfolio optimization, saddlepoint problems, quadratic programming, semidefinite programming. 1.
Two InteriorPoint Methods for Nonlinear
 J. Optim. Theory Appl
, 1999
"... . Two interiorpoint algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P complementarity problems. The proof of the polynomial complexity of the first method requires that the problem satisfies a scaled Lipschitz condition. When specialized to the monotone co ..."
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Cited by 2 (1 self)
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. Two interiorpoint algorithms using a wide neighborhood of the central path are proposed to solve nonlinear P complementarity problems. The proof of the polynomial complexity of the first method requires that the problem satisfies a scaled Lipschitz condition. When specialized to the monotone complementarity problems, the results of the first method are similar to the ones in Ref. 1. The second method is quite different from the first in that the proof of its global convergence does not require the scaled Lipschitz assumption. At each step of this algorithm, however, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. Key Words. Interiorpoint algorithms, nonlinear P complementarity problems, polynomial complexity, scaled Lipschitz condition. 2 1. Introduction Consider the complementarity problem (CP), that is, finding a pair (x; u) 2 R n \Theta R n such that u = F (x); (x; u) 0 and x T u = 0; where F ...
Global Linear And Local Quadratic Convergence Of A LongStep AdaptiveMode Interior Point Method For Some Monotone Variational Inequality Problems
, 1996
"... . An interior point method is proposed to solve variational inequality problems for monotone functions and polyhedral sets. The method has the following advantages. 1. Given an initial interior feasible solution with duality gap ¯ 0 , the algorithm requires at most O[n log(¯ 0 =ffl)] iterations to ..."
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Cited by 2 (0 self)
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. An interior point method is proposed to solve variational inequality problems for monotone functions and polyhedral sets. The method has the following advantages. 1. Given an initial interior feasible solution with duality gap ¯ 0 , the algorithm requires at most O[n log(¯ 0 =ffl)] iterations to obtain an ffloptimal solution. 2. The rate of convergence of the duality gap is qquadratic. 3. At each iteration, a longstep improvement based on a line search is allowed. 4. The algorithm can automatically transfer from a linear mode to a quadratic mode to accelerate the local convergence. Keywords: Polynomial Complexity of Algorithms, Interior Point Methods, Monotone Variational Inequality Problems, Rate of Convergence. 1 The research is partially supported by Grant RP930033 of National University of Singapore. 2 Department of Decision Sciences. Email: fbasunj@nus.sg. 3 Department of Mathematics. Email: matzgy@nus.sg. 1 Introduction Given a function F : IR n ! IR n and a nonem...
A Quadratically Convergent Polynomial LongStep Algorithm For A Class Of Nonlinear Monotone Complementarity Problems
, 1999
"... . Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worstcase complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper pre ..."
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Cited by 1 (0 self)
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. Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worstcase complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper presents an algorithm which is both longstep and polynomial. In addition, the sequence generated by the algorithm, as well as the corresponding complementarity gap, converges quadratically. The proof of the polynomial complexity requires that the monotone mapping satisfies a scaled Lipschitz condition, while the quadratic rate of convergence is derived under the assumptions that the problem has a strictly complementary solution and that the Jacobian of the mapping satisfies certain regularity conditions. Keywords: Complexity of Algorithms, Interior Point Methods, Monotone Complementarity Problems, Rate of Convergence. 1 The research is partially supported by Grant RP930033 of National Universi...