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Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones
 University of Tsukuba
, 2006
"... We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [5] depends on the optimization theory of convex logbarrier functions, our approach is based on the paper of Monteiro and Pang [17], where a vast set of co ..."
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We study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones. While the analysis in [5] depends on the optimization theory of convex logbarrier functions, our approach is based on the paper of Monteiro and Pang [17], where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices. As an application of the results, we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects. Key words. Complementarity problem, symmetric cone, homogeneous algorithm, existence of trajectory, interior point method 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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. 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 ...
Infeasible Start Semidefinite Programming Algorithms Via SelfDual Embeddings
, 1997
"... The development of algorithms for semidefinite programming is an active research area, based on extensions of interior point methods for linear programming. As semidefinite programming duality theory is weaker than that of linear programming, only partial information can be obtained in some cases of ..."
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The development of algorithms for semidefinite programming is an active research area, based on extensions of interior point methods for linear programming. As semidefinite programming duality theory is weaker than that of linear programming, only partial information can be obtained in some cases of infeasibility, nonzero optimal duality gaps, etc. Infeasible start algorithms have been proposed which yield different kinds of information about the solution. In this paper a comprehensive treatment of a specific initialization strategy is presented, namely selfdual embedding, where the original primal and dual problems are embedded in a larger problem with a known interior feasible starting point. A framework for infeasible start algorithms with the best obtainable complexity bound is thus presented. The information that can be obtained in case of infeasibility, unboundedness, etc., is stated clearly. Important unresolved issues are discussed.
Interactive manipulation of articulated objects with geometry awareness
 In Proceedings of IEEE International Conference on Robotics and Automation
, 1999
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A new pathfollowing algorithm for nonlinear P∗ complementarity problems
 Computational Optimization and Applications
, 2005
"... Abstract. Based on the recent theoretical results of Zhao and Li [Math. Oper. Res., 26 (2001), pp. 119146], we present in this paper a new pathfollowing method for nonlinear P ∗ complementarity problems. Different from most existing interiorpoint algorithms that are based on the central path, thi ..."
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Abstract. Based on the recent theoretical results of Zhao and Li [Math. Oper. Res., 26 (2001), pp. 119146], we present in this paper a new pathfollowing method for nonlinear P ∗ complementarity problems. Different from most existing interiorpoint algorithms that are based on the central path, this algorithm tracks the “regularized central path ” which exists for any continuous P ∗ problem. It turns out that the algorithm is globally convergent for any P ∗ problem provided that its solution set is nonempty. By different choices of the parameters in the algorithm, the iterative sequence can approach to different types of points of the solution set. Moreover, local superlinear convergence of this algorithm can also be achieved under certain conditions. Key words. Nonlinear complementarity problems, pathfollowing algorithms, regularized central path, Tikhonov regularization, P∗mappings.
An Interior Point Approach to Quadratic and Parametric Quadratic Optimization
, 2004
"... In this thesis sensitivity analysis for quadratic optimization problems is studied. In sensitivity analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in ..."
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In this thesis sensitivity analysis for quadratic optimization problems is studied. In sensitivity analysis, which is often referred to as parametric optimization or parametric programming, a perturbation parameter is introduced into the optimization problem, which means that the coefficients in the objective function of the problem and in the righthandside of the constraints are perturbed. First, we describe quadratic programming problems and their parametric versions. Second, the theory for finding solutions of the parametric problems is developed. We also present an algorithm for solving such problems. In the implementation part, the implementation of the quadratic optimization solver is made. For that purpose, we extend the linear interior point package McIPM to solve quadratic problems. The quadratic solver is tested on the problems from the Maros and Mészáros test set. Finally, we implement the algorithm for parametric quadratic optimization. It utilizes the quadratic solver to solve auxiliary problems. We present numerical results produced by our parametric optimization package.
Complementarity Problems over Symmetric Cones: A Survey of Recent Developments in Several Aspects
, 2010
"... The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the S ..."
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The complementarity problem over a symmetric cone (that we call the Symmetric Cone Complementarity Problem, or the SCCP) has received much attention of researchers in the last decade. Most of studies done on the SCCP can be categorized into the three research themes, interior point methods for the SCCP, merit or smoothing function methods for the SCCP, and various properties of the SCCP. In this paper, we will provide a brief survey on the recent developments on these three themes.
GeometricallyAware Interactive Object Manipulation
, 2000
"... This paper describes formulation and management of constraints, and a nonlinear optimization algorithm that together enable interactive geometrically aware manipulation of articulated objects. Going beyond purely kinematic or dynamic approaches, our solution method directly employs geometric const ..."
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This paper describes formulation and management of constraints, and a nonlinear optimization algorithm that together enable interactive geometrically aware manipulation of articulated objects. Going beyond purely kinematic or dynamic approaches, our solution method directly employs geometric constraints to ensure noninterpenetration during object manipulation. We present the formulation of the inequality constraints used to ensure nonpenetration, describe how to manage the set of active inequality constraints as objects move, and show how these results are combined with a nonlinear optimization algorithm to achieve interactive geometrically aware object manipulation. Our optimization algorithm handles equality and inequality constraints and does not restrict object topology. It is an efficient iterative algorithm, quadratically convergent, with each iteration bounded by O#n nz #L##, where n nz #L# is the number of nonzeros in L, a Cholesky factor of a sparse matrix. Keywords:...