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137
A trust region method based on interior point techniques for nonlinear programming
 Mathematical Programming
, 1996
"... Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direc ..."
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Cited by 105 (18 self)
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Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primaldual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, SQP iteration, barrier method, trust region method.
Fundamentals of Spherical Parameterization for 3D Meshes
 PROCEEDINGS OF THE 2006 SYMPOSIUM ON INTERACTIVE 3D GRAPHICS AND GAMES, MARCH 1417, 2006
, 2003
"... Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the ..."
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Cited by 104 (25 self)
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Parametrization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parametrizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity do not overlap. Satisfying the nonoverlapping requirement is the most difficult and critical component of this process. We present a generalization of the method of barycentric coordinates for planar parametrization which solves the spherical parametrization problem, prove its correctness by establishing a connection to spectral graph theory and describe efficient numerical methods for computing these parametrizations.
Newton's Method For Large BoundConstrained Optimization Problems
 SIAM JOURNAL ON OPTIMIZATION
, 1998
"... We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and superlinea ..."
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Cited by 76 (4 self)
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We analyze a trust region version of Newton's method for boundconstrained problems. Our approach relies on the geometry of the feasible set, not on the particular representation in terms of constraints. The convergence theory holds for linearlyconstrained problems, and yields global and superlinear convergence without assuming neither strict complementarity nor linear independence of the active constraints. We also show that the convergence theory leads to an efficient implementation for large boundconstrained problems.
An interior point algorithm for large scale nonlinear programming
 SIAM Journal on Optimization
, 1999
"... The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of t ..."
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Cited by 72 (17 self)
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The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primaldual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests. Key words: constrained optimization, interior point method, largescale optimization, nonlinear programming, primal method, primaldual method, successive quadratic programming, trust region method.
TrustRegion InteriorPoint Algorithms For Minimization Problems With Simple Bounds
 SIAM J. Control and Optimization
, 1995
"... . Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model a ..."
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Cited by 48 (19 self)
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. Two trustregion interiorpoint algorithms for the solution of minimization problems with simple bounds are analyzed and tested. The algorithms scale the local model in a way similar to Coleman and Li [1]. The first algorithm is more usual in that the trust region and the local quadratic model are consistently scaled. The second algorithm proposed here uses an unscaled trust region. A global convergence result for these algorithms is given and dogleg and conjugategradient algorithms to compute trial steps are introduced. Some numerical examples that show the advantages of the second algorithm are presented. Keywords. trustregion methods, interiorpoint algorithms, DikinKarmarkar ellipsoid, Coleman and Li affine scaling, simple bounds. AMS subject classification. 49M37, 90C20, 90C30 1. Introduction. In this note we consider the boxconstrained minimization problem minimize f(x) subject to a x b; (1) where x 2 IR n , a 2 (IR [ f\Gamma1g) n , b 2 (IR [ f+1g) n and...
Fast sweeping methods for static hamiltonjacobi equations
 Society for Industrial and Applied Mathematics
, 2005
"... Abstract. We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimiz ..."
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Cited by 41 (4 self)
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Abstract. We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions.
A Subspace, Interior, and Conjugate Gradient Method for LargeScale BoundConstrained Minimization Problems
 SIAM Journal on Scientific Computing
, 1999
"... A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergenc ..."
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Cited by 35 (1 self)
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A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its fullspace version.
Reconstructing The Unknown Local Volatility Function
 Journal of Computational Finance
, 1998
"... Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphas ..."
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Cited by 34 (5 self)
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Using market European option prices, a method for computing a smooth local volatility function in a 1factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.
Global Methods For Nonlinear Complementarity Problems
 MATH. OPER. RES
, 1994
"... Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinea ..."
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Cited by 28 (1 self)
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Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations approach, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a boundconstrained nonlinear least squares problem. Algorithms based on this formulation are applicable to general nonlinear complementarity problems, can be started from any nonnegative starting point, and each iteration only requires the solution of systems of linear equations. Convergence to a solution of the nonlinear complementarity problem is guaranteed under reasonable regularity assumptions. The converge rate is Qlinear, Qsuperlinear, or Qquadratic, depending on the tolerances used to solve the subproblems.
A Discriminative Learning Framework with Pairwise Constraints for Video Object Classification
 In Proc. of CVPR
, 2004
"... In video object classification, insufficient labeled data may at times be easily augmented with pairwise constraints on sample points, i.e, whether they are in the same class or not. In this paper, we proposed a discriminative learning approach which incorporates pairwise constraints into a conventi ..."
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Cited by 27 (5 self)
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In video object classification, insufficient labeled data may at times be easily augmented with pairwise constraints on sample points, i.e, whether they are in the same class or not. In this paper, we proposed a discriminative learning approach which incorporates pairwise constraints into a conventional marginbased learning framework. The proposed approach offers several advantages over existing approaches dealing with pairwise constraints. First, as opposed to learning distance metrics, the new approach derives its classification power by directly modeling the decision boundary. Second, most previous work handles labeled data by converting them to pairwise constraints and thus leads to much more computation. The proposed approach can handle pairwise constraints together with labeled data so that the computation is greatly reduced. The proposed approach is evaluated on a people classification task with two surveillance video datasets.