197 research outputs found
On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems
By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems
Hybrid Newton-type method for a class of semismooth equations
In this paper, we present a hybrid method for the solution of a class of composite semismooth equations encountered frequently in applications. The method is obtained by combining a generalized finite-difference Newton method to an inexpensive direct search method. We prove that, under standard assumptions, the method is globally convergent with a local rate of convergence which is superlinear or quadratic. We report also several numerical results obtained applying the method to suitable reformulations of well-known nonlinear complementarity problem
A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems
10.1137/S1052623400379620SIAM Journal on Optimization143783-80
A full approximation scheme multilevel method for nonlinear variational inequalities
We present the full approximation scheme constraint decomposition (FASCD)
multilevel method for solving variational inequalities (VIs). FASCD is a common
extension of both the full approximation scheme (FAS) multigrid technique for
nonlinear partial differential equations, due to A.~Brandt, and the constraint
decomposition (CD) method introduced by X.-C.~Tai for VIs arising in
optimization. We extend the CD idea by exploiting the telescoping nature of
certain function space subset decompositions arising from multilevel mesh
hierarchies. When a reduced-space (active set) Newton method is applied as a
smoother, with work proportional to the number of unknowns on a given mesh
level, FASCD V-cycles exhibit nearly mesh-independent convergence rates, and
full multigrid cycles are optimal solvers. The example problems include
differential operators which are symmetric linear, nonsymmetric linear, and
nonlinear, in unilateral and bilateral VI problems.Comment: 25 pages, 9 figure
A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem
In this paper, we consider the tensor eigenvalue complementarity problem
which is closely related to the optimality conditions for polynomial
optimization, as well as a class of differential inclusions with nonconvex
processes. By introducing an NCP-function, we reformulate the tensor eigenvalue
complementarity problem as a system of nonlinear equations. We show that this
function is strongly semismooth but not differentiable, in which case the
classical smoothing methods cannot apply. Furthermore, we propose a damped
semismooth Newton method for tensor eigenvalue complementarity problem. A new
procedure to evaluate an element of the generalized Jocobian is given, which
turns out to be an element of the B-subdifferential under mild assumptions. As
a result, the convergence of the damped semismooth Newton method is guaranteed
by existing results. The numerical experiments also show that our method is
efficient and promising
On the Finite Termination of An Entropy Function Based Smoothing Newton Method for Vertical Linear Complementarity Problems
By using a smooth entropy function to approximate the non-smooth max-type function, a vertical
linear complementarity problem (VLCP) can be treated as a family of parameterized smooth
equations. A Newton-type method with a testing procedure is proposed to solve such
a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite
number of iterations, under some conditions milder than those assumed in literature. Some
computational results are included to illustrate the potential of this approach
The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function
AbstractThe nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. By introducing a new smoothing NCP-function, the problem is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is proposed for solving the nonlinear complementarity problem with P0-function (P0-NCP) based on the new smoothing NCP-function. The proposed algorithm solves only one linear system of equations and performs only one line search per iteration. Without requiring strict complementarity assumption at the P0-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions
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