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138
The PATH Solver: A NonMonotone Stabilization Scheme for Mixed Complementarity Problems
 OPTIMIZATION METHODS AND SOFTWARE
, 1995
"... The Path solver is an implementation of a stabilized Newton method for the solution of the Mixed Complementarity Problem. The stabilization scheme employs a pathgeneration procedure which is used to construct a piecewiselinear path from the current point to the Newton point; a step length acceptan ..."
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Cited by 148 (33 self)
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The Path solver is an implementation of a stabilized Newton method for the solution of the Mixed Complementarity Problem. The stabilization scheme employs a pathgeneration procedure which is used to construct a piecewiselinear path from the current point to the Newton point; a step length acceptance criterion and a nonmonotone pathsearch are then used to choose the next iterate. The algorithm is shown to be globally convergent under assumptions which generalize those required to obtain similar results in the smooth case. Several implementation issues are discussed, and extensive computational results obtained from problems commonly found in the literature are given.
A Semismooth Equation Approach To The Solution Of Nonlinear Complementarity Problems
, 1995
"... In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the n ..."
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Cited by 79 (9 self)
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In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.
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 74 (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.
A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm
, 1997
"... We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem ..."
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Cited by 73 (6 self)
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We investigate the properties of a new merit function which allows us to reduce a nonlinear complementarity problem to an unconstrained global minimization one. Assuming that the complementarity problem is defined by a P 0 function we prove that every stationary point of the unconstrained problem is a global solution; furthermore, if the complementarity problem is defined by a uniform P function, the level sets of the merit function are bounded. The properties of the new merit function are compared with those of the MangasarianSolodov's implicit Lagrangian and Fukushima's regularized gap function. We also introduce a new, simple, activeset local method for the solution of complementarity problems and show how this local algorithm can be made globally convergent by using the new merit function.
Asymptotic analysis of stochastic programs, Annals of Operations Research 30
 169–186. Shapiro / Journal of Multivariate Analysis 100 (2009) 936–945 945
, 1991
"... In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptoti ..."
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Cited by 61 (13 self)
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In this paper we discuss a general approach to studying asymptotic properties of statistical estimators in stochastic programming. The approach is based on an extended delta method and appears to be particularly suitable for deriving asymptotics of the optimal value of stochastic programs. Asymptotic analysis of the optimal value will be presented in detail. Asymptotic properties of the corresponding optimal solutions are briefly discussed.
Optimization Problems with perturbations, A guided tour
 SIAM REVIEW
, 1996
"... This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and app ..."
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Cited by 46 (10 self)
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This paper presents an overview of some recent and significant progress in the theory of optimization with perturbations. We put the emphasis on methods based on upper and lower estimates of the value of the perturbed problems. These methods allow to compute expansions of the value function and approximate solutions in situations where the set of Lagrange multipliers may be unbounded, or even empty. We give rather complete results for nonlinear programming problems, and describe some partial extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.
Characterizations of strong regularity for variational inequalities over polyhedral convex sets
 SIAM J. OPTIMIZATION
, 1996
"... Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critica ..."
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Cited by 46 (15 self)
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Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson’s notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new “critical face” condition and in other ways. The consequences for complementarity problems are worked out as a special case. Application is also made to standard nonlinear programming problems with parameters that include the canonical perturbations. In that framework a new characterization of strong regularity is obtained for the variational inequality associated with the KarushKuhnTucker conditions.
A Penalized FischerBurmeister NcpFunction: Theoretical Investigation And Numerical Results
, 1997
"... We introduce a new NCPfunction that reformulates a nonlinear complementarity problem as a system of semismooth equations \Phi(x) = 0. The new NCPfunction possesses all the nice properties of the FischerBurmeister function for local convergence. In addition, its natural merit function \Psi(x) = ..."
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Cited by 43 (12 self)
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We introduce a new NCPfunction that reformulates a nonlinear complementarity problem as a system of semismooth equations \Phi(x) = 0. The new NCPfunction possesses all the nice properties of the FischerBurmeister function for local convergence. In addition, its natural merit function \Psi(x) = 1 2 \Phi(x) T \Phi(x) has all the nice features of the KanzowYamashitaFukushima merit function for global convergence. In particular, the merit function has bounded level sets for a monotone complementarity problem with a strictly feasible point. This property allows the existing semismooth Newton methods to solve this important class of complementarity problems without additional assumptions. We investigate the properties of a semismooth Newtontype method based on the new NCPfunction and apply the method to a large class of complementarity problems. The numerical results indicate that the new algorithm is extremely promising.
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 41 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
On The Accurate Identification Of Active Constraints
, 1996
"... : We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an ..."
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Cited by 39 (7 self)
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: We consider nonlinear programs with inequality constraints, and we focus on the problem of identifying those constraints which will be active at an isolated local solution. The correct identification of active constraints is important from both a theoretical and a practical point of view. Such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained one which can be more easily dealt with. We present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers. As an example of application of the new technique we present a local active set Newtontype algorithm for the solution of general inequality constrained problems for which Qquadratic convergence of the primal variables can be proved under very weak conditions. We also present extensions to variational inequalities. Ke...