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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 super ..."
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Cited by 102 (5 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.
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 id ..."
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Cited by 61 (9 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.
A Truncated Newton Algorithm for Large Scale Box Constrained Optimization
 SIAM Journal on Optimization
"... : A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the approximate solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict com ..."
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Cited by 7 (0 self)
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: A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the approximate solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The algorithm proposed is based on a simple, smooth unconstrained reformulation of the bound constrained problem and may produce a sequence of points that are not feasible. Numerical results are reported. Key Words: Bound constrained problem, penalty function, Newton method, conjugate gradient, nonmonotone line search. 1 The postscript file of the TR 1599 DIS is available at http://www.dis.uniroma1.it/~facchinei 2 F. Facchinei, S. Lucidi, L. Palagi 1 Introduction We are concerned with the solution of simple bound constrained minimization problems of the form min f(x), s.t. l # x # u, (PB) where the objective function f is su#ciently smooth, l and u are cons...
An algorithm for the nonlinear programming problem of the railway Yield Management
"... In this paper we consider the problem of solving a special class of nonlinear constrained optimization problems. The study of this class of problems has been motivated by a practical application, namely the railway yield management problem. The aim of this paper is to define a nonlinear minimization ..."
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In this paper we consider the problem of solving a special class of nonlinear constrained optimization problems. The study of this class of problems has been motivated by a practical application, namely the railway yield management problem. The aim of this paper is to define a nonlinear minimization algorithm which exploits as much as possible the structure of the problems under consideration. The approach followed is to transform the original constrained problem into the unconstrained minimization of a continuously differentiable merit function whose unconstrained minimizers give a solution of the original constrained problem. In comparison with the merit functions proposed before in the literature, this new merit function has the distinguishing feature of taking full advantage of the particular structure of the constraints of the original problem. Furthermore we show also that it is possible to define descent search directions which are particularly well suited for the proposed merit...
An Unconstrained Minimization Approach To The Solution Of Optimization Problems With Simple Bounds
"... : A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity ..."
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: A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The theory presented covers Newton and QuasiNewton methods, allows rapid changes in the active set estimate and is based on a smooth unconstrained reformulation of the bound constrained problem. Key Words: Bound constrained problem, penalty function, Newton method, nonmonotone line search, strict complementarity. 2 F. Facchinei and S. Lucidi 1 Introduction We are concerned with the solution of simple bound constrained minimization problems of the form min f(x); s:t: l x u; (PB) where the objective function f is smooth, l and u are constant vectors, and the inequalities are valid componentwise. This paper is the continuation of [13], where a wide class of differentiable exa...