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Representations Of QuasiNewton Matrices And Their Use In Limited Memory Methods
, 1994
"... We derive compact representations of BFGS and symmetric rankone matrices for optimization. These representations allow us to efficiently implement limited memory methods for large constrained optimization problems. In particular, we discuss how to compute projections of limited memory matrices onto ..."
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Cited by 103 (8 self)
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We derive compact representations of BFGS and symmetric rankone matrices for optimization. These representations allow us to efficiently implement limited memory methods for large constrained optimization problems. In particular, we discuss how to compute projections of limited memory matrices onto subspaces. We also present a compact representation of the matrices generated by Broyden's update for solving systems of nonlinear equations. Key words: QuasiNewton method, constrained optimization, limited memory method, largescale optimization. Abbreviated title: Representation of quasiNewton matrices. 1. Introduction. Limited memory quasiNewton methods are known to be effective techniques for solving certain classes of largescale unconstrained optimization problems (Buckley and Le Nir (1983), Liu and Nocedal (1989), Gilbert and Lemar'echal (1989)) . They make simple approximations of Hessian matrices, which are often good enough to provide a fast rate of linear convergence, and re...
Revising Hull and Box Consistency
 INT. CONF. ON LOGIC PROGRAMMING
, 1999
"... Most intervalbased solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consist ..."
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Cited by 76 (13 self)
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Most intervalbased solvers in the constraint logic programming framework are based on either hull consistency or box consistency (or a variation of these ones) to narrow domains of variables involved in continuous constraint systems. This paper rst presents HC4, an algorithm to enforce hull consistency without decomposing complex constraints into primitives. Next, an extended denition for box consistency is given and the resulting consistency is shown to subsume hull consistency. Finally, BC4, a new algorithm to eciently enforce box consistency is described, that replaces BC3the original solely Newtonbased algorithm to achieve box consistencyby an algorithm based on HC4 and BC3 taking care of the number of occurrences of each variable in a constraint. BC4 is then shown to signicantly outperform both HC3 (the original algorithm enforcing hull consistency by decomposing constraints) and BC3. 1 Introduction Finite representation of numbers precludes computers from exactly solv...
Complete search in continuous global optimization and constraint satisfaction, Acta Numerica 13
, 2004
"... A chapter for ..."
Automatic Differentiation Of Advanced CFD Codes For Multidisciplinary Design
 Journal on Computing Systems in Engineering
, 1992
"... This paper addresses one such synergism for computa ..."
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Cited by 22 (16 self)
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This paper addresses one such synergism for computa
Progress in the Solving of a Circuit Design Problem
 JOURNAL OF GLOBAL OPTIMIZATION
, 2001
"... A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algori ..."
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Cited by 9 (2 self)
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A new branchandprune algorithm for globally solving nonlinear systems is proposed. The pruning technique combines a multidimensional interval Newton method with the constraint satisfaction algorithm HC4 [1]. The main contributions of this paper are the finegrained interaction between both algorithms which avoids some unnecessary computation,and the description of HC4 in terms of a chain rule for constraints’ projections. Our algorithm is experimentally compared with two global methods from Ratschek and Rokne [17] and from Puget and Van Hentenryck [16] on Ebers and Moll’ circuit design problem [6]. An interval enclosure of the solution with a precision of twelve significant digits is computed in four minutes, providing an improvement factor of five on the same machine.
Extending an algebraic modeling language to support constraint programming
 INFORMS Journal on Computing
, 2001
"... Abstract. Although algebraic modeling languages are widely used in linear and nonlinear programming applications, their use for combinatorial or discrete optimization has largely been limited to developing integer linear programming models for solution by generalpurpose branchandbound procedures. ..."
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Cited by 8 (3 self)
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Abstract. Although algebraic modeling languages are widely used in linear and nonlinear programming applications, their use for combinatorial or discrete optimization has largely been limited to developing integer linear programming models for solution by generalpurpose branchandbound procedures. Yet much of a modeling language’s underlying structure for expressing integer programs is equally useful for describing more general combinatorial optimization constructs. Constraint programming solvers offer an alternative approach to solving combinatorial optimization problems, in which natural combinatorial constructs are addressed directly within the solution procedure. Hence the growing popularity of constraint programming motivates a variety of extensions to algebraic modeling languages for the purpose of describing combinatorial problems and conveying them to solvers. We examine some of these language extensions along with the significant changes in solver interface design that they require. In particular, we describe how several useful combinatorial features have been added to the AMPL modeling language and how AMPL’s generalpurpose solver interface has been adapted accordingly. As an illustration of a solver connection, we provide examples from an AMPL driver for ILOG Solver. This work has been supported in part by Bell Laboratories and by grants DMI9414487
A Note on Efficient Computation of the Gradient in Semidefinite Programming
, 1999
"... In the GoemansWilliamson semidefinite relaxation of MAXCUT, the gradient of the dual barrier objective function has a term of the form diag(Z 1 ), where Z is the slack matrix. The purpose of this note is to show that this term can be computed in time and space proportional to the time and space f ..."
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Cited by 5 (0 self)
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In the GoemansWilliamson semidefinite relaxation of MAXCUT, the gradient of the dual barrier objective function has a term of the form diag(Z 1 ), where Z is the slack matrix. The purpose of this note is to show that this term can be computed in time and space proportional to the time and space for computing a sparse Cholesky factor of Z using an algorithm due to Erisman and Tinney. The algorithm for computing the term can also be derived from automatic differentiation in backward mode.
Tangent Linear and Adjoint Biogeochemical Models
"... . Adjoint models are powerful tools for inverse modeling. They are increasingly being used in meteorology and oceanography for sensitivity studies, data assimilation, and parameter estimation. Covering the range from simple box models to sophisticated General Circulation Models, they eciently compu ..."
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Cited by 4 (0 self)
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. Adjoint models are powerful tools for inverse modeling. They are increasingly being used in meteorology and oceanography for sensitivity studies, data assimilation, and parameter estimation. Covering the range from simple box models to sophisticated General Circulation Models, they eciently compute the sensitivity of a few model output variables with respect to arbitrarily many input variables. To the contrary, tangent linear models eciently compute the model output perturbation resulting from an initial input perturbation. Mathematically, both models evaluate the rst derivate or Jacobian matrix of the mapping dened by the original model. Eciency is an important issue for sophisticated models and in practice often determine whether a problem is solvable or not. We discuss here the advantages of tangent linear and adjoint models, as well as when to use either of them. The construction of adjoint and tangent linear models by hand is tedious and error prone. Computational Dierentiat...
On Combining Computational Differentiation and Toolkits for Parallel Scientific Computing
, 2000
"... . Automatic dierentiation is a powerful technique for evaluating derivatives of functions given in the form of a highlevel programming language such as Fortran, C, or C++. The program is treated as a potentially very long sequence of elementary statements to which the chain rule of dierential c ..."
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Cited by 3 (2 self)
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. Automatic dierentiation is a powerful technique for evaluating derivatives of functions given in the form of a highlevel programming language such as Fortran, C, or C++. The program is treated as a potentially very long sequence of elementary statements to which the chain rule of dierential calculus is applied over and over again. Combining automatic dierentiation and the organizational structure of toolkits for parallel scientic computing provides a mechanism for evaluating derivatives by exploiting mathematical insight on a higher level. In these toolkits, algorithmic structures such as BLASlike operations, linear and nonlinear solvers, or integrators for ordinary dierential equations can be identied by their standardized interfaces and recognized as highlevel mathematical objects rather than as a sequence of elementary statements. In this note, the dierentiation of a linear solver with respect to some parameter vector is taken as an example. Mathematical in...