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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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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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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
Operator Overloading as an Enabling Technology for Automatic Differentiation
, 1993
"... We present an example of the science that is enabled by objectoriented programming techniques. Scientific computation often needs derivatives for solving nonlinear systems such as those arising in many PDE algorithms, optimization, parameter identification, stiff ordinary differential equations, or ..."
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We present an example of the science that is enabled by objectoriented programming techniques. Scientific computation often needs derivatives for solving nonlinear systems such as those arising in many PDE algorithms, optimization, parameter identification, stiff ordinary differential equations, or sensitivity analysis. Automatic differentiation computes derivatives accurately and efficiently by applying the chain rule to each arithmetic operation or elementary function. Operator overloading enables the techniques of either the forward or the reverse mode of automatic differentiation to be applied to realworld scientific problems. We illustrate automatic differentiation with an example drawn from a model of unsaturated flow in a porous medium. The problem arises from planning for the longterm storage of radioactive waste. 1 Introduction Scientific computation often needs derivatives for solving nonlinear partial differential equations. One such problem currently under investigation...
A Modeling Interface to NonLinear Programming Solvers  An instance: xMPS, the extended MPS format
, 2000
"... We present a ModelerOptimizer Interface (MOI) for general closed form NonLinear Programs (NLP), which can be used to to transfer NLPs in a clear and simple manner between optimization components in a distributed environment. We demonstrate how this interface allows rst order derivative informat ..."
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We present a ModelerOptimizer Interface (MOI) for general closed form NonLinear Programs (NLP), which can be used to to transfer NLPs in a clear and simple manner between optimization components in a distributed environment. We demonstrate how this interface allows rst order derivative information to be easily calculated on the optimizer's side, using automatic dierentiation, hence removing the bottleneck of communicating derivative information between the modeler and the optimizer. We also show how this interface directly corresponds to a le format for NLPs, the extended MPS format (xMPS). This format directly extends the standard MPS le format for linear and mixed integer programs to include NLPs and permits a standardized way of transferring benchmark problems. The format spares the modeler the tedious task of calculating derivative information with minimal extra work required by the optimizer and thus increases eciency. This work was originally done at Maximal Sof...
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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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.
Evaluating Gradients in Optimal Control  Continuous Adjoints versus Automatic
 J. Optim. Theory Appl
, 2002
"... This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim at providing the optimization software not only with the discre ..."
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This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim at providing the optimization software not only with the discrete objective functional, but also with its gradient. The objective gradient can be computed either from forward (sensitivity) or backward (adjoint) information.
Application of automatic dierentiation to groundwater transport codes
 Preprint MCSP4410594, Mathematics and Computer Science Division, Argonne National Laboratory
, 1994
"... Abstract. Automatic dierentiation is a technique for generating ecient and reliable derivative codes from computer programs with minimal human eort. Derivatives of model output with respect to input are obtained exactly. No intrinsic limits to program length or complexity exist for this procedure. C ..."
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Abstract. Automatic dierentiation is a technique for generating ecient and reliable derivative codes from computer programs with minimal human eort. Derivatives of model output with respect to input are obtained exactly. No intrinsic limits to program length or complexity exist for this procedure. Calculation of derivatives of complex numerical models is required in system optimization, parameter identication, and systems identication. We report on our experiences with the ADIFOR (Automatic Dierentiation of Fortran) tool on a twodimensional groundwater ow and contaminant transport niteelement model, ISOQUAD, and a threedimensional contaminant transport niteelement model, TLS3D. Derivative values and computational times for the automatic dierentiation procedure are compared with values obtained from the divided dierences and handwritten analytic approaches. The automatic dierentiation tool ADIFOR produced derivative codes that calculated exact derivatives in typically almost an order of magnitude less CPU time than what is required for the imprecise divided dierences method for both the two and threedimensional codes. We also comment on the benet of automatic dierentiation technology with respect to accelerating the transfer of general techniques developed for using water resource computer models (such as optimal design, sensitivity analysis, and inverse modeling problems) to eld applications. 1
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 4 (3 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...
Computational Differentiation and Multidisciplinary Design
 IN INVERSE PROBLEMS AND OPTIMAL DESIGN IN INDUSTRY
, 1994
"... Multidisciplinary Design Optimization (MDO) by means of formal sensitivity analysis requires that each singlediscipline analysis code supply not only the output functions for the (usually constrained) optimization process and other discipline analysis inputs, but also the derivatives of all of thes ..."
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Multidisciplinary Design Optimization (MDO) by means of formal sensitivity analysis requires that each singlediscipline analysis code supply not only the output functions for the (usually constrained) optimization process and other discipline analysis inputs, but also the derivatives of all of these output functions with respect to its input variables. Computational differentiation techniques and automatic aifferentiation tools enable MDO by providing accurate and efficient derivatives of computer programs with little human effort. We discuss the principles behind automatic differentiation and give a brief overview of automatic differentiation tools and how they can be employed judiciously, for example, for sparse Jacobians and to exploit parallelism. We show how, and under what circumstances, automatic differentiation applied to iterative solvers delivers the mathematically desired derivatives. We then show how derivatives that can now be feasibly obtained by computational differenti...
Computational Divided Differencing and DividedDifference Arithmetics
, 2000
"... Tools for computational differentiation transform a program that computes a numerical function F (x) into a related program that computes F 0 (x) (the derivative of F ). This paper describes how techniques similar to those used in computationaldifferentiation tools can be used to implement other pr ..."
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Tools for computational differentiation transform a program that computes a numerical function F (x) into a related program that computes F 0 (x) (the derivative of F ). This paper describes how techniques similar to those used in computationaldifferentiation tools can be used to implement other program transformations  in particular, a variety of transformations for computational divided differencing . The specific technical contributions of the paper are as follows: It presents a program transformation that, given a numerical function F (x) de ned by a program, creates a program that computes F [x0 ; x1 ], the first divided difference of F(x), where F [x0 ; x1 ] def = F (x 0 ) F (x 1 ) x 0 x 1 if x0 6= x1 d dz F (z); evaluated at z = x0 if x0 = x1 It shows how computational first divided differencing generalizes computational differentiation. It presents a second program transformation that permits the creation of higherorder divided differences of a numerical function de ...