## The structure of conservative gradient fields

The classical Clarke subdifferential alone is inadequate for understanding automatic differentiation in nonsmooth contexts. Instead, we can sometimes rely on enlarged generalized gradients called “conservative fields”, defined through the natural path-wise chain rule: one application is the convergence analysis of gradient-based deep learning algorithms. In the semi-algebraic case, we show that all conservative fields are … Read more

## Automatic Differentiation of the Open CASCADE Technology CAD System and its coupling with an Adjoint CFD Solver

Automatic Differentiation (AD) is applied to the open-source CAD system Open CASCADE Technology using the AD software tool ADOL-C (Automatic Differentiation by OverLoading in C++). The differentiated CAD system is coupled with a discrete adjoint CFD solver, thus providing the first example of a complete differentiated design chain built from generic, multi-purpose tools. The design … Read more

## JuMP: A modeling language for mathematical optimization

JuMP is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. JuMP takes advantage of advanced features of the Julia programming language to offer unique functionality while achieving performance on par with commercial modeling tools for standard … Read more

## Some notes on applying computational divided differencing in optimization

We consider the problem of accurate computation of the finite difference $f(\x+\s)-f(\x)$ when $\Vert\s\Vert$ is very small. Direct evaluation of this difference in floating point arithmetic succumbs to cancellation error and yields 0 when $\s$ is sufficiently small. Nonetheless, accurate computation of this finite difference is required by many optimization algorithms for a “sufficient decrease” … Read more

## A framework for automated PDE-constrained optimisation

A generic framework for the solution of PDE-constrained optimisation problems based on the FEniCS system is presented. Its main features are an intuitive mathematical interface, a high degree of automation, and an efficient implementation of the generated adjoint model. The framework is based upon the extension of a domain-specific language for variational problems to cleanly … Read more

## Linearizing the Method of Conjugate Gradients

The method of conjugate gradients (CG) is widely used for the iterative solution of large sparse systems of equations $Ax=b$, where $A\in\Re^{n\times n}$ is symmetric positive definite. Let $x_k$ denote the $k$–th iterate of CG. In this paper we obtain an expression for $J_k$, the Jacobian matrix of $x_k$ with respect to $b$. We use … Read more

## On Stable Piecewise Linearization and Generalized Algorithmic Differentiation

It is shown how functions that are defined by evaluation programs involving the absolute value function (besides smooth elementals), can be approximated locally by piecewise-linear models in the style of algorithmic, or automatic, differentiation (AD). The model can be generated by a minor modification of standard AD tools and it is Lipschitz continuous with respect … Read more

## Adjoint Broyden a la GMRES

It is shown that a compact storage implementation of a quasi-Newton method based on the adjoint Broyden update reduces in the affine case exactly to the well established GMRES procedure. Generally, storage and linear algebra effort per step are small multiples of n k, where n is the number of variables and k the number … Read more

## Symbolic-Algebraic Computations in a Modeling Language for Mathematical Programming

AMPL is a language and environment for expressing and manipulating mathematical programming problems, i.e., minimizing or maximizing an algebraic objective function subject to algebraic constraints. AMPL permits separating a model, i.e., a symbolic representation of a class of problems, from the data required to specify a particular problem instance. Once AMPL has a problem instance, … Read more