117 research outputs found
Forward-Mode Automatic Differentiation in Julia
We present ForwardDiff, a Julia package for forward-mode automatic
differentiation (AD) featuring performance competitive with low-level languages
like C++. Unlike recently developed AD tools in other popular high-level
languages such as Python and MATLAB, ForwardDiff takes advantage of
just-in-time (JIT) compilation to transparently recompile AD-unaware user code,
enabling efficient support for higher-order differentiation and differentiation
using custom number types (including complex numbers). For gradient and
Jacobian calculations, ForwardDiff provides a variant of vector-forward mode
that avoids expensive heap allocation and makes better use of memory bandwidth
than traditional vector mode. In our numerical experiments, we demonstrate that
for nontrivially large dimensions, ForwardDiff's gradient computations can be
faster than a reverse-mode implementation from the Python-based autograd
package. We also illustrate how ForwardDiff is used effectively within JuMP, a
modeling language for optimization. According to our usage statistics, 41
unique repositories on GitHub depend on ForwardDiff, with users from diverse
fields such as astronomy, optimization, finite element analysis, and
statistics.
This document is an extended abstract that has been accepted for presentation
at the AD2016 7th International Conference on Algorithmic Differentiation.Comment: 4 page
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
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