Differentiating through cone programs
Project description
diffcp
diffcp
is a Python package for computing the derivative of a convex cone program, with respect to its problem data. The derivative is implemented as an abstract linear map, with methods for its forward application and its adjoint.
The implementation is based on the calculations in our paper Differentiating through a cone program.
Installation
diffcp
is available on Pip.
pip install diffcp
The requirements are:
If you require Python 2 support, please file a Github issue.
Cone programs
diffcp
differentiates through a primaldual cone program pair. The primal problem must be expressed as
minimize c'x
subject to Ax + s = b
s in K
where x
and s
are variables, A
, b
and c
are the usersupplied problem data, and K
is a userdefined convex cone. The corresponding dual problem is
minimize b'y
subject to A'y + c == 0
y in K^*
with dual variable y
.
Usage
diffcp
exposes the function
solve_and_derivative(A, b, c, cone_dict, warm_start=None, **kwargs).
This function returns a primaldual solution x
, y
, and s
, along with
functions for evaluating the derivative and its adjoint (transpose). These
functions respectively compute right and left multiplication of the derivative
of the solution map at A
, b
, and c
by a vector.
Arguments
The arguments A
, b
, and c
correspond to the problem data of a cone program.
A
must be a SciPy sparse CSC matrix.b
andc
must be NumPy arrays.cone_dict
is a dictionary that defines the convex coneK
.warm_start
is an optional tuple(x, y, s)
at which to warmstart SCS.**kwargs
are keyword arguments to forward to SCS (e.g.,verbose=True
).
These inputs must conform to the SCS convention for problem data. The keys in cone_dict
correspond to the cones, with
diffcp.ZERO
for the zero cone,diffcp.POS
for the positive orthant,diffcp.SOC
for a product of SOC cones,diffcp.PSD
for a product of PSD cones, anddiffcp.EXP
for a product of exponential cones.
The values in cone_dict
denote the sizes of each cone; the values of diffcp.SOC
, diffcp.PSD
, and diffcp.EXP
should be lists. The order of the rows of A
must match the ordering of the cones given above. For more details, consult the SCS documentation.
Return value
The function solve_and_derivative
returns a tuple
(x, y, s, derivative, adjoint_derivative)

x
,y
, ands
are a primaldual solution. 
derivative
is a function that applies the derivative at(A, b, c)
to perturbationsdA
,db
,dc
. It has the signaturederivative(dA, db, dc) > dx, dy, ds
, wheredA
is a SciPy sparse CSC matrix with the same sparsity pattern asA
, anddb
anddc
are NumPy arrays.dx
,dy
, andds
are NumPy arrays, approximating the change in the primaldual solution due to the perturbation. 
adjoint_derivative
is a function that applies the adjoint of the derivative to perturbationsdx
,dy
,ds
. It has the signatureadjoint_derivative(dx, dy, ds) > dA, db, dc
, wheredx
,dy
, andds
are NumPy arrays.
Example
import numpy as np from scipy import sparse import diffcp cone_dict = { diffcp.ZERO: 3, diffcp.POS: 3, diffcp.SOC: [5] } m = 3 + 3 + 5 n = 5 A, b, c = diffcp.utils.random_cone_prog(m, n, cone_dict) x, y, s, D, DT = diffcp.solve_and_derivative(A, b, c, cone_dict) # evaluate the derivative nonzeros = A.nonzero() data = 1e4 * np.random.randn(A.size) dA = sparse.csc_matrix((data, nonzeros), shape=A.shape) db = 1e4 * np.random.randn(m) dc = 1e4 * np.random.randn(n) dx, dy, ds = D(dA, db, dc) # evaluate the adjoint of the derivative dx = c dy = np.zeros(m) ds = np.zeros(m) dA, db, dc = DT(dx, dy, ds)
For more examples, including the SDP example described in the paper, see the examples
directory.
Citing
If you wish to cite diffcp
, please use the following BibTex:
@article{diffcp2019,
author = {Agrawal, A. and Barratt, S. and Boyd, S. and Busseti, E. and Moursi, W.},
title = {Differentiation Through a Cone Program},
journal = {arXiv preprint arXiv:1904.09043},
year = {2019},
}
@misc{diffcp,
author = {Agrawal, A. and Barratt, S. and Boyd, S. and Busseti, E. and Moursi, W.},
title = {{diffcp}: differentiation of a cone program, version 1.0},
howpublished = {\url{https://github.com/cvxgrp/diffcp}},
year = 2019
}
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