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# 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 PyPI, as a source distribution. Install it with

pip install diffcp


You will need a C++11-capable compiler to build diffcp.

diffcp requires:

diffcp uses Eigen; Eigen operations can be automatically vectorized by compilers. To enable vectorization, install with

MARCH_NATIVE=1 pip install diffcp


OpenMP can be enabled by passing extra arguments to your compiler. For example, on linux, you can tell gcc to activate the OpenMP extension by specifying the flag "-fopenmp":

OPENMP_FLAG="-fopenmp" pip install diffcp


To enable both vectorization and OpenMP (on linux), use

MARCH_NATIVE=1 OPENMP_FLAG="-fopenmp" pip install diffcp


### Cone programs

diffcp differentiates through a primal-dual 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 user-supplied problem data, and K is a user-defined 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, solver=None, **kwargs).


This function returns a primal-dual 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. The solver argument determines which solver to use; the available solvers are solver="SCS" and solver="ECOS". If no solver is specified, diffcp will choose the solver itself. In the case that the problem is not solved, i.e. the solver fails for some reason, we will raise a SolverError Exception.

#### 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 and c must be NumPy arrays.
• cone_dict is a dictionary that defines the convex cone K.
• warm_start is an optional tuple (x, y, s) at which to warm-start. (Note: this is only available for the SCS solver).
• **kwargs are keyword arguments to forward to the solver (e.g., verbose=False).

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, and
• diffcp.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, and s are a primal-dual solution.

• derivative is a function that applies the derivative at (A, b, c) to perturbations dA, db, dc. It has the signature derivative(dA, db, dc) -> dx, dy, ds, where dA is a SciPy sparse CSC matrix with the same sparsity pattern as A, and db and dc are NumPy arrays. dx, dy, and ds are NumPy arrays, approximating the change in the primal-dual solution due to the perturbation.

• adjoint_derivative is a function that applies the adjoint of the derivative to perturbations dx, dy, ds. It has the signature adjoint_derivative(dx, dy, ds) -> dA, db, dc, where dx, dy, and ds 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 = 1e-4 * np.random.randn(A.size)
dA = sparse.csc_matrix((data, nonzeros), shape=A.shape)
db = 1e-4 * np.random.randn(m)
dc = 1e-4 * 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        = {Differentiating through a Cone Program},
journal      = {Journal of Applied and Numerical Optimization},
year         = {2019},
volume       = {1},
number       = {2},
pages        = {107--115},
}

@misc{diffcp,
author       = {Agrawal, A. and Barratt, S. and Boyd, S. and Busseti, E. and Moursi, W.},
title        = {{diffcp}: differentiating through a cone program, version 1.0},
howpublished = {\url{https://github.com/cvxgrp/diffcp}},
year         = 2019
}


The following thesis concurrently derived the mathematics behind differentiating cone programs.

@phdthesis{amos2019differentiable,
author       = {Brandon Amos},
title        = {{Differentiable Optimization-Based Modeling for Machine Learning}},
school       = {Carnegie Mellon University},
year         = 2019,
month        = May,
}


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