Python interface for COSMO.jl

This is a thin Python-wrapper around the Julia package COSMO.jl. COSMO is a general purpose solver for convex conic optimisation problems of the form:

with decision variables x ϵ R^n, s ϵ R^m and data matrices P=P'>=0, q ϵ R^n, A ϵ R^(m×n), and b ϵ R^m.

Example • Installation • Main documentation • Main repository

Installation

The wrapper makes a call to Julia via the pyjulia interface. To set this up:

On the Julia side:

1. Install the Julia programming language (v1.5+ recommended) [Julia Download page]

2. Open the Julia REPL and install the Julia package COSMO.jl: (type ]): pkg> add COSMO (at least v0.7.7+)

On the Python side:

3. Install pyjulia, the interface that lets you call Julia code from Python: python3 -m pip install julia

4. In Python run import julia followed by julia.install() to finish the pyjulia setup.

5. Install this package: git clone [email protected]:oxfordcontrol/cosmo-python.git (for now, later via pip / conda)

Example

This is a quick example to show the syntax of the interface. You can also use the example to verify that the steps in Installation were successful. Assume that we want to solve the following quadratic program:

minimize 1/2 x' [4, 1; 1 2] x + [1;1]' x
s.t.     A x == b 
         x >= 0

import cosmopy as cosmo
import numpy as np
from scipy import sparse

# define the problem
P = sparse.csc_matrix([[4., 1], [1, 2]])
q = np.array([1., 1])
A = sparse.csc_matrix([[1., 1], [1, 0], [0, 1], [-1., -1], [-1, 0], [0, -1]])
b = np.array([1, 0.7, 0.7, -1, 0, 0])
cone = {"l" : 6 }

# create a solver model
model = cosmo.Model()

# setup the model with the problem data and some optional solver settings
model.setup(P = P, q = q, A = A, b = b, cone = cone, verbose = True, eps_abs = 1e-5, max_iter = 4000)

# optional: warm starting of x
x = np.array([1., 0])
model.warm_start(x = x)

# solve the problem
model.optimize()

# query solution info
obj_val = model.get_objective_value() # optimal objective vale
status = model.get_status() # solution status
iter = model.get_iter() # number of iterations until convergence
times = model.get_times() # a dictionary of timings 
x_opt = model.get_x() # query the optimal primal variable x_opt

print("Solved with objective value: ", obj_val, " in", times["solver_time"], "s.")

More examples can be found in /examples.

Documentation

These notes only refer to the usage of this interface. For a more general overview take a look at the Documentation of the Julia package.

A Model is a thin Python class that wraps a COSMO.Model-type in Julia.

import cosmopy as cosmo
model = cosmo.Model()

The function setup copies the problem data and settings to the model:

def setup(P = None, q = None, A = None, b = None, cone = None, l = None, u = None, settings**)

The input to the function should be:

• P, A: scipy.sparse.csc_matrix. Note that for PSD constraints the off-diagonals of A have to be scaled appropriately (see details below).
• q, b: np.array
• cone: A dictionary that holds the dimensions of the conic constraints in the following order and corresponding to the rows in A (following SCS convention):

key	value	constraint	corresponds to
"f"	number of equality constraints	zero cone	COSMO.ZeroSet
"l"	number of inequality constraints	nonegative orthant	COSMO.Nonnegatives
"q"	list of SOC sizes	second order cone(s)	COSMO.SecondOrderCone
"s"	list of SDP sizes	psd cone	COSMO.PsdConeTriangle
"ep"	number of primal exp cones	exp cone (p)	COSMO.ExponentialCone
"ed"	number of dual exp cones	exp cone (d)	COSMO.DualExponentialCone
"p"	list of power cone parameters (neg value for dual)	3d-power cone	COSMO.PowerCone and COSMO.DualPowerCone
"b"	number of intervall constraints l <= s <= u	box constraint	COSMO.Box(l, u)

So if you want to create a problem with 2 equality constraints, 3 inequality constraints, 2 SOC-constraints of dim 3, 1 PSD-constraint for a 3x3 matrix, 1 PSD-constraint for a 4x4 matrix, 2 primal exponential cones, 1 dual exponential cone, 2 primal power cones with exponent 0.3 and 0.4, one dual power cone with exponent 0.5 and a box constraint of dim 3, define cone as follows:

cone = {"f" : 2, "l" : 3, "q" : [3, 3], "s" : [6, 10], "ep" : 2, "ed": 1, "p" : [0.3, 0.4, -0.5], "b" : 3}

• l, u: np.array boundary vectors defining the box constraint l <= s <= u. len(u) = len(l) have to correspond to the corresponding cone entry cone["b"]

The solver settings can be passed into setup as key-value arguments. A list of available solver settings can be found here. The only difference is that settings related to the kkt_solver and to the merge_strategy keys have to be passed as strings. So if you want to configure COSMO to use 5000 max iterations, the QDLDL solver for the linear system and the ParentChild clique merging strategy pass the following:

model.setup(..., max_iter = 5000, kkt_solver = "QDLDLKKTSolver", merge_strategy = "ParentChildMerge")

Before we attempt to solve the problem, we can provide COSMO with initial guesses for the primal variable x and dual variable y:

def warm_start(self, x=None, y=None):

The model.optimize() function calls the Julia equivalent: COSMO.optimize!(model) and solves the problem.

The following functions can be used to query the results:

def get_objective_value(self): #optimal objective value

def get_x(self): #optimal primal variable

def get_y(self): #optimal dual variable

def get_s(self): #optimal slack variable

def get_status(self): #solution status

def get_iter(self): #number of iterations of algorithm

def get_times(self): #dict with timing information

Performance

We advise to read the Performance Tips page. In particular, when COSMO is called from a python script, e.g. python3 solve_problem.py Julia will just-in-time compile the solver code which will slow down the overall execution. For larger problems it is advisable to solve a mini problem first to trigger the JIT-compilation and get full performance on the subsequent solve of the actual problem .

Licence

This project is licensed under the Apache License - see the LICENSE.md file for details.