ex_5_1.py

"""
Copyright 2013 Steven Diamond

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""

from __future__ import division

from multiprocessing import Pool

import numpy as np
from pylab import axis, plot, show, title, xlabel, ylabel

from cvxpy import Minimize, Parameter, Problem, Variable, quad_form

# Taken from CVX website http://cvxr.com/cvx/examples/
# Exercise 5.1d: Sensitivity analysis for a simple QCQP
# Ported from cvx matlab to cvxpy by Misrab Faizullah-Khan
# Original comments below

# Boyd & Vandenberghe, "Convex Optimization"
# Joelle Skaf - 08/29/05
# (a figure is generated)
#
# Let p_star(u) denote the optimal value of:
#           minimize    x^2 + 1
#               s.t.    (x-2)(x-2)<=u
# Finds p_star(u) and plots it versus u

u = Parameter()
x = Variable()

objective = Minimize( quad_form(x,1) + 1 )
constraint = [ quad_form(x,1) - 6*x + 8 <= u ]
p = Problem(objective, constraint)

# Assign a value to gamma and find the optimal x.
def get_x(u_value):
    u.value = u_value
    result = p.solve()
    return x.value

u_values = np.linspace(-0.9,10,num=50)
# Serial computation.
x_values = [get_x(value) for value in u_values]

# Parallel computation.
pool = Pool(processes = 4)
x_values = pool.map(get_x, u_values)

# Plot the tradeoff curve
plot(u_values, x_values)
# label
title('Sensitivity Analysis: p*(u) vs u')
xlabel('u')
ylabel('p*(u)')
axis([-2, 10, -1, 3])
show()