# Chapter 6: Optimization¶

Robert Johansson

Source code listings for Numerical Python - Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib (ISBN 978-1-484242-45-2).

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import matplotlib
matplotlib.rcParams['mathtext.fontset'] = 'stix'
matplotlib.rcParams['font.family'] = 'serif'
matplotlib.rcParams['font.sans-serif'] = 'stix'

In [2]:
import numpy as np

In [3]:
import sympy

In [4]:
sympy.init_printing()

In [5]:
from scipy import optimize

In [6]:
import cvxopt

In [7]:
from __future__ import division


## Univariate¶

In [8]:
r, h = sympy.symbols("r, h")

In [9]:
Area = 2 * sympy.pi * r**2 + 2 * sympy.pi * r * h

In [10]:
Volume = sympy.pi * r**2 * h

In [11]:
h_r = sympy.solve(Volume - 1)[0]

In [12]:
Area_r = Area.subs(h_r)

In [13]:
rsol = sympy.solve(Area_r.diff(r))[0]

In [14]:
rsol

Out[14]:
$$\frac{2^{\frac{2}{3}}}{2 \sqrt[3]{\pi}}$$
In [15]:
_.evalf()

Out[15]:
$$0.541926070139289$$
In [16]:
# verify that the second derivative is positive, so that rsol is a minimum
Area_r.diff(r, 2).subs(r, rsol)

Out[16]:
$$12 \pi$$
In [17]:
Area_r.subs(r, rsol)

Out[17]:
$$3 \sqrt[3]{2} \sqrt[3]{\pi}$$
In [18]:
_.evalf()

Out[18]:
$$5.53581044593209$$
In [19]:
def f(r):
return 2 * np.pi * r**2 + 2 / r

In [20]:
r_min = optimize.brent(f, brack=(0.1, 4))

In [21]:
r_min

Out[21]:
$$0.541926077256$$
In [22]:
f(r_min)

Out[22]:
$$5.53581044593$$
In [23]:
optimize.minimize_scalar(f, bracket=(0.1, 4))

Out[23]:
     fun: 5.5358104459320856
nfev: 19
nit: 15
success: True
x: 0.54192607725571351
In [24]:
r = np.linspace(0, 2, 100)[1:]

In [25]:
fig, ax = plt.subplots(figsize=(8, 4))

ax.plot(r, f(r), lw=2, color='b')
ax.plot(r_min, f(r_min), 'r*', markersize=15)
ax.set_title(r"$f(r) = 2\pi r^2+2/r$", fontsize=18)
ax.set_xlabel(r"$r$", fontsize=18)
ax.set_xticks([0, 0.5, 1, 1.5, 2])
ax.set_ylim(0, 30)

fig.tight_layout()
fig.savefig('ch6-univariate-optimization-example.pdf')


## Two-dimensional¶

In [26]:
x1, x2 = sympy.symbols("x_1, x_2")

In [27]:
f_sym = (x1-1)**4 + 5 * (x2-1)**2 - 2*x1*x2

In [28]:
fprime_sym = [f_sym.diff(x_) for x_ in (x1, x2)]

In [29]:
# Gradient
sympy.Matrix(fprime_sym)

Out[29]:
$$\left[\begin{matrix}- 2 x_{2} + 4 \left(x_{1} - 1\right)^{3}\\- 2 x_{1} + 10 x_{2} - 10\end{matrix}\right]$$
In [30]:
fhess_sym = [[f_sym.diff(x1_, x2_) for x1_ in (x1, x2)] for x2_ in (x1, x2)]

In [31]:
# Hessian
sympy.Matrix(fhess_sym)

Out[31]:
$$\left[\begin{matrix}12 \left(x_{1} - 1\right)^{2} & -2\\-2 & 10\end{matrix}\right]$$
In [32]:
f_lmbda = sympy.lambdify((x1, x2), f_sym, 'numpy')

In [33]:
fprime_lmbda = sympy.lambdify((x1, x2), fprime_sym, 'numpy')

In [34]:
fhess_lmbda = sympy.lambdify((x1, x2), fhess_sym, 'numpy')

In [35]:
def func_XY_X_Y(f):
"""
Wrapper for f(X) -> f(X[0], X[1])
"""
return lambda X: np.array(f(X[0], X[1]))

In [36]:
f = func_XY_X_Y(f_lmbda)

In [37]:
fprime = func_XY_X_Y(fprime_lmbda)

In [38]:
fhess = func_XY_X_Y(fhess_lmbda)

In [39]:
X_opt = optimize.fmin_ncg(f, (0, 0), fprime=fprime, fhess=fhess)

Optimization terminated successfully.
Current function value: -3.867223
Iterations: 8
Function evaluations: 10
Hessian evaluations: 8

In [40]:
X_opt

Out[40]:
array([ 1.88292613,  1.37658523])
In [41]:
fig, ax = plt.subplots(figsize=(6, 4))
x_ = y_ = np.linspace(-1, 4, 100)
X, Y = np.meshgrid(x_, y_)
c = ax.contour(X, Y, f_lmbda(X, Y), 50)
ax.plot(X_opt[0], X_opt[1], 'r*', markersize=15)
ax.set_xlabel(r"$x_1$", fontsize=18)
ax.set_ylabel(r"$x_2$", fontsize=18)
plt.colorbar(c, ax=ax)
fig.tight_layout()
fig.savefig('ch6-examaple-two-dim.pdf');


## Brute force search for initial point¶

In [42]:
def f(X):
x, y = X
return (4 * np.sin(np.pi * x) + 6 * np.sin(np.pi * y)) + (x - 1)**2 + (y - 1)**2

In [43]:
x_start = optimize.brute(f, (slice(-3, 5, 0.5), slice(-3, 5, 0.5)), finish=None)

In [44]:
x_start

Out[44]:
array([ 1.5,  1.5])
In [45]:
f(x_start)

Out[45]:
$$-9.5$$
In [46]:
x_opt = optimize.fmin_bfgs(f, x_start)

Optimization terminated successfully.
Current function value: -9.520229
Iterations: 4
Function evaluations: 28

In [47]:
x_opt

Out[47]:
array([ 1.47586906,  1.48365787])
In [48]:
f(x_opt)

Out[48]:
$$-9.52022927306$$
In [49]:
def func_X_Y_to_XY(f, X, Y):
s = np.shape(X)
return f(np.vstack([X.ravel(), Y.ravel()])).reshape(*s)

In [50]:
fig, ax = plt.subplots(figsize=(6, 4))
x_ = y_ = np.linspace(-3, 5, 100)
X, Y = np.meshgrid(x_, y_)
c = ax.contour(X, Y, func_X_Y_to_XY(f, X, Y), 25)
ax.plot(x_opt[0], x_opt[1], 'r*', markersize=15)
ax.set_xlabel(r"$x_1$", fontsize=18)
ax.set_ylabel(r"$x_2$", fontsize=18)
plt.colorbar(c, ax=ax)
fig.tight_layout()
fig.savefig('ch6-example-2d-many-minima.pdf');


## Nonlinear least square¶

In [51]:
def f(x, beta0, beta1, beta2):
return beta0 + beta1 * np.exp(-beta2 * x**2)

In [52]:
beta = (0.25, 0.75, 0.5)

In [53]:
xdata = np.linspace(0, 5, 50)

In [54]:
y = f(xdata, *beta)

In [55]:
ydata = y + 0.05 * np.random.randn(len(xdata))

In [56]:
def g(beta):
return ydata - f(xdata, *beta)

In [57]:
beta_start = (1, 1, 1)

In [58]:
beta_opt, beta_cov = optimize.leastsq(g, beta_start)

In [59]:
beta_opt

Out[59]:
array([ 0.25498249,  0.7693897 ,  0.52778771])
In [60]:
fig, ax = plt.subplots()

ax.scatter(xdata, ydata, label="samples")
ax.plot(xdata, y, 'r', lw=2, label="true model")
ax.plot(xdata, f(xdata, *beta_opt), 'b', lw=2, label="fitted model")
ax.set_xlim(0, 5)
ax.set_xlabel(r"$x$", fontsize=18)
ax.set_ylabel(r"$f(x, \beta)$", fontsize=18)
ax.legend()
fig.tight_layout()
fig.savefig('ch6-nonlinear-least-square.pdf')

In [61]:
beta_opt, beta_cov = optimize.curve_fit(f, xdata, ydata)

In [62]:
beta_opt

Out[62]:
array([ 0.25498249,  0.7693897 ,  0.52778771])

## Constrained optimization¶

### Bounds¶

In [63]:
def f(X):
x, y = X
return (x-1)**2 + (y-1)**2

In [64]:
x_opt = optimize.minimize(f, [0, 0], method='BFGS').x

In [65]:
bnd_x1, bnd_x2 = (2, 3), (0, 2)

In [66]:
x_cons_opt = optimize.minimize(f, [0, 0], method='L-BFGS-B', bounds=[bnd_x1, bnd_x2]).x

In [ ]:
fig, ax = plt.subplots(figsize=(6, 4))
x_ = y_ = np.linspace(-1, 3, 100)
X, Y = np.meshgrid(x_, y_)
c = ax.contour(X, Y, func_X_Y_to_XY(f, X, Y), 50)
ax.plot(x_opt[0], x_opt[1], 'b*', markersize=15)
ax.plot(x_cons_opt[0], x_cons_opt[1], 'r*', markersize=15)
bound_rect = plt.Rectangle((bnd_x1[0], bnd_x2[0]),
bnd_x1[1] - bnd_x1[0], bnd_x2[1] - bnd_x2[0],
facecolor="grey")
ax.set_xlabel(r"$x_1$", fontsize=18)
ax.set_ylabel(r"$x_2$", fontsize=18)
plt.colorbar(c, ax=ax)

fig.tight_layout()
fig.savefig('ch6-example-constraint-bound.pdf');


## Lagrange multiplier¶

In [ ]:
x = x1, x2, x3, l = sympy.symbols("x_1, x_2, x_3, lambda")

In [ ]:
f = x1 * x2 * x3

In [ ]:
g = 2 * (x1 * x2 + x2 * x3 + x3 * x1) - 1

In [ ]:
L = f + l * g

In [ ]:
grad_L = [sympy.diff(L, x_) for x_ in x]

In [ ]:
sols = sympy.solve(grad_L)
sols

In [ ]:
g.subs(sols[0])

In [ ]:
f.subs(sols[0])

In [ ]:
def f(X):
return -X[0] * X[1] * X[2]

In [ ]:
def g(X):
return 2 * (X[0]*X[1] + X[1] * X[2] + X[2] * X[0]) - 1

In [ ]:
constraints = [dict(type='eq', fun=g)]

In [ ]:
result = optimize.minimize(f, [0.5, 1, 1.5], method='SLSQP', constraints=constraints)

In [ ]:
result

In [ ]:
result.x


## Inequality constraints¶

In [ ]:
def f(X):
return (X[0] - 1)**2 + (X[1] - 1)**2

def g(X):
return X[1] - 1.75 - (X[0] - 0.75)**4

In [ ]:
%time x_opt = optimize.minimize(f, (0, 0), method='BFGS').x

In [ ]:
constraints = [dict(type='ineq', fun=g)]

In [ ]:
%time x_cons_opt = optimize.minimize(f, (0, 0), method='SLSQP', constraints=constraints).x

In [ ]:
%time x_cons_opt = optimize.minimize(f, (0, 0), method='COBYLA', constraints=constraints).x

In [ ]:
fig, ax = plt.subplots(figsize=(6, 4))
x_ = y_ = np.linspace(-1, 3, 100)
X, Y = np.meshgrid(x_, y_)
c = ax.contour(X, Y, func_X_Y_to_XY(f, X, Y), 50)
ax.plot(x_opt[0], x_opt[1], 'b*', markersize=15)

ax.plot(x_, 1.75 + (x_-0.75)**4, 'k-', markersize=15)
ax.fill_between(x_, 1.75 + (x_-0.75)**4, 3, color="grey")
ax.plot(x_cons_opt[0], x_cons_opt[1], 'r*', markersize=15)

ax.set_ylim(-1, 3)
ax.set_xlabel(r"$x_0$", fontsize=18)
ax.set_ylabel(r"$x_1$", fontsize=18)
plt.colorbar(c, ax=ax)

fig.tight_layout()
fig.savefig('ch6-example-constraint-inequality.pdf');


## Linear programming¶

In [ ]:
c = np.array([-1.0, 2.0, -3.0])

A = np.array([[ 1.0, 1.0, 0.0],
[-1.0, 3.0, 0.0],
[ 0.0, -1.0, 1.0]])

b = np.array([1.0, 2.0, 3.0])

In [ ]:
A_ = cvxopt.matrix(A)
b_ = cvxopt.matrix(b)
c_ = cvxopt.matrix(c)

In [ ]:
sol = cvxopt.solvers.lp(c_, A_, b_)

In [ ]:
x = np.array(sol['x'])

In [ ]:
x

In [ ]:
sol

In [ ]:
sol['primal objective']


## Quandratic problem with cvxopt¶

$\min \frac{1}{2}x^TPx + q^T x$

$G x \leq h$

For example, let's solve the problem

min $f(x_1, x_2) = (x_1 - 1)^2 + (x_2 - 1)^2 =$

$x_1^2 -2x_1 + 1 + x_2^2 - 2x_2 + 1 =$

$x_1^2 + x_2^2 - 2x_1 - 2x_2 + 2 =$

$= \frac{1}{2} x^T P x - q^T x + 2$

and

$\frac{3}{4} x_1 + x_2 \geq 3$, $x_1 \geq 0$

where

$P = 2 [[1, 0], [0, 1]]$ and $q = [-2, -2]$

and

$G = [[-3/4, -1], [-1, 0]]$ and $h = [-3, 0]$

In [ ]:
from cvxopt import matrix, solvers

P = 2 * np.array([[1.0, 0.0],
[0.0, 1.0]])
q = np.array([-2.0, -2.0])

G = np.array([[-0.75, -1.0],
[-1.0,  0.0]])

h = np.array([-3.0, 0.0])

In [ ]:
_P = cvxopt.matrix(P)
_q = cvxopt.matrix(q)
_G = cvxopt.matrix(G)
_h = cvxopt.matrix(h)

In [ ]:
%time sol = solvers.qp(_P, _q, _G, _h)

In [ ]:
# sol

In [ ]:
x = sol['x']

In [ ]:
x = np.array(x)

In [ ]:
sol['primal objective'] + 2

In [ ]:
fig, ax = plt.subplots(figsize=(6, 4))
x_ = y_ = np.linspace(-1, 3, 100)
X, Y = np.meshgrid(x_, y_)
c = ax.contour(X, Y, func_X_Y_to_XY(f, X, Y), 50)

y_ = (h[0] - G[0, 0] * x_)/G[0, 1]
ax.plot(x_, y_, 'k')
ax.add_patch(plt.Rectangle((0, 3), 4, 3, angle=angle, facecolor="grey"))

ax.plot(x_opt[0], x_opt[1], 'b*', markersize=15)
ax.plot(x[0], x[1], 'r*', markersize=15)

ax.set_ylim(-1, 3)
ax.set_xlim(-1, 3)
ax.set_xlabel(r"$x_1$", fontsize=18)
ax.set_ylabel(r"$x_2$", fontsize=18)
plt.colorbar(c, ax=ax)

fig.tight_layout()


# Arbitrary function callback API¶

In [ ]:
from cvxopt import modeling

In [ ]:
x1 = modeling.variable(1, "x1")
x2 = modeling.variable(1, "x2")

In [ ]:
x1 * x2 + x2

In [ ]:
help(modeling)

In [ ]:
help(solvers)

In [ ]:
help(solvers.cp)

In [ ]:
CVXPY

cvxpy.org

In [ ]:


In [ ]:


In [ ]:


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## Versions¶

In [ ]:
%reload_ext version_information

In [ ]:
%version_information numpy, scipy, cvxopt, sympy, matplotlib

In [ ]: