2.7. Mathematical optimization: finding minima of functions¶
Authors: Gaël Varoquaux
Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. In this context, the function is called cost function, or objective function, or energy.
Here, we are interested in using scipy.optimize
for blackbox
optimization: we do not rely on the mathematical expression of the
function that we are optimizing. Note that this expression can often be
used for more efficient, non blackbox, optimization.
See also
References
Mathematical optimization is very … mathematical. If you want performance, it really pays to read the books:
Convex Optimization by Boyd and Vandenberghe (pdf available free online).
Numerical Optimization, by Nocedal and Wright. Detailed reference on gradient descent methods.
Practical Methods of Optimization by Fletcher: good at handwaving explanations.
2.7.1. Knowing your problem¶
Not all optimization problems are equal. Knowing your problem enables you to choose the right tool.
2.7.1.1. Convex versus nonconvex optimization¶
A convex function:

A nonconvex function 
Optimizing convex functions is easy. Optimizing nonconvex functions can be very hard.
Note
It can be proven that for a convex function a local minimum is also a global minimum. Then, in some sense, the minimum is unique.
2.7.1.2. Smooth and nonsmooth problems¶
A smooth function: The gradient is defined everywhere, and is a continuous function 
A nonsmooth function 
Optimizing smooth functions is easier (true in the context of blackbox optimization, otherwise Linear Programming is an example of methods which deal very efficiently with piecewise linear functions).
2.7.1.3. Noisy versus exact cost functions¶
Noisy (blue) and nonnoisy (green) functions 
2.7.1.4. Constraints¶
Optimizations under constraints Here: 
2.7.2. A review of the different optimizers¶
2.7.2.1. Getting started: 1D optimization¶
Let’s get started by finding the minimum of the scalar function
. scipy.optimize.minimize_scalar()
uses
Brent’s method to find the minimum of a function:
>>> import numpy as np
>>> import scipy as sp
>>> def f(x):
... return np.exp((x  0.5)**2)
>>> result = sp.optimize.minimize_scalar(f)
>>> result.success # check if solver was successful
True
>>> x_min = result.x
>>> x_min
0.50...
>>> x_min  0.5
5.8...e09
Note
You can use different solvers using the parameter method
.
Note
scipy.optimize.minimize_scalar()
can also be used for optimization
constrained to an interval using the parameter bounds
.
2.7.2.2. Gradient based methods¶
Some intuitions about gradient descent¶
Here we focus on intuitions, not code. Code will follow.
Gradient descent basically consists in taking small steps in the direction of the gradient, that is the direction of the steepest descent.
A wellconditioned quadratic function. 

An illconditioned quadratic function. The core problem of gradientmethods on illconditioned problems is that the gradient tends not to point in the direction of the minimum. 
We can see that very anisotropic (illconditioned) functions are harder to optimize.
Also, it clearly can be advantageous to take bigger steps. This is done in gradient descent code using a line search.
A wellconditioned quadratic function. 

An illconditioned quadratic function. 

An illconditioned nonquadratic function. 

An illconditioned very nonquadratic function. 
The more a function looks like a quadratic function (elliptic isocurves), the easier it is to optimize.
Conjugate gradient descent¶
The gradient descent algorithms above are toys not to be used on real problems.
As can be seen from the above experiments, one of the problems of the simple gradient descent algorithms, is that it tends to oscillate across a valley, each time following the direction of the gradient, that makes it cross the valley. The conjugate gradient solves this problem by adding a friction term: each step depends on the two last values of the gradient and sharp turns are reduced.
An illconditioned nonquadratic function. 

An illconditioned very nonquadratic function. 
SciPy provides scipy.optimize.minimize()
to find the minimum of scalar
functions of one or more variables. The simple conjugate gradient method can
be used by setting the parameter method
to CG
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> sp.optimize.minimize(f, [2, 1], method="CG")
message: Optimization terminated successfully.
success: True
status: 0
fun: 1.650...e11
x: [ 1.000e+00 1.000e+00]
nit: 13
jac: [6.15...e06 2.53...e07]
nfev: 81
njev: 27
Gradient methods need the Jacobian (gradient) of the function. They can compute it numerically, but will perform better if you can pass them the gradient:
>>> def jacobian(x):
... return np.array((2*.5*(1  x[0])  4*x[0]*(x[1]  x[0]**2), 2*(x[1]  x[0]**2)))
>>> sp.optimize.minimize(f, [2, 1], method="CG", jac=jacobian)
message: Optimization terminated successfully.
success: True
status: 0
fun: 2.95786...e14
x: [ 1.000e+00 1.000e+00]
nit: 8
jac: [ 7.183e07 2.990e07]
nfev: 16
njev: 16
Note that the function has only been evaluated 27 times, compared to 108 without the gradient.
2.7.2.3. Newton and quasinewton methods¶
Newton methods: using the Hessian (2nd differential)¶
Newton methods use a local quadratic approximation to compute the jump direction. For this purpose, they rely on the 2 first derivative of the function: the gradient and the Hessian.
An illconditioned quadratic function: Note that, as the quadratic approximation is exact, the Newton method is blazing fast 

An illconditioned nonquadratic function: Here we are optimizing a Gaussian, which is always below its quadratic approximation. As a result, the Newton method overshoots and leads to oscillations. 

An illconditioned very nonquadratic function: 
In SciPy, you can use the Newton method by setting method
to NewtonCG in
scipy.optimize.minimize()
. Here, CG refers to the fact that an internal
inversion of the Hessian is performed by conjugate gradient
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> def jacobian(x):
... return np.array((2*.5*(1  x[0])  4*x[0]*(x[1]  x[0]**2), 2*(x[1]  x[0]**2)))
>>> sp.optimize.minimize(f, [2,1], method="NewtonCG", jac=jacobian)
message: Optimization terminated successfully.
success: True
status: 0
fun: 1.5601357400786612e15
x: [ 1.000e+00 1.000e+00]
nit: 10
jac: [ 1.058e07 7.483e08]
nfev: 11
njev: 33
nhev: 0
Note that compared to a conjugate gradient (above), Newton’s method has required less function evaluations, but more gradient evaluations, as it uses it to approximate the Hessian. Let’s compute the Hessian and pass it to the algorithm:
>>> def hessian(x): # Computed with sympy
... return np.array(((1  4*x[1] + 12*x[0]**2, 4*x[0]), (4*x[0], 2)))
>>> sp.optimize.minimize(f, [2,1], method="NewtonCG", jac=jacobian, hess=hessian)
message: Optimization terminated successfully.
success: True
status: 0
fun: 1.6277298383706738e15
x: [ 1.000e+00 1.000e+00]
nit: 10
jac: [ 1.110e07 7.781e08]
nfev: 11
njev: 11
nhev: 10
Note
At very highdimension, the inversion of the Hessian can be costly and unstable (large scale > 250).
Note
Newton optimizers should not to be confused with Newton’s root finding
method, based on the same principles, scipy.optimize.newton()
.
QuasiNewton methods: approximating the Hessian on the fly¶
BFGS: BFGS (BroydenFletcherGoldfarbShanno algorithm) refines at each step an approximation of the Hessian.
2.7.3. Full code examples¶
2.7.4. Examples for the mathematical optimization chapter¶
Finding a minimum in a flat neighborhood
Constraint optimization: visualizing the geometry
Plotting the comparison of optimizers
Gallery generated by SphinxGallery
An illconditioned quadratic function: On a exactly quadratic function, BFGS is not as fast as Newton’s method, but still very fast. 

An illconditioned nonquadratic function: Here BFGS does better than Newton, as its empirical estimate of the curvature is better than that given by the Hessian. 

An illconditioned very nonquadratic function: 
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> def jacobian(x):
... return np.array((2*.5*(1  x[0])  4*x[0]*(x[1]  x[0]**2), 2*(x[1]  x[0]**2)))
>>> sp.optimize.minimize(f, [2, 1], method="BFGS", jac=jacobian)
message: Optimization terminated successfully.
success: True
status: 0
fun: 2.630637192365927e16
x: [ 1.000e+00 1.000e+00]
nit: 8
jac: [ 6.709e08 3.222e08]
hess_inv: [[ 9.999e01 2.000e+00]
[ 2.000e+00 4.499e+00]]
nfev: 10
njev: 10
LBFGS: Limitedmemory BFGS Sits between BFGS and conjugate gradient: in very high dimensions (> 250) the Hessian matrix is too costly to compute and invert. LBFGS keeps a lowrank version. In addition, box bounds are also supported by LBFGSB:
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> def jacobian(x):
... return np.array((2*.5*(1  x[0])  4*x[0]*(x[1]  x[0]**2), 2*(x[1]  x[0]**2)))
>>> sp.optimize.minimize(f, [2, 2], method="LBFGSB", jac=jacobian)
message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
success: True
status: 0
fun: 1.4417677473...e15
x: [ 1.000e+00 1.000e+00]
nit: 16
jac: [ 1.023e07 2.593e08]
nfev: 17
njev: 17
hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
2.7.4.12. Gradientless methods¶
A shooting method: the Powell algorithm¶
Almost a gradient approach
An illconditioned quadratic function: Powell’s method isn’t too sensitive to local illconditionning in low dimensions 

An illconditioned very nonquadratic function: 
Simplex method: the NelderMead¶
The NelderMead algorithms is a generalization of dichotomy approaches to highdimensional spaces. The algorithm works by refining a simplex, the generalization of intervals and triangles to highdimensional spaces, to bracket the minimum.
Strong points: it is robust to noise, as it does not rely on computing gradients. Thus it can work on functions that are not locally smooth such as experimental data points, as long as they display a largescale bellshape behavior. However it is slower than gradientbased methods on smooth, nonnoisy functions.
An illconditioned nonquadratic function: 

An illconditioned very nonquadratic function: 
Using the NelderMead solver in scipy.optimize.minimize()
:
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> sp.optimize.minimize(f, [2, 1], method="NelderMead")
message: Optimization terminated successfully.
success: True
status: 0
fun: 1.11527915993744e10
x: [ 1.000e+00 1.000e+00]
nit: 58
nfev: 111
final_simplex: (array([[ 1.000e+00, 1.000e+00],
[ 1.000e+00, 1.000e+00],
[ 1.000e+00, 1.000e+00]]), array([ 1.115e10, 1.537e10, 4.988e10]))
2.7.4.13. Global optimizers¶
If your problem does not admit a unique local minimum (which can be hard to test unless the function is convex), and you do not have prior information to initialize the optimization close to the solution, you may need a global optimizer.
Brute force: a grid search¶
scipy.optimize.brute()
evaluates the function on a given grid of
parameters and returns the parameters corresponding to the minimum
value. The parameters are specified with ranges given to
numpy.mgrid
. By default, 20 steps are taken in each direction:
>>> def f(x): # The rosenbrock function
... return .5*(1  x[0])**2 + (x[1]  x[0]**2)**2
>>> sp.optimize.brute(f, ((1, 2), (1, 2)))
array([1.0000..., 1.0000...])
2.7.5. Practical guide to optimization with SciPy¶
2.7.5.1. Choosing a method¶
All methods are exposed as the method
argument of
scipy.optimize.minimize()
.
 Without knowledge of the gradient:
In general, prefer BFGS or LBFGS, even if you have to approximate numerically gradients. These are also the default if you omit the parameter
method
 depending if the problem has constraints or boundsOn wellconditioned problems, Powell and NelderMead, both gradientfree methods, work well in high dimension, but they collapse for illconditioned problems.
 With knowledge of the gradient:
BFGS or LBFGS.
Computational overhead of BFGS is larger than that LBFGS, itself larger than that of conjugate gradient. On the other side, BFGS usually needs less function evaluations than CG. Thus conjugate gradient method is better than BFGS at optimizing computationally cheap functions.
 With the Hessian:
If you can compute the Hessian, prefer the Newton method (NewtonCG or TCG).
 If you have noisy measurements:
Use NelderMead or Powell.
2.7.5.2. Making your optimizer faster¶
Choose the right method (see above), do compute analytically the gradient and Hessian, if you can.
Use preconditionning when possible.
Choose your initialization points wisely. For instance, if you are running many similar optimizations, warmrestart one with the results of another.
Relax the tolerance if you don’t need precision using the parameter
tol
.
2.7.5.3. Computing gradients¶
Computing gradients, and even more Hessians, is very tedious but worth the effort. Symbolic computation with Sympy may come in handy.
Warning
A very common source of optimization not converging well is human
error in the computation of the gradient. You can use
scipy.optimize.check_grad()
to check that your gradient is
correct. It returns the norm of the different between the gradient
given, and a gradient computed numerically:
>>> sp.optimize.check_grad(f, jacobian, [2, 1])
2.384185791015625e07
See also scipy.optimize.approx_fprime()
to find your errors.
2.7.5.4. Synthetic exercices¶
2.7.6. Special case: nonlinear leastsquares¶
2.7.6.1. Minimizing the norm of a vector function¶
Least square problems, minimizing the norm of a vector function, have a
specific structure that can be used in the Levenberg–Marquardt algorithm
implemented in scipy.optimize.leastsq()
.
Lets try to minimize the norm of the following vectorial function:
>>> def f(x):
... return np.arctan(x)  np.arctan(np.linspace(0, 1, len(x)))
>>> x0 = np.zeros(10)
>>> sp.optimize.leastsq(f, x0)
(array([0. , 0.11111111, 0.22222222, 0.33333333, 0.44444444,
0.55555556, 0.66666667, 0.77777778, 0.88888889, 1. ]), 2)
This took 67 function evaluations (check it with ‘full_output=1’). What if we compute the norm ourselves and use a good generic optimizer (BFGS):
>>> def g(x):
... return np.sum(f(x)**2)
>>> result = sp.optimize.minimize(g, x0, method="BFGS")
>>> result.fun
2.6940...e11
BFGS needs more function calls, and gives a less precise result.
Note
leastsq is interesting compared to BFGS only if the dimensionality of the output vector is large, and larger than the number of parameters to optimize.
Warning
If the function is linear, this is a linearalgebra problem, and
should be solved with scipy.linalg.lstsq()
.
2.7.6.2. Curve fitting¶
Least square problems occur often when fitting a nonlinear to data.
While it is possible to construct our optimization problem ourselves,
SciPy provides a helper function for this purpose:
scipy.optimize.curve_fit()
:
>>> def f(t, omega, phi):
... return np.cos(omega * t + phi)
>>> x = np.linspace(0, 3, 50)
>>> rng = np.random.default_rng(27446968)
>>> y = f(x, 1.5, 1) + .1*rng.normal(size=50)
>>> sp.optimize.curve_fit(f, x, y)
(array([1.4812..., 0.9999...]), array([[ 0.0003..., 0.0004...],
[0.0004..., 0.0010...]]))
2.7.7. Optimization with constraints¶
2.7.7.1. Box bounds¶
Box bounds correspond to limiting each of the individual parameters of
the optimization. Note that some problems that are not originally written
as box bounds can be rewritten as such via change of variables. Both
scipy.optimize.minimize_scalar()
and scipy.optimize.minimize()
support bound constraints with the parameter bounds
:
>>> def f(x):
... return np.sqrt((x[0]  3)**2 + (x[1]  2)**2)
>>> sp.optimize.minimize(f, np.array([0, 0]), bounds=((1.5, 1.5), (1.5, 1.5)))
message: CONVERGENCE: NORM_OF_PROJECTED_GRADIENT_<=_PGTOL
success: True
status: 0
fun: 1.5811388300841898
x: [ 1.500e+00 1.500e+00]
nit: 2
jac: [9.487e01 3.162e01]
nfev: 9
njev: 3
hess_inv: <2x2 LbfgsInvHessProduct with dtype=float64>
2.7.7.2. General constraints¶
Equality and inequality constraints specified as functions: and .
scipy.optimize.fmin_slsqp()
Sequential least square programming: equality and inequality constraints:>>> def f(x): ... return np.sqrt((x[0]  3)**2 + (x[1]  2)**2) >>> def constraint(x): ... return np.atleast_1d(1.5  np.sum(np.abs(x))) >>> x0 = np.array([0, 0]) >>> sp.optimize.minimize(f, x0, constraints={"fun": constraint, "type": "ineq"}) message: Optimization terminated successfully success: True status: 0 fun: 2.4748737350439685 x: [ 1.250e+00 2.500e01] nit: 5 jac: [7.071e01 7.071e01] nfev: 15 njev: 5
Warning
The above problem is known as the Lasso problem in statistics, and there exist very efficient solvers for it (for instance in scikitlearn). In general do not use generic solvers when specific ones exist.
2.7.8. Full code examples¶
2.7.9. Examples for the mathematical optimization chapter¶
Finding a minimum in a flat neighborhood
Constraint optimization: visualizing the geometry
Plotting the comparison of optimizers