SG++
optimization.py

On this page, we look at an example application of the sgpp::optimization module.

Identical versions of the example are given in all languages currently supported by SG++: C++, Python, Java, and MATLAB.

For instructions on how to run the example, please see Installation and Usage.

The example interpolates a bivariate test function like the tutorial.cpp (Start Here) example. However, we use B-splines here instead to obtain a smoother interpolant. The resulting sparse grid function is then minimized with the method of steepest descent. For comparison, we also minimize the objective function with Nelder-Mead's method.

First, we import pysgpp and the required modules.

## First, we import pysgpp and the required modules.
import pysgpp
import math
import sys

The function $$f\colon [0, 1]^d \to \mathbb{R}$$ to be minimized is called objective function and has to derive from pysgpp.OptScalarFunction. In the constructor, we give the dimensionality of the domain (in this case $$d = 2$$). The eval method evaluates the objective function and returns the function value $$f(\vec{x})$$ for a given point $$\vec{x} \in [0, 1]^d$$.

class ExampleFunction(pysgpp.OptScalarFunction):
"""Example objective function from the title of my Master's thesis."""
def __init__(self):
super(ExampleFunction, self).__init__(2)
def eval(self, x):
"""Evaluates the function."""
return math.sin(8.0 * x[0]) + math.sin(7.0 * x[1])
def printLine():
print "----------------------------------------" + \

We have to disable OpenMP within pysgpp since it interferes with SWIG's director feature.

print "sgpp::optimization example program started.\n"
# increase verbosity of the output
pysgpp.OptPrinter.getInstance().setVerbosity(2)

Here, we set define some parameters: objective function, dimensionality, B-spline degree, maximal number of grid points, and adaptivity.

# objective function
f = ExampleFunction()
# dimension of domain
d = f.getNumberOfParameters()
# B-spline degree
p = 3
# maximal number of grid points
N = 30
gamma = 0.95

First, we define a grid with modified B-spline basis functions and an iterative grid generator, which can generate the grid adaptively.

grid = pysgpp.Grid.createModBsplineGrid(d, p)
gridGen = pysgpp.OptIterativeGridGeneratorRitterNovak(f, grid, N, gamma)

With the iterative grid generator, we generate adaptively a sparse grid.

printLine()
print "Generating grid...\n"
if not gridGen.generate():
print "Grid generation failed, exiting."
sys.exit(1)

Then, we hierarchize the function values to get hierarchical B-spline coefficients of the B-spline sparse grid interpolant $$\tilde{f}\colon [0, 1]^d \to \mathbb{R}$$.

printLine()
print "Hierarchizing...\n"
functionValues = gridGen.getFunctionValues()
coeffs = pysgpp.DataVector(len(functionValues))
hierSLE = pysgpp.OptHierarchisationSLE(grid)
sleSolver = pysgpp.OptAutoSLESolver()
# solve linear system
if not sleSolver.solve(hierSLE, gridGen.getFunctionValues(), coeffs):
print "Solving failed, exiting."
sys.exit(1)

We define the interpolant $$\tilde{f}$$ and its gradient $$\nabla\tilde{f}$$ for use with the gradient method (steepest descent). Of course, one can also use other optimization algorithms from sgpp::optimization::optimizer.

printLine()
print "Optimizing smooth interpolant...\n"
ft = pysgpp.OptInterpolantScalarFunction(grid, coeffs)
x0 = pysgpp.DataVector(d)

The gradient method needs a starting point. We use a point of our adaptively generated sparse grid as starting point. More specifically, we use the point with the smallest (most promising) function value and save it in x0.

gridStorage = gridGen.getGrid().getStorage()
# index of grid point with minimal function value
x0Index = 0
fX0 = functionValues[0]
for i in range(1, len(functionValues)):
if functionValues[i] < fX0:
fX0 = functionValues[i]
x0Index = i
x0 = gridStorage.getCoordinates(gridStorage.getPoint(x0Index));
ftX0 = ft.eval(x0)
print "x0 = {}".format(x0)
print "f(x0) = {:.6g}, ft(x0) = {:.6g}\n".format(fX0, ftX0)

We apply the gradient method and print the results.

fXOpt = f.eval(xOpt)
print "\nxOpt = {}".format(xOpt)
print "f(xOpt) = {:.6g}, ft(xOpt) = {:.6g}\n".format(fXOpt, ftXOpt)

For comparison, we apply the classical gradient-free Nelder-Mead method directly to the objective function $$f$$.

printLine()
print "Optimizing objective function (for comparison)...\n"
ftXOptNM = ft.eval(xOptNM)
print "\nxOptNM = {}".format(xOptNM)
print "f(xOptNM) = {:.6g}, ft(xOptNM) = {:.6g}\n".format(fXOptNM, ftXOptNM)
printLine()
print "\nsgpp::optimization example program terminated."

The example program outputs the following results:

sgpp::optimization example program started.

--------------------------------------------------------------------------------
Generating grid...

100.0% (N = 29, k = 3)
Done in 3ms.
--------------------------------------------------------------------------------
Hierarchizing...

Solving linear system (automatic method)...
estimated nnz ratio: 59.8%
constructing matrix (100.0%)
nnz ratio: 58.0%
Done in 0ms.
Done in 1ms.
--------------------------------------------------------------------------------
Optimizing smooth interpolant...

x0 = [0.625, 0.75]
f(x0) = -1.81786, ft(x0) = -1.81786

9 steps, f(x) = -2.000780
Done in 1ms.

xOpt = [0.589526, 0.673268]
f(xOpt) = -1.99999, ft(xOpt) = -2.00078

--------------------------------------------------------------------------------
Optimizing objective function (for comparison)...


We see that both the gradient-based optimization of the smooth sparse grid interpolant and the gradient-free optimization of the objective function find reasonable approximations of the minimum, which lies at $$(3\pi/16, 3\pi/14) \approx (0.58904862, 0.67319843)$$.