This example demonstrates the optimization of an objective function \( f\) with additional constraints.

The inequality constraints are specified via a function \( g\), the equality constraints are specified via a function \( h\).

Set the verbosity of the printed output. Increase the argument for more information about the process of solving linear system.

Load the predefined test problem G04. G04 consists of the following:

Generate a regular sparse grid of level five with modified B-Splines of degree five as basis functions.

const size_t p = 5;
const size_t d = problem.getObjectiveFunction().getNumberOfParameters();
std::unique_ptr<sgpp::base::Grid> grid(sgpp::base::Grid::createModBsplineGrid(d, p));
const size_t l = 5;

Initialize vector \( x \) that will contain the grid points and auxiliary vectors that will contain \( g(x) \) and \( h(x) \) for the hierarchisation. Initialize further a vector and two matrices that will contain the hierarchisation coefficients \( f_{\alpha} , g_{\alpha} \) and \( h_{\alpha} \).

const size_t N = grid->getSize();
const size_t mG = g.getNumberOfComponents();
const size_t mH = h.getNumberOfComponents();
sgpp::base::DataVector x(d), gx(mG), hx(mH);
sgpp::base::DataMatrix gAlpha(N, mG);
sgpp::base::DataMatrix hAlpha(N, mH);

Prepare the hierarchisation by filling \( f_{\alpha}, g_{\alpha}\) and \( h_{\alpha} \) with the function values \( f(x), g(x)\) and \( h(x)\) for every grid point x.

sgpp::base::GridStorage& gridStorage = grid->getStorage();
for (size_t i = 0; i < N; i++) {
x = gridStorage.getCoordinates(gridStorage[i]);
fAlpha[i] = f.eval(x);
g.eval(x, gx);
gAlpha.setRow(i, gx);
h.eval(x, hx);
hAlpha.setRow(i, hx);

Perform the hierarchisation.

std::unique_ptr<sgpp::optimization::OperationMultipleHierarchisation> hierOp(

Using the coefficients \( f_{\alpha}, g_{\alpha}\) and \( h_{\alpha} \) define the interpolant functions \(\tilde{f}, \tilde{g}\) and \(\tilde{h}\) as well as their gradients.

Define an augmented lagrangian method to solve the constrained optimization problem using the interpolants \(\tilde{f}, \tilde{g}\) and \(\tilde{h}\) as well as their gradients as arguments.

sgpp::optimization::optimizer::AugmentedLagrangian optimizer(ft, ftGradient, gt, gtGradient, ht,

Create a feasible starting point \(x_0 \) and apply the augmented lagrangian method defined above.

const sgpp::base::DataVector x0(optimizer.findFeasiblePoint());

Get the optimum point \( x_{Opt}\) and the calculated approximation to the optimum point \( x_{Opt}^*\).

const sgpp::base::DataVector xOptStar(optimizer.getOptimalPoint());

Evaluate \( f,g\) and \(h\) as well as their gradients in \( \lbrace x, x_{Opt}, x_{Opt}^{*} \rbrace \)

for (const sgpp::base::DataVector& x : {xOpt, x0, xOptStar}) {
const double fx = f.eval(x);
g.eval(x, gx);
h.eval(x, hx);
const double ftx = ft.eval(x);
gt.eval(x, gtx);
ht.eval(x, htx);

Print the evaluation point, and the values of \( f, g, h\) as well as the values \(\tilde{f}, \tilde{g}\) and \(\tilde{h}\) evaluated in the evaluation point.

std::cout << "x = " << x.toString() << "\n";
std::cout << "f = " << fx << ", g = " << gx.toString() << ", h = " << hx.toString() << "\n";
std::cout << "ft = " << ftx << ", gt = " << gtx.toString() << ", ht = " << htx.toString()
<< "\n";
std::cout << "\n";
return 0;

When executed this example produces the following output
(remark that the used example problem G04 has no equality constraints, so h is empty)

Solving linear system (automatic method)...
Done in 11124ms.
Solving linear system (automatic method)...
Done in 4013ms.
Solving linear system (automatic method)...
Done in 3746ms.
Optimizing (Augmented Lagrangian)...
Done in 157ms.
Optimizing (Augmented Lagrangian)...
Done in 7290ms.
x = [8.07474373435067976912e-03, 6.41782974255440154254e-03, 1.61029433421242457181e-01, 9.96927758912899086852e-01, 5.50146141071411132195e-01]
f = -30665.5, g = [-4.26325641456060111523e-14, -9.19999999999999573674e+01, -1.11594996910731083517e+01, -8.84050030892689164830e+00, -4.99999999999988631316e+00, -1.13686837721616029739e-13], h = []
ft = -30665.5, gt = [2.97498166580675220360e-14, -9.20000000000000710543e+01, -1.11594988391265399486e+01, -8.84050116087346715688e+00, -4.99999999999987476684e+00, -1.14863590187006103699e-13], ht = []

x = [4.19202291911377755707e-01, 1.35037410213593056518e-01, 7.12332125450077957574e-01, 3.49062366127655510084e-01, 1.19669759364614158859e-01]
f = -26846.1, g = [-1.68542663525514058165e+00, -9.03145733647448594184e+01, -9.76861095224427344874e+00, -1.02313890477557265513e+01, -3.30920437708866188586e+00, -1.69079562291133811414e+00], h = []
ft = -26846.1, gt = [-1.68542663525515057366e+00, -9.03145733647449446835e+01, -9.76861091582623330964e+00, -1.02313890841737737958e+01, -3.30920437708865167181e+00, -1.69079562291134677388e+00], ht = []

x = [0.00000000000000000000e+00, 1.33215347999698752179e-01, 6.77166052213670210946e-01, 9.99970269638240760735e-01, 9.02152104816860589409e-01]
f = -26818.1, g = [2.56624090454906195191e-01, -9.22566240904549061952e+01, -7.46102616932890327917e+00, -1.25389738306710967208e+01, -5.21058637424896886614e-01, -4.47894136257510311339e+00], h = []
ft = -26818.1, gt = [2.56624090454927789029e-01, -9.22566240904548919843e+01, -7.46102618953152685322e+00, -1.25389738104684500541e+01, -5.21058637424886450518e-01, -4.47894136257504182907e+00], ht = []