// Old implementation of Jacobian
DenseMatrix BSplineBasis::evalBasisJacobianOld(DenseVector &x) const
{
// Jacobian basis matrix
DenseMatrix J; J.setZero(getNumBasisFunctions(), numVariables);
// Calculate partial derivatives
for (unsigned int i = 0; i < numVariables; i++)
{
// One column in basis jacobian
DenseVector bi; bi.setOnes(1);
for (unsigned int j = 0; j < numVariables; j++)
{
DenseVector temp = bi;
DenseVector xi;
if (j == i)
{
// Differentiated basis
xi = bases.at(j).evaluateFirstDerivative(x(j));
}
else
{
// Normal basis
xi = bases.at(j).evaluate(x(j));
}
bi = kroneckerProduct(temp, xi);
}
// Fill out column
J.block(0,i,bi.rows(),1) = bi.block(0,0,bi.rows(),1);
}
return J;
}
double RBFSpline::eval(DenseVector x) const
{
std::vector<double> y;
for(int i=0; i<x.rows(); i++)
y.push_back(x(i));
return eval(y);
}
DenseVector BilinearRelaxationTest::bilinearFunction(DenseVector x)
{
assert(x.rows() == 2);
DenseVector y; y.setZero(1);
y(0) = x(0)*x(1);
return y;
}
DenseVector Michalewicz::michalewiczFunction(DenseVector x)
{
assert(x.rows() == 2);
double pi = atan(1)*4;
DenseVector y; y.setZero(1);
y(0) = -sin(x(0))*pow(sin(x(0)*x(0)/pi), 20) -sin(x(1))*pow(sin(2*x(1)*x(1)/pi), 20);
return y;
}
/*
* Calculate coefficients of B-spline representing a multivariate polynomial
*
* The polynomial f(x), with x in R^n, has m terms on the form
* f(x) = c(0)*x(0)^E(0,0)*x(1)^E(0,1)*...*x(n-1)^E(0,n-1)
* +c(1)*x(0)^E(1,0)*x(1)^E(1,1)*...*x(n-1)^E(1,n-1)
* +...
* +c(m-1)*x(0)^E(m-1,0)*x(1)^E(m-1,1)*...*x(n-1)^E(m-1,n-1)
* where c in R^m is a vector with coefficients for each of the m terms,
* and E in N^(mxn) is a matrix with the exponents of each variable in each of the m terms,
* e.g. the first row of E defines the first term with variable exponents E(0,0) to E(0,n-1).
*
* Note: E must be a matrix of nonnegative integers
*/
DenseMatrix getBSplineBasisCoefficients(DenseVector c, DenseMatrix E, std::vector<double> lb, std::vector<double> ub)
{
unsigned int dim = E.cols();
unsigned int terms = E.rows();
assert(dim >= 1); // At least one variable
assert(terms >= 1); // At least one term (assumes that c is a column vector)
assert(terms == c.rows());
assert(dim == lb.size());
assert(dim == ub.size());
// Get highest power of each variable
DenseVector powers = E.colwise().maxCoeff();
// Store in std vector
std::vector<unsigned int> powers2;
for (unsigned int i = 0; i < powers.size(); ++i)
powers2.push_back(powers(i));
// Calculate tensor product transformation matrix T
DenseMatrix T = getTransformationMatrix(powers2, lb, ub);
// Compute power basis coefficients (lambda vector)
SparseMatrix L(T.cols(),1);
L.setZero();
for (unsigned int i = 0; i < terms; i++)
{
SparseMatrix Li(1,1);
Li.insert(0,0) = 1;
for (unsigned int j = 0; j < dim; j++)
{
int e = E(i,j);
SparseVector li(powers(j)+1);
li.reserve(1);
li.insert(e) = 1;
SparseMatrix temp = Li;
Li = kroneckerProduct(temp, li);
}
L += c(i)*Li;
}
// Compute B-spline coefficients
DenseMatrix C = T*L;
return C;
}
ConstraintPtr ConstraintQuadratic::getConvexRelaxation()
{
if (!constraintConvex)
{
// Temp fix
bool isQuadConvex = false;
assert(isQuadConvex);
return this->clone();
// Temp fix end
// TODO: this code has not been used or tested in a long while!
Eigen::EigenSolver<DenseMatrix> es(H);
DenseVector eigs = es.eigenvalues().real();
double max_alpha = 0;
for (int i = 0; i < eigs.rows(); i++)
{
cout <<"eig nr " << i << " = " << eigs(i) << endl;
max_alpha = std::max(max_alpha, -0.5 * eigs(i));
}
if (max_alpha != 0)
{
cout << "Max alpha = " << max_alpha <<endl;
std::vector<double> alpha;
for (unsigned int i = 0; i < variables.size(); i++)
{
alpha.push_back(max_alpha);
}
cout << "Sour about that negative hessian." <<endl;
//ConstraintPtr quad_org(this);
//ConstraintPtr quad_org(this->clone());
//return new ConstraintDecoratorQuadraticConvexRelaxation(quad_org,alpha);
}
}
// Constraint is convex
return this->clone(false);
}
double Michalewicz::michalewiczFunction(DenseVector x, unsigned int m = 10)
{
assert(x.rows() == 2);
double pi = atan(1)*4;
return -sin(x(0))*pow(sin(x(0)*x(0)/pi), 2*m) -sin(x(1))*pow(sin(2*x(1)*x(1)/pi), 2*m);
}
ConstraintQuadratic::ConstraintQuadratic(std::vector<VariablePtr> variables, DenseMatrix A, DenseMatrix b, double c, double lb, double ub)
: Constraint(variables), A(A), b(b), c(c)
{
assert(A.cols() == (int)variables.size());
assert(A.rows() == b.rows());
assert(b.cols() == 1);
numConstraints = 1;
this->lb.push_back(lb);
this->ub.push_back(ub);
jacobianCalculated = true;
hessianCalculated = true;
constraintLinear = false;
constraintConvex = false;
convexRelaxationAvailable = true;
nnzJacobian = A.rows();
nnzHessian = 0;
constraintName = "Constraint Quadratic";
// // Check for parameters for NaN
// for (int i = 0; i < A.rows(); i++)
// {
// for (int j = 0; j < A.cols(); j++)
// {
// bool nanA = false;
// if (A(i,j) != A(i,j)) nanA = true;
// assert(nanA == false);
// }
// bool nanb = false;
// if (b(i) != b(i)) nanb = true;
// assert(nanb == false);
// }
// Calculate and store Hessian
H = A.transpose() + A;
// H is symmetric so fill out lower left triangle only
for (int row = 0; row < H.rows(); row++)
{
for (int col = 0; col <= row; col++)
{
if (H(row,col) != 0)
{
nnzHessian++;
}
}
}
// Check convexity using Hessian
Eigen::EigenSolver<DenseMatrix> es(H);
DenseVector eigs = es.eigenvalues().real();
double minEigVal = 0;
for (int i = 0; i < eigs.rows(); i++)
{
if (eigs(i) < minEigVal) minEigVal = eigs(i);
}
if (minEigVal >= 0 && lb <= -INF)
{
constraintConvex = true;
}
// Note could also check that max. eigen value <= 0 and ub = INF
checkConstraintSanity();
}
double sixHumpCamelBack(DenseVector x)
{
assert(x.rows() == 2);
return (4 - 2.1*x(0)*x(0) + (1/3.)*x(0)*x(0)*x(0)*x(0))*x(0)*x(0) + x(0)*x(1) + (-4 + 4*x(1)*x(1))*x(1)*x(1);
}