void SQPInternal::init(){
// Call the init method of the base class
NLPSolverInternal::init();
// Read options
maxiter_ = getOption("maxiter");
maxiter_ls_ = getOption("maxiter_ls");
c1_ = getOption("c1");
beta_ = getOption("beta");
merit_memsize_ = getOption("merit_memory");
lbfgs_memory_ = getOption("lbfgs_memory");
tol_pr_ = getOption("tol_pr");
tol_du_ = getOption("tol_du");
regularize_ = getOption("regularize");
if(getOption("hessian_approximation")=="exact")
hess_mode_ = HESS_EXACT;
else if(getOption("hessian_approximation")=="limited-memory")
hess_mode_ = HESS_BFGS;
if (hess_mode_== HESS_EXACT && H_.isNull()) {
if (!getOption("generate_hessian")){
casadi_error("SQPInternal::evaluate: you set option 'hessian_approximation' to 'exact', but no hessian was supplied. Try with option \"generate_hessian\".");
}
}
// If the Hessian is generated, we use exact approximation by default
if (bool(getOption("generate_hessian"))){
setOption("hessian_approximation", "exact");
}
// Allocate a QP solver
CRSSparsity H_sparsity = hess_mode_==HESS_EXACT ? H_.output().sparsity() : sp_dense(n_,n_);
H_sparsity = H_sparsity + DMatrix::eye(n_).sparsity();
CRSSparsity A_sparsity = J_.isNull() ? CRSSparsity(0,n_,false) : J_.output().sparsity();
QPSolverCreator qp_solver_creator = getOption("qp_solver");
qp_solver_ = qp_solver_creator(H_sparsity,A_sparsity);
// Set options if provided
if(hasSetOption("qp_solver_options")){
Dictionary qp_solver_options = getOption("qp_solver_options");
qp_solver_.setOption(qp_solver_options);
}
qp_solver_.init();
// Lagrange multipliers of the NLP
mu_.resize(m_);
mu_x_.resize(n_);
// Lagrange gradient in the next iterate
gLag_.resize(n_);
gLag_old_.resize(n_);
// Current linearization point
x_.resize(n_);
x_cand_.resize(n_);
x_old_.resize(n_);
// Constraint function value
gk_.resize(m_);
gk_cand_.resize(m_);
// Hessian approximation
Bk_ = DMatrix(H_sparsity);
// Jacobian
Jk_ = DMatrix(A_sparsity);
// Bounds of the QP
qp_LBA_.resize(m_);
qp_UBA_.resize(m_);
qp_LBX_.resize(n_);
qp_UBX_.resize(n_);
// QP solution
dx_.resize(n_);
qp_DUAL_X_.resize(n_);
qp_DUAL_A_.resize(m_);
// Gradient of the objective
gf_.resize(n_);
// Create Hessian update function
if(hess_mode_ == HESS_BFGS){
// Create expressions corresponding to Bk, x, x_old, gLag and gLag_old
SXMatrix Bk = ssym("Bk",H_sparsity);
SXMatrix x = ssym("x",input(NLP_X_INIT).sparsity());
SXMatrix x_old = ssym("x",x.sparsity());
SXMatrix gLag = ssym("gLag",x.sparsity());
SXMatrix gLag_old = ssym("gLag_old",x.sparsity());
SXMatrix sk = x - x_old;
SXMatrix yk = gLag - gLag_old;
SXMatrix qk = mul(Bk, sk);
// Calculating theta
SXMatrix skBksk = inner_prod(sk, qk);
SXMatrix omega = if_else(inner_prod(yk, sk) < 0.2 * inner_prod(sk, qk),
0.8 * skBksk / (skBksk - inner_prod(sk, yk)),
1);
yk = omega * yk + (1 - omega) * qk;
SXMatrix theta = 1. / inner_prod(sk, yk);
SXMatrix phi = 1. / inner_prod(qk, sk);
SXMatrix Bk_new = Bk + theta * mul(yk, trans(yk)) - phi * mul(qk, trans(qk));
// Inputs of the BFGS update function
vector<SXMatrix> bfgs_in(BFGS_NUM_IN);
bfgs_in[BFGS_BK] = Bk;
bfgs_in[BFGS_X] = x;
bfgs_in[BFGS_X_OLD] = x_old;
bfgs_in[BFGS_GLAG] = gLag;
bfgs_in[BFGS_GLAG_OLD] = gLag_old;
bfgs_ = SXFunction(bfgs_in,Bk_new);
bfgs_.setOption("number_of_fwd_dir",0);
bfgs_.setOption("number_of_adj_dir",0);
bfgs_.init();
// Initial Hessian approximation
B_init_ = DMatrix::eye(n_);
}
// Header
if(bool(getOption("print_header"))){
cout << "-------------------------------------------" << endl;
cout << "This is CasADi::SQPMethod." << endl;
switch (hess_mode_) {
case HESS_EXACT:
cout << "Using exact Hessian" << endl;
break;
case HESS_BFGS:
cout << "Using limited memory BFGS Hessian approximation" << endl;
break;
}
cout << endl;
cout << "Number of variables: " << setw(9) << n_ << endl;
cout << "Number of constraints: " << setw(9) << m_ << endl;
cout << "Number of nonzeros in constraint Jacobian: " << setw(9) << A_sparsity.size() << endl;
cout << "Number of nonzeros in Lagrangian Hessian: " << setw(9) << H_sparsity.size() << endl;
cout << endl;
}
}
void LiftedSQPInternal::init(){
// Call the init method of the base class
NlpSolverInternal::init();
// Number of lifted variables
nv = getOption("num_lifted");
if(verbose_){
cout << "Initializing SQP method with " << nx_ << " variables and " << ng_ << " constraints." << endl;
cout << "Lifting " << nv << " variables." << endl;
if(gauss_newton_){
cout << "Gauss-Newton objective with " << F_.input().numel() << " terms." << endl;
}
}
// Read options
max_iter_ = getOption("max_iter");
max_iter_ls_ = getOption("max_iter_ls");
toldx_ = getOption("toldx");
tolgl_ = getOption("tolgl");
sigma_ = getOption("sigma");
rho_ = getOption("rho");
mu_safety_ = getOption("mu_safety");
eta_ = getOption("eta");
tau_ = getOption("tau");
// Assume SXFunction for now
SXFunction ffcn = shared_cast<SXFunction>(F_);
casadi_assert(!ffcn.isNull());
SXFunction gfcn = shared_cast<SXFunction>(G_);
casadi_assert(!gfcn.isNull());
// Extract the free variables and split into independent and dependent variables
SX x = ffcn.inputExpr(0);
int nx = x.size();
nu = nx-nv;
SX u = x[Slice(0,nu)];
SX v = x[Slice(nu,nu+nv)];
// Extract the constraint equations and split into constraints and definitions of dependent variables
SX f1 = ffcn.outputExpr(0);
int nf1 = f1.numel();
SX g = gfcn.outputExpr(0);
int nf2 = g.numel()-nv;
SX v_eq = g(Slice(0,nv));
SX f2 = g(Slice(nv,nv+nf2));
// Definition of v
SX v_def = v_eq + v;
// Objective function
SX f;
// Multipliers
SX lam_x, lam_g, lam_f2;
if(gauss_newton_){
// Least square objective
f = inner_prod(f1,f1)/2;
} else {
// Scalar objective function
f = f1;
// Lagrange multipliers for the simple bounds on u
SX lam_u = ssym("lam_u",nu);
// Lagrange multipliers for the simple bounds on v
SX lam_v = ssym("lam_v",nv);
// Lagrange multipliers for the simple bounds on x
lam_x = vertcat(lam_u,lam_v);
// Lagrange multipliers corresponding to the definition of the dependent variables
SX lam_v_eq = ssym("lam_v_eq",nv);
// Lagrange multipliers for the nonlinear constraints that aren't eliminated
lam_f2 = ssym("lam_f2",nf2);
if(verbose_){
cout << "Allocated intermediate variables." << endl;
}
// Lagrange multipliers for constraints
lam_g = vertcat(lam_v_eq,lam_f2);
// Lagrangian function
SX lag = f + inner_prod(lam_x,x);
if(!f2.empty()) lag += inner_prod(lam_f2,f2);
if(!v.empty()) lag += inner_prod(lam_v_eq,v_def);
// Gradient of the Lagrangian
SX lgrad = casadi::gradient(lag,x);
if(!v.empty()) lgrad -= vertcat(SX::zeros(nu),lam_v_eq); // Put here to ensure that lgrad is of the form "h_extended -v_extended"
makeDense(lgrad);
if(verbose_){
cout << "Generated the gradient of the Lagrangian." << endl;
}
// Condensed gradient of the Lagrangian
f1 = lgrad[Slice(0,nu)];
nf1 = nu;
// Gradient of h
SX v_eq_grad = lgrad[Slice(nu,nu+nv)];
// Reverse lam_v_eq and v_eq_grad
SX v_eq_grad_reversed = v_eq_grad;
copy(v_eq_grad.rbegin(),v_eq_grad.rend(),v_eq_grad_reversed.begin());
SX lam_v_eq_reversed = lam_v_eq;
copy(lam_v_eq.rbegin(),lam_v_eq.rend(),lam_v_eq_reversed.begin());
// Augment h and lam_v_eq
v_eq.append(v_eq_grad_reversed);
v.append(lam_v_eq_reversed);
}
// Residual function G
SXVector G_in(G_NUM_IN);
G_in[G_X] = x;
G_in[G_LAM_X] = lam_x;
G_in[G_LAM_G] = lam_g;
SXVector G_out(G_NUM_OUT);
G_out[G_D] = v_eq;
G_out[G_G] = g;
G_out[G_F] = f;
rfcn_ = SXFunction(G_in,G_out);
rfcn_.setOption("number_of_fwd_dir",0);
rfcn_.setOption("number_of_adj_dir",0);
rfcn_.setOption("live_variables",true);
rfcn_.init();
if(verbose_){
cout << "Generated residual function ( " << shared_cast<SXFunction>(rfcn_).getAlgorithmSize() << " nodes)." << endl;
}
// Difference vector d
SX d = ssym("d",nv);
if(!gauss_newton_){
vector<SX> dg = ssym("dg",nv).data();
reverse(dg.begin(),dg.end());
d.append(dg);
}
// Substitute out the v from the h
SX d_def = (v_eq + v)-d;
SXVector ex(3);
ex[0] = f1;
ex[1] = f2;
ex[2] = f;
substituteInPlace(v, d_def, ex, false);
SX f1_z = ex[0];
SX f2_z = ex[1];
SX f_z = ex[2];
// Modified function Z
enum ZIn{Z_U,Z_D,Z_LAM_X,Z_LAM_F2,Z_NUM_IN};
SXVector zfcn_in(Z_NUM_IN);
zfcn_in[Z_U] = u;
zfcn_in[Z_D] = d;
zfcn_in[Z_LAM_X] = lam_x;
zfcn_in[Z_LAM_F2] = lam_f2;
enum ZOut{Z_D_DEF,Z_F12,Z_NUM_OUT};
SXVector zfcn_out(Z_NUM_OUT);
zfcn_out[Z_D_DEF] = d_def;
zfcn_out[Z_F12] = vertcat(f1_z,f2_z);
SXFunction zfcn(zfcn_in,zfcn_out);
zfcn.init();
if(verbose_){
cout << "Generated reconstruction function ( " << zfcn.getAlgorithmSize() << " nodes)." << endl;
}
// Matrix A and B in lifted Newton
SX B = zfcn.jac(Z_U,Z_F12);
SX B1 = B(Slice(0,nf1),Slice(0,B.size2()));
SX B2 = B(Slice(nf1,B.size1()),Slice(0,B.size2()));
if(verbose_){
cout << "Formed B1 (dimension " << B1.size1() << "-by-" << B1.size2() << ", "<< B1.size() << " nonzeros) " <<
"and B2 (dimension " << B2.size1() << "-by-" << B2.size2() << ", "<< B2.size() << " nonzeros)." << endl;
}
// Step in u
SX du = ssym("du",nu);
SX dlam_f2 = ssym("dlam_f2",lam_f2.sparsity());
SX b1 = f1_z;
SX b2 = f2_z;
SX e;
if(nv > 0){
// Directional derivative of Z
vector<vector<SX> > Z_fwdSeed(2,zfcn_in);
vector<vector<SX> > Z_fwdSens(2,zfcn_out);
vector<vector<SX> > Z_adjSeed;
vector<vector<SX> > Z_adjSens;
Z_fwdSeed[0][Z_U].setZero();
Z_fwdSeed[0][Z_D] = -d;
Z_fwdSeed[0][Z_LAM_X].setZero();
Z_fwdSeed[0][Z_LAM_F2].setZero();
Z_fwdSeed[1][Z_U] = du;
Z_fwdSeed[1][Z_D] = -d;
Z_fwdSeed[1][Z_LAM_X].setZero();
Z_fwdSeed[1][Z_LAM_F2] = dlam_f2;
zfcn.eval(zfcn_in,zfcn_out,Z_fwdSeed,Z_fwdSens,Z_adjSeed,Z_adjSens);
b1 += Z_fwdSens[0][Z_F12](Slice(0,nf1));
b2 += Z_fwdSens[0][Z_F12](Slice(nf1,B.size1()));
e = Z_fwdSens[1][Z_D_DEF];
}
if(verbose_){
cout << "Formed b1 (dimension " << b1.size1() << "-by-" << b1.size2() << ", "<< b1.size() << " nonzeros) " <<
"and b2 (dimension " << b2.size1() << "-by-" << b2.size2() << ", "<< b2.size() << " nonzeros)." << endl;
}
// Generate Gauss-Newton Hessian
if(gauss_newton_){
b1 = mul(trans(B1),b1);
B1 = mul(trans(B1),B1);
if(verbose_){
cout << "Gauss Newton Hessian (dimension " << B1.size1() << "-by-" << B1.size2() << ", "<< B1.size() << " nonzeros)." << endl;
}
}
// Make sure b1 and b2 are dense vectors
makeDense(b1);
makeDense(b2);
// Quadratic approximation
SXVector lfcn_in(LIN_NUM_IN);
lfcn_in[LIN_X] = x;
lfcn_in[LIN_D] = d;
lfcn_in[LIN_LAM_X] = lam_x;
lfcn_in[LIN_LAM_G] = lam_g;
SXVector lfcn_out(LIN_NUM_OUT);
lfcn_out[LIN_F1] = b1;
lfcn_out[LIN_J1] = B1;
lfcn_out[LIN_F2] = b2;
lfcn_out[LIN_J2] = B2;
lfcn_ = SXFunction(lfcn_in,lfcn_out);
// lfcn_.setOption("verbose",true);
lfcn_.setOption("number_of_fwd_dir",0);
lfcn_.setOption("number_of_adj_dir",0);
lfcn_.setOption("live_variables",true);
lfcn_.init();
if(verbose_){
cout << "Generated linearization function ( " << shared_cast<SXFunction>(lfcn_).getAlgorithmSize() << " nodes)." << endl;
}
// Step expansion
SXVector efcn_in(EXP_NUM_IN);
copy(lfcn_in.begin(),lfcn_in.end(),efcn_in.begin());
efcn_in[EXP_DU] = du;
efcn_in[EXP_DLAM_F2] = dlam_f2;
efcn_ = SXFunction(efcn_in,e);
efcn_.setOption("number_of_fwd_dir",0);
efcn_.setOption("number_of_adj_dir",0);
efcn_.setOption("live_variables",true);
efcn_.init();
if(verbose_){
cout << "Generated step expansion function ( " << shared_cast<SXFunction>(efcn_).getAlgorithmSize() << " nodes)." << endl;
}
// Current guess for the primal solution
DMatrix &x_k = output(NLP_SOLVER_X);
// Current guess for the dual solution
DMatrix &lam_x_k = output(NLP_SOLVER_LAM_X);
DMatrix &lam_g_k = output(NLP_SOLVER_LAM_G);
// Allocate a QP solver
QpSolverCreator qp_solver_creator = getOption("qp_solver");
qp_solver_ = qp_solver_creator(B1.sparsity(),B2.sparsity());
// Set options if provided
if(hasSetOption("qp_solver_options")){
Dictionary qp_solver_options = getOption("qp_solver_options");
qp_solver_.setOption(qp_solver_options);
}
// Initialize the QP solver
qp_solver_.init();
if(verbose_){
cout << "Allocated QP solver." << endl;
}
// Residual
d_k_ = DMatrix(d.sparsity(),0);
// Primal step
dx_k_ = DMatrix(x_k.sparsity());
// Dual step
dlam_x_k_ = DMatrix(lam_x_k.sparsity());
dlam_g_k_ = DMatrix(lam_g_k.sparsity());
}