vector<sp_mat> build_E_blocks(const Simulator * sim,
const Discretizer * disc,
double gamma,
bool include_oob){
Points points = disc->get_spatial_nodes();
uint N;
if(include_oob)
N = disc->number_of_all_nodes();
else
N = disc->number_of_spatial_nodes();
uint A = sim->num_actions();
uint Ad = sim->dim_actions();
mat actions = sim->get_actions();
assert(size(A,Ad) == size(actions));
uint num_samples = 25;
vector<sp_mat> E_blocks;
sp_mat I = speye(N,N);
vec u;
ElementDist P,Q;
for(uint i = 0; i < A; i++){
P = ElementDist(N,N);
u = actions.row(i).t();
cout << "\tAction [" << i << "]: u = " << u.t();
for(uint j = 0; j < num_samples; j++){
P += sim->transition_matrix(disc,u,include_oob);
}
P /= (double) num_samples;
E_blocks.push_back(I - gamma * P);
}
assert(A == E_blocks.size());
return E_blocks;
}
vector<sp_mat> build_hallway_blocks(const uint N,
const double p_stick,
const double gamma){
// Build the transition matrices for -1,0,1
vector<sp_mat> blocks;
// Left
sp_mat P_left = build_hallway_P(N,p_stick,-1);
blocks.push_back(speye(N,N) - gamma * P_left);
// Stay
blocks.push_back((1-gamma) * speye(N,N));
//Right
sp_mat P_right = build_hallway_P(N,p_stick,1);
blocks.push_back(speye(N,N) - gamma * P_right);
return blocks;
}
sp_mat build_hallway_P(const uint N,
const double p_stick,
const int action){
// Build the hallway transition matrix associated with
// and action
assert(action == -1 or action == 1);
sp_mat P = p_stick * speye(N,N)
+ spdiag((1-p_stick) * ones<vec>(N-1),action);
if(action < 0)
P(0,N-1) = (1 - p_stick);
else
P(N-1,0) = (1 - p_stick);
return P;
}
void newton_matrix(ode_workspace *odews)
{
// Create a Newton matrix from the given step gamma and Jacobian in W
cs *M, *eye;
if (odews->mdeclared)
{
cs_nfree(odews->N);
}
else
{
odews->mdeclared = 1;
}
eye = speye(odews->W->J->m);
M = cs_add(eye, odews->W->J, 1, -odews->dt);
cs_spfree(eye);
odews->N = cs_lu(M, odews->S, 1);
cs_spfree(M);
}
sp_mat Device1D::qCMatFunct(mat phin, mat phip, bool Equilibrum) {
if ( phin.n_elem != sumPoint || ( phip.n_elem != sumPoint ) )
std::cerr << "Error: input wrong phin/phip size!" << std::endl;
sp_mat cp=speye(sumPoint, sumPoint);
if (Equilibrum == true ) {
for ( int i = 0; i < bandGapArray.size(); i++) {
phip(i) = bandGapArray[i] - phin(i);
}
}
int k=0;
for ( int i=0; i < materialList.size(); i++) {
for ( int j=0; j < nyList[i]; j++) {
if (typeList[i] == Semiconductor) {
cp(k + 1, k + 1) = materialList[i].quantumCapa((double)phin(k), (double)phip(k)); //TODO: why here used to be k+1, then "cp(0, 0) = cp(1, 1);"
} else if (typeList[i]==Dielectric) {
};
k++;
};
};
return ( cp );
}
SolverResult KojimaSolver::solve(const LCP & lcp,
vec & x,
vec & y) const{
superlu_opts opts;
opts.equilibrate = true;
opts.permutation = superlu_opts::COLAMD;
opts.refine = superlu_opts::REF_SINGLE;
vec q = lcp.q;
sp_mat M = lcp.M + regularizer * speye(size(lcp.M));
uint N = q.n_elem;
assert(N == x.n_elem);
assert(N == y.n_elem);
// Figure out what is free (-inf,inf)
// and what is bound to be non-negative [0,inf)
bvec free_vars = lcp.free_vars;
uvec bound_idx = find(0 == free_vars);
uvec free_idx = find(1 == free_vars);
assert(N == bound_idx.n_elem + free_idx.n_elem);
uint NB = bound_idx.n_elem; // number of bound vars
uint NF = free_idx.n_elem; // number of free vars
/*
In what follows, the primal variables x are partitioned into
free variables and bound variables x = [f;b]'
Likewise, the dual variables are partitioned into y = [0,s]'
*/
/* The Newton system:
[M_ff M_fb 0][df] [M_f x + q_f]
[M_bf M_bb -I][db] + [M_b x + q_b - s]
[0 S B][dv] [u1 + VBe]
Where "M_ff" is the free-free block
Overwrite the S,B diagonals every iteration*/
// Split M matrix into blocks based on free and bound indicies
block_sp_mat M_part = sp_partition(M,free_idx,bound_idx);
sp_mat M_recon = block_mat(M_part);
assert(PRETTY_SMALL > norm(M_recon - M));
vec qf = q(free_idx);
vec qb = q(bound_idx);
vec b = x(bound_idx);
vec f = x(free_idx);
vec s = y(bound_idx);
// Build the Newton matrix
vector<vector<sp_mat>> block_G;
block_G.push_back(block_sp_vec{sp_mat(),sp_mat(),sp_mat(NB,NB)});
block_G.push_back(block_sp_vec{-M_part[0][0],-M_part[0][1],sp_mat()});
block_G.push_back(block_sp_vec{-M_part[1][0],-M_part[1][1],speye(NB,NB)});
// Start iteration
double mean_comp, steplen;
double sigma = initial_sigma;
uint iter;
for(iter = 0; iter < max_iter; iter++){
if(verbose or iter_verbose)
cout << "---Iteration " << iter << "---" << endl;
assert(all(0 == y(free_idx)));
// Mean complementarity
mean_comp = dot(b,s) / (double) NB;
if(mean_comp < comp_thresh)
break;
block_G[0][1] = spdiag(s);
block_G[0][2] = spdiag(b);
sp_mat G = block_mat(block_G);
assert(size(N + NB,N + NB) == size(G));
// Form RHS from residual and complementarity
vec h = vec(N + NB);
vec res_f = M_part[0][0]*f + M_part[0][1]*b + qf;
vec res_b = M_part[1][0]*f + M_part[1][1]*b + qb - s;
h.head(NB) = sigma * mean_comp - b % s;
h.subvec(NB,size(res_f)) = res_f;
h.tail(NB) = res_b;
//Archiver arch;
//arch.add_sp_mat("G",G);
//arch.add_vec("h",h);
//arch.write("test.sys");
// Solve and extract directions
vec dir = spsolve(G,h,"superlu",opts);
assert((N+NB) == dir.n_elem);
vec df = dir.head(NF);
vec db = dir.subvec(NF,N-1);
assert(NB == db.n_elem);
vec ds = dir.tail(NB);
vec dir_recon = join_vert(df,join_vert(db,ds));
assert(PRETTY_SMALL > norm(dir_recon-dir));
steplen = steplen_heuristic(b,s,db,ds,0.9);
sigma = sigma_heuristic(sigma,steplen);
f += steplen * df;
b += steplen * db;
s += steplen * ds;
if(verbose){
double res = norm(join_vert(res_f,res_b));
cout <<"\t Mean complementarity: " << mean_comp
<<"\n\t Residual norm: " << res
<<"\n\t |df|: " << norm(df)
<<"\n\t |db|: " << norm(db)
<<"\n\t |ds|: " << norm(ds)
<<"\n\t Step length: " << steplen
<<"\n\t Centering sigma: " << sigma << endl;
}
}
if(verbose){
cout << "Finished"
<<"\n\t Final mean complementarity: " << mean_comp << endl;
}
x(free_idx) = f;
x(bound_idx) = b;
y(free_idx).fill(0);
y(bound_idx) = s;
return SolverResult(x,y,iter);
}
SolverResult ProjectiveSolver::solve(const PLCP & plcp,
vec & x,
vec & y,
vec & w) const{
sp_mat P = plcp.P;
sp_mat U = plcp.U;
vec q = plcp.q;
uint N = P.n_rows;
uint K = P.n_cols;
assert(size(P.t()) == size(U));
bvec free_vars = plcp.free_vars;
uvec bound_idx = find(0 == free_vars);
uvec free_idx = find(1 == free_vars);
assert(N == bound_idx.n_elem + free_idx.n_elem);
uint NB = bound_idx.n_elem; // number of bound vars
uint NF = free_idx.n_elem; // number of free vars
assert(NF == accu(conv_to<uvec>::from(free_vars)));
assert(N == x.n_elem);
assert(N == y.n_elem);
assert(K == w.n_elem);
assert(ALMOST_ZERO > norm(y(free_idx)));
if(verbose)
cout << "Free variables: \t" << NF << endl
<< "Non-neg variables:\t" << NB << endl;
sp_mat J = join_vert(sp_mat(NF,NB),speye(NB,NB));
assert(size(N,NB) == size(J));
assert(PRETTY_SMALL > norm(y(free_idx)));
if(verbose)
cout << "Forming pre-computed products..." << endl;
mat PtP = mat(P.t() * P);
assert(size(K,K) == size(PtP));
mat PtPU = PtP*U;
assert(size(K,N) == size(PtPU));
mat Pt_PtPU = P.t() - PtPU;
assert(size(K,N) == size(Pt_PtPU));
mat PtPUP = PtPU * P;
assert(size(K,K) == size(PtPUP));
vec Ptq = P.t() * q;
if(verbose)
cout << "Done..." << endl;
double sigma = initial_sigma;
uint iter;
double mean_comp;
for(iter = 0; iter < max_iter; iter++){
if(verbose or iter_verbose)
cout << "---Iteration " << iter << "---" << endl;
// Mean complementarity
vec s = y(bound_idx);
vec b = x(bound_idx);
mean_comp = dot(s,b) / (double) NB;
if(mean_comp < comp_thresh)
break;
// Generate reduced Netwon system
mat C = s+b;
vec g = sigma * mean_comp - s % b;
assert(NB == g.n_elem);
// NB: A,G,and h have opposite sign from python version
mat A = Pt_PtPU * J * spdiag(1.0 / C);
assert(size(K,NB) == size(A));
mat G = PtPUP + (A * spdiag(s)) * J.t() * P;
assert(size(K,K) == size(G));
vec Ptr = P.t() * (J * s) - PtPU*x - Ptq;
vec h = Ptr + A*g;
assert(K == h.n_elem);
// Options don't make much difference
vec dw = arma::solve(G+1e-15*eye(K,K),h,
solve_opts::equilibrate);
assert(K == dw.n_elem);
// Recover dy
vec Pdw = P * dw;
vec JtPdw = J.t() * Pdw;
assert(NB == JtPdw.n_elem);
vec ds = (g - s % JtPdw) / C;
assert(NB == ds.n_elem);
// Recover dx
vec dx = (J * ds) + (Pdw);
assert(N == dx.n_elem);
double steplen = steplen_heuristic(x(bound_idx),s,dx(bound_idx),ds,0.9);
sigma = sigma_heuristic(sigma,steplen);
x += steplen * dx;
s += steplen * ds;
y(bound_idx) = s;
w += steplen * dw;
if(verbose){
cout <<"\tMean complementarity: " << mean_comp
<<"\n\tStep length: " << steplen
<<"\n\tCentering sigma: " << sigma << endl;
}
}
if(verbose){
cout << "Finished"
<<"\n\t Final mean complementarity: " << mean_comp << endl;
}
return SolverResult(x,y,iter);
}
PLCP augment_plcp(const PLCP & original,
vec & x,
vec & y,
vec & w,
double scale){
uint N = original.P.n_rows;
uint K = original.P.n_cols;
assert(size(K,N) == size(original.U));
sp_mat P = sp_mat(original.P);
double I_norm = norm(speye(K,K) - P.t() * P);
if(norm(I_norm) >= PRETTY_SMALL){
cerr << "Error: P does not look orthogonal ("
<< I_norm << ")..." << endl;
}
assert(I_norm < PRETTY_SMALL);
sp_mat U = sp_mat(original.U);
vec q = vec(original.q);
assert(all(q(find(1 == original.free_vars)) <= 0));
vec q_neg = min(zeros<vec>(N),q);
vec q_pos = max(zeros<vec>(N),q);
x = ones<vec>(N) - q_neg;
y = ones<vec>(N) + q_pos;
y(find(1 == original.free_vars)).fill(0);
assert(norm(q_pos(find(1 == original.free_vars))) < ALMOST_ZERO);
assert(N == x.n_elem);
assert(N == y.n_elem);
assert(all(x >= 0));
assert(all(y >= 0));
vec res = x - y + q;
w = spsolve(P.t()*P + 1e-15*speye(K,K),P.t()*(x - y + q));
vec w_res = P * w - res;
if(norm(w_res) >= PRETTY_SMALL){
cerr << "Error: Reduced vector w residual large ("
<< w_res << ")..." << endl;
}
assert(norm(w_res) < PRETTY_SMALL);
vec b = P.t()*x - U*x - w;
assert(K == b.n_elem);
P.resize(N+1,K+1);
U.resize(K+1,N+1);
q.resize(N+1);
P(N,K) = 1.0;
U(span(0,K-1),N) = b;
U(K,N) = scale;
q(N) = 0;
x.resize(N+1);
y.resize(N+1);
w.resize(K+1);
x(N) = 1;
y(N) = scale;
w(K) = 1.0 - scale;
bvec free_vars = bvec(N+1);
free_vars.head(N) = original.free_vars;
free_vars(N) = 0;
return PLCP(P,U,q,free_vars);
}
