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jsketcher/web/app/math/noptim.js
Val Erastov 9aec197419 minimization magic
2015-02-19 01:12:17 -08:00

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optim = {};
//Added strong wolfe condition to numeric's uncmin
optim.bfgs_ = function(f,x0,tol,gradient,maxit,callback,options) {
var grad = numeric.gradient;
if(typeof options === "undefined") { options = {}; }
if(typeof tol === "undefined") { tol = 1e-8; }
if(typeof gradient === "undefined") { gradient = function(x) { return grad(f,x); }; }
if(typeof maxit === "undefined") maxit = 1000;
x0 = numeric.clone(x0);
var n = x0.length;
var f0 = f(x0),f1,df0;
if(isNaN(f0)) throw new Error('uncmin: f(x0) is a NaN!');
var max = Math.max, norm2 = numeric.norm2;
tol = max(tol,numeric.epsilon);
var step,g0,g1,H1 = options.Hinv || numeric.identity(n);
var dot = numeric.dot, inv = numeric.inv, sub = numeric.sub, add = numeric.add, ten = numeric.tensor, div = numeric.div, mul = numeric.mul;
var all = numeric.all, isfinite = numeric.isFinite, neg = numeric.neg;
var it=0,i,s,x1,y,Hy,Hs,ys,i0,t,nstep,t1,t2;
var msg = "";
g0 = gradient(x0);
while(it<maxit) {
if(typeof callback === "function") { if(callback(it,x0,f0,g0,H1)) { msg = "Callback returned true"; break; } }
if(!all(isfinite(g0))) { msg = "Gradient has Infinity or NaN"; break; }
step = neg(dot(H1,g0));
if(!all(isfinite(step))) { msg = "Search direction has Infinity or NaN"; break; }
nstep = norm2(step);
if(nstep < tol) { msg="Newton step smaller than tol"; break; }
t = 1;
df0 = dot(g0,step);
// line search
x1 = x0;
var tL = 0;
var tR = 100;
while(it < maxit) {
if(t*nstep < tol) { break; }
s = mul(step,t);
x1 = add(x0,s);
f1 = f(x1);
//Nocadel, 3.7(a,b)
if(f1-f0 >= 0.1*t*df0 || isNaN(f1)) {
tR = t;
t = (tL + tR) * 0.5;
++it;
continue;
} else {
var slope = dot(gradient(x1), step);
if (slope <= 0.9 * Math.abs(df0)){
break;
}else if ( slope >= 0.9 * df0) {
tR = t;
t = (tL+ tR) * 0.5;
continue;
}else{
tL = t;
t = (tL+ tR)*0.5;
continue;
}
}
break;
}
if(t*nstep < tol) { msg = "Line search step size smaller than tol"; break; }
if(it === maxit) { msg = "maxit reached during line search"; break; }
g1 = gradient(x1);
y = sub(g1,g0);
ys = dot(y,s);
Hy = dot(H1,y);
// BFGS update on H1
H1 = sub(add(H1,
mul(
(ys+dot(y,Hy))/(ys*ys),
ten(s,s) )),
div(add(ten(Hy,s),ten(s,Hy)),ys));
x0 = x1;
f0 = f1;
g0 = g1;
++it;
}
return {solution: x0, f: f0, gradient: g0, invHessian: H1, iterations:it, message: msg};
};
optim.bfgs = function(f,x0,tol,gradient,maxit,callback,options) {
var grad = numeric.gradient;
if(typeof options === "undefined") { options = {}; }
if(typeof tol === "undefined") { tol = 1e-8; }
if(typeof gradient === "undefined") { gradient = function(x) { return grad(f,x); }; }
if(typeof maxit === "undefined") maxit = 1000;
x0 = numeric.clone(x0);
var n = x0.length;
var f0 = f(x0),f1,df0;
if(isNaN(f0)) throw new Error('uncmin: f(x0) is a NaN!');
var max = Math.max, norm2 = numeric.norm2;
tol = max(tol,numeric.epsilon);
var step,g0,g1,H1 = options.Hinv || numeric.identity(n);
var dot = numeric.dot, inv = numeric.inv, sub = numeric.sub, add = numeric.add, ten = numeric.tensor, div = numeric.div, mul = numeric.mul;
var all = numeric.all, isfinite = numeric.isFinite, neg = numeric.neg;
var it=0,i,s,x1,y,Hy,Hs,ys,i0,t,nstep,t1,t2;
var msg = "";
g0 = gradient(x0);
while(it<maxit) {
if(typeof callback === "function") { if(callback(it,x0,f0,g0,H1)) { msg = "Callback returned true"; break; } }
if(!all(isfinite(g0))) { msg = "Gradient has Infinity or NaN"; break; }
step = neg(dot(H1,g0));
if(!all(isfinite(step))) { msg = "Search direction has Infinity or NaN"; break; }
nstep = norm2(step);
if(nstep < tol) { msg="Newton step smaller than tol"; break; }
df0 = dot(g0,step);
// line search
t1 = 0.0;
f1 = f0;
t2 = 1.0;
s = mul(step,t2);
x1 = add(x0,s);
var f2 = f(x1);
t3 = 2.0;
s = mul(step,t3);
x1 = add(x0,s);
var f3 = f(x1);
var tMax = 1e23;
while( (f2 > f1 || f2 > f3) && it < maxit) {
if(t*nstep < tol) { break; }
if (f2 > f1) {
//If f2 is greater than f1 then we shorten alpha2 and alpha3 closer to f1
//Effectively both are shortened by a factor of two.
t3 = t2;
f3 = f2;
t2 = t2 / 2;
s = mul(step,t2);
x1 = add(x0,s);
f2 = f(x1);
}
else if (f2 > f3) {
if (t3 >= tMax)
break;
//If f2 is greater than f3 then we increase alpha2 and alpha3 away from f1
//Effectively both are lengthened by a factor of two.
t2 = t3;
f2 = f3;
t3 = t3 * 2;
s = mul(step,t3);
x1 = add(x0,s);
f3 = f(x1);
}
it ++;
}
//Get the alpha for the minimum f of the quadratic approximation
var ts = t2 + ((t2-t1)*(f1-f3))/(3*(f1-2*f2+f3));
//Guarantee that the new alphaStar is within the bracket
if (ts >= t3 || ts <= t1)
ts = t2;
if (ts > tMax)
ts = tMax;
if (ts != ts)
ts = 0.;
//Take a final step to alphaStar
s = mul(step,ts);
x1 = add(x0,s);
f1 = f(x1);
if(t*nstep < tol) { msg = "Line search step size smaller than tol"; break; }
if(it === maxit) { msg = "maxit reached during line search"; break; }
g1 = gradient(x1);
y = sub(g1,g0);
ys = dot(y,s);
Hy = dot(H1,y);
// BFGS update on H1
H1 = sub(add(H1,
mul(
(ys+dot(y,Hy))/(ys*ys),
ten(s,s) )),
div(add(ten(Hy,s),ten(s,Hy)),ys));
x0 = x1;
f0 = f1;
g0 = g1;
++it;
}
return {solution: x0, f: f0, gradient: g0, invHessian: H1, iterations:it, message: msg};
};
optim.bfgs_updater = function(gradient, x0) {
var n = x0.length;
var max = Math.max, norm2 = numeric.norm2;
var g0,g1,H1 = numeric.identity(n);
var dot = numeric.dot, inv = numeric.inv, sub = numeric.sub, add = numeric.add, ten = numeric.tensor, div = numeric.div, mul = numeric.mul;
var all = numeric.all, isfinite = numeric.isFinite, neg = numeric.neg;
var y,Hy,Hs,ys;
var msg = "";
var g0 = gradient(x0);
function step() {
return neg(dot(H1,g0));
}
function update(x, real_step) {
var s = real_step;
g1 = gradient(x);
y = sub(g1,g0);
ys = dot(y,s);
Hy = dot(H1,y);
// BFGS update on H1
H1 = sub(add(H1,
mul(
(ys+dot(y,Hy))/(ys*ys),
ten(s,s) )),
div(add(ten(Hy,s),ten(s,Hy)),ys));
g0 = g1;
}
return {step:step, update:update};
};
optim.inv = function inv(x) {
var s = numeric.dim(x), abs = Math.abs, m = s[0], n = s[1];
var A = numeric.clone(x), Ai, Aj;
var I = numeric.identity(m), Ii, Ij;
var i,j,k,x;
for(j=0;j<n;++j) {
var i0 = -1;
var v0 = -1;
for(i=j;i!==m;++i) { k = abs(A[i][j]); if(k>v0) { i0 = i; v0 = k; } }
Aj = A[i0]; A[i0] = A[j]; A[j] = Aj;
Ij = I[i0]; I[i0] = I[j]; I[j] = Ij;
x = Aj[j];
if (x === 0) {
//console.log("CAN' INVERSE MATRIX");
x = 1e-32
}
for(k=j;k!==n;++k) Aj[k] /= x;
for(k=n-1;k!==-1;--k) Ij[k] /= x;
for(i=m-1;i!==-1;--i) {
if(i!==j) {
Ai = A[i];
Ii = I[i];
x = Ai[j];
for(k=j+1;k!==n;++k) Ai[k] -= Aj[k]*x;
for(k=n-1;k>0;--k) { Ii[k] -= Ij[k]*x; --k; Ii[k] -= Ij[k]*x; }
if(k===0) Ii[0] -= Ij[0]*x;
}
}
}
return I;
}
// this is Gauss-Newton least square algorithm with trust region(dog leg) control/
optim.dog_leg = function(subsys) {
var tolg=1e-80, tolx=1e-80, tolf=1e-10;
var xsize = subsys.params.length;
var csize = subsys.constraints.length;
if (xsize == 0)
return 'Success';
var vec = TCAD.math._arr;
var mx = TCAD.math._matrix;
var n = numeric;
var x = vec(xsize);
var x_new = vec(xsize);
var fx = vec(csize);
var fx_new = vec(csize);
var Jx = mx(csize, xsize);
var Jx_new = mx(csize, xsize);
var g = vec(xsize);
var h_sd = vec(xsize);
var h_gn = vec(xsize);
var h_dl = vec(xsize);
var r0 = vec(csize);
var err;
subsys.fillParams(x);
// subsys.setParams(vec(xsize));
// subsys.calcResidual(r0);
subsys.setParams(x);
err = subsys.calcResidual(fx);
subsys.fillJacobian(Jx);
function lls(A, b) {
var At = n.transpose(A);
var J = n.dot(At, A);
var r = n.dot(At, b);;
return nocadel_10_18(J, r);
}
function nocadel_10_18(A, r) {
//10.18
var usv = n.svd(A);
var x = vec(xsize);
for (var i = 0; i < usv.S.length; ++i) {
var u = usv.U[i];
var v = usv.V[i];
var b = usv.S[i];
if (b != 0) {
var _t = n.mul(v, n.dot(u, r) / b);
x = n.add(x, _t);
}
}
return x;
}
function lsolve(A, b) {
// if (csize < xsize) {
// var At = n.transpose(A);
// var J = n.dot(At, A);
// var r = n.dot(At, b);;
// return n.solve(J, r);
// } else {
// return n.solve(A, b);
// }
var At = n.transpose(A);
var res = n.dot(n.dot(At, optim.inv(n.dot(A, At)) ), b);
return res;
}
function lusolve(A, b) {
var At = n.transpose(A);
var A = n.dot(At, A);
var b = n.dot(At, b);
return n.solve(A, b, true);
}
g = n.dot(n.transpose(Jx), n.mul(fx, -1));
// get the infinity norm fx_inf and g_inf
var g_inf = n.norminf(g);
var fx_inf = n.norminf(fx);
var maxIterNumber = 100 * xsize;
var divergingLim = 1e6*err + 1e12;
var delta=0.1;
var alpha=0.;
var nu=2.;
var iter=0, stop=0, reduce=0;
while (stop === 0) {
// check if finished
if (fx_inf <= tolf || err <= tolf) // Success
stop = 1;
else if (g_inf <= tolg)
stop = 2;
else if (delta <= tolx*(tolx + n.norm2(x)))
stop = 2;
else if (iter >= maxIterNumber)
stop = 4;
else if (err > divergingLim || err != err) { // check for diverging and NaN
stop = 6;
}
else {
// get the steepest descent direction
alpha = n.norm2Squared(g)/n.norm2Squared(n.dot(Jx, g));
h_sd = n.mul(g, alpha);
// get the gauss-newton step
//h_gn = n.solve(Jx, n.mul(fx, -1));
h_gn = lsolve(Jx, n.mul(fx, -1));
//LU-Decomposition
// h_gn = lusolve(Jx, n.mul(fx, -1));
//Conjugate gradient method
//h_gn = optim.cg(Jx, h_gn, n.mul(fx, -1), 1e-8, maxIterNumber);
//solve linear problem using svd formula to get the gauss-newton step
//h_gn = lls(Jx, n.mul(fx, -1));
var rel_error = n.norm2(n.add(n.dot(Jx, h_gn), fx)) / n.norm2(fx);
if (rel_error > 1e15)
break;
// compute the dogleg step
if (n.norm2(h_gn) < delta) {
h_dl = n.clone(h_gn);
if (n.norm2(h_dl) <= tolx*(tolx + n.norm2(x))) {
stop = 5;
break;
}
}
else if (alpha*n.norm2(g) >= delta) {
h_dl = n.mul( h_sd, delta/(alpha*n.norm2(g)));
}
else {
//compute beta
var beta = 0;
var b = n.sub(h_gn, h_sd);
var bb = Math.abs(n.dot(b, b));
var gb = Math.abs(n.dot(h_sd,b));
var c = (delta + n.norm2(h_sd))*(delta - n.norm2(h_sd));
if (gb > 0)
beta = c / (gb + Math.sqrt(gb * gb + c * bb));
else
beta = (Math.sqrt(gb * gb + c * bb) - gb)/bb;
// and update h_dl and dL with beta
h_dl = n.add(h_sd, n.mul(beta,b));
}
}
// see if we are already finished
if (stop)
break;
// get the new values
var err_new;
x_new = n.add(x, h_dl);
subsys.setParams(x_new);
err_new = subsys.calcResidual(fx_new);
subsys.fillJacobian(Jx_new);
// calculate the linear model and the update ratio
var dL = err - 0.5* n.norm2Squared(n.add(fx, n.dot(Jx, h_dl)));
var dF = err - err_new;
var rho = dL/dF;
if (dF > 0 && dL > 0) {
x = n.clone(x_new);
Jx = n.clone(Jx_new);
fx = n.clone(fx_new);
err = err_new;
g = n.dot(n.transpose(Jx), n.mul(fx, -1));
// get infinity norms
g_inf = n.norminf(g);
fx_inf = n.norminf(fx);
}
else
rho = -1;
// update delta
if (Math.abs(rho-1.) < 0.2 && n.norm2(h_dl) > delta/3. && reduce <= 0) {
delta = 3*delta;
nu = 2;
reduce = 0;
}
else if (rho < 0.25) {
delta = delta/nu;
nu = 2*nu;
reduce = 2;
}
else
reduce--;
// count this iteration and start again
iter++;
}
return (stop == 1) ? 'Success' : 'Failed';
};
optim.cg = function(A, x, b, tol, maxIt) {
var _ = numeric;
var tr = _.transpose;
var At = tr(A);
if (A.length != A[0].length) {
var A = _.dot(At, A);
var b = _.dot(At, b);
At = tr(A);
}
var r = _.sub(_.dot(A, x), b);
var p = _.mul(r, -1);
var rr = _.dotVV(r, r);
var a;
var _rr;
var beta;
for (var i = 0; i < maxIt; ++i) {
if (_.norm2(r) <= tol) break;
var Axp =_.dot(A, p);
a = rr / _.dotVV(Axp, p);
x = _.add(x, _.mul(p, a));
r = _.add(r, _.mul(Axp, a));
_rr = rr;
rr = _.dotVV(r, r);
beta = rr / _rr;
p = _.add(_.mul(r, -1), _.mul(p, beta));
}
// console.log("liner problem solved in " + i);
return x;
};
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