minimize.js

/** 
 * from https://github.com/bijection/g9/blob/master/src/minimize.js
 * 
 * MIT License
 * 
 * Copyright (c) 2016
 * 
 * Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
 * 
 * The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
 * 
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
 *
 */

function norm2(x){
    return Math.sqrt(x.reduce((a,b)=>a+b*b,0))
}

function identity(n){
    var ret = Array(n)
    for (var i = 0; i < n; i++) {
        ret[i] = Array(n)
        for (var j = 0; j < n; j++) ret[i][j] = +(i == j) ;
    }
    return ret
}

function neg(x){
    return x.map(a=>-a)
}

function dot(a,b){
    if (typeof a[0] !== 'number'){
        return a.map(x=>dot(x,b))
    }
    return a.reduce((x,y,i) => x+y*b[i],0)
}

function sub(a,b){
    if(typeof a[0] !== 'number'){
        return a.map((c,i)=>sub(c,b[i]))
    }
    return a.map((c,i)=>c-b[i])
}

function add(a,b){
    if(typeof a[0] !== 'number'){
        return a.map((c,i)=>add(c,b[i]))
    }
    return a.map((c,i)=>c+b[i])
}

function div(a,b){
    return a.map(c=>c.map(d=>d/b))
}

function mul(a,b){
    if(typeof a[0] !== 'number'){
        return a.map(c=>mul(c,b))
    }
    return a.map(c=>c*b)
}

function ten(a,b){
    return a.map((c,i)=>mul(b,c))
}

function isZero(a){
    for (var i = 0; i < a.length; i++) {
        if(a[i]!== 0) return false
    }
    return true
}

// Adapted from the numeric.js gradient and uncmin functions
// Numeric Javascript
// Copyright (C) 2011 by Sébastien Loisel

// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:

// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.

// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

export function gradient(f,x) {
    var dim = x.length, f1 = f(x);
    if(isNaN(f1)) throw new Error('The gradient at ['+x.join(' ')+'] is NaN!');
    var {max, abs, min} = Math
    var tempX = x.slice(0), grad = Array(dim);
    for(var i=0; i<dim; i++) {
        var delta = max(1e-6*f1, 1e-8);
        for (var k = 0;; k++) {
            if(k == 20) throw new Error("Gradient failed at index "+i+" of ["+x.join(' ')+"]");
            tempX[i] = x[i]+delta;
            var f0 = f(tempX);
            tempX[i] = x[i]-delta;
            var f2 = f(tempX);
            tempX[i] = x[i];
            if(!(isNaN(f0) || isNaN(f2))) {
                grad[i] = (f0-f2)/(2*delta)
                var t0 = x[i]-delta;
                var t1 = x[i];
                var t2 = x[i]+delta;
                var d1 = (f0-f1)/delta;
                var d2 = (f1-f2)/delta;
                var err = min(max(abs(d1-grad[i]),abs(d2-grad[i]),abs(d1-d2)),delta);
                var normalize = max(abs(grad[i]),abs(f0),abs(f1),abs(f2),abs(t0),abs(t1),abs(t2),1e-8);
                if(err/normalize < 1e-3) break; //break if this index is done
            }
            delta /= 16
        }
    }
    return grad;
}

export function minimize(f, x0, end_on_line_search,tol=1e-8,maxit=1000) {


    tol = Math.max(tol,2e-16);
    var grad = a => gradient(f,a)

    x0 = x0.slice(0)
    var g0 = grad(x0)
    var f0 = f(x0)
    if(isNaN(f0)) throw new Error('minimize: f(x0) is a NaN!');
    var n = x0.length;
    var H1 = identity(n)
    
    for(var it = 0; it<maxit; it++) {
        if(!g0.every(isFinite)) { var msg = "Gradient has Infinity or NaN"; break; }
        var step = neg(dot(H1,g0));
        if(!step.every(isFinite)) { var msg = "Search direction has Infinity or NaN"; break; }
        var nstep = norm2(step);
        if(nstep < tol) { var msg="Newton step smaller than tol"; break; }
        var t = 1;
        var df0 = dot(g0,step);
        // line search
        var x1 = x0;
        var s;
        for(;it < maxit && t*nstep >= tol; it++) {
            s = mul(step,t);
            x1 = add(x0,s);
            var f1 = f(x1);
            if(!(f1-f0 >= 0.1*t*df0 || isNaN(f1))) break;
            t *= 0.5;
        }
        if(t*nstep < tol && end_on_line_search) {var msg = "Line search step size smaller than tol"; break; }
        if(it === maxit) { var msg = "maxit reached during line search"; break; }
        var g1 = grad(x1);
        var y = sub(g1,g0);
        var ys = dot(y,s);
        var Hy = dot(H1,y);
        H1 = sub(add(H1,mul(ten(s,s),(ys+dot(y,Hy))/(ys*ys))),div(add(ten(Hy,s),ten(s,Hy)),ys));
        x0 = x1;
        f0 = f1;
        g0 = g1;
    }

    return {solution: x0, f: f0, gradient: g0, invHessian: H1, iterations:it, message: msg};
}

export function findPhaseChange(f, known_true, known_false){
    while(Math.abs(known_true - known_false) > 1e-3){
        var mid = (known_true + known_false) / 2
        f(mid) ? known_true = mid : known_false = mid
    }
    return (known_true + known_false) / 2
}


// var x = [4,26,2,8]
// minimize(norm2, x).f *1e9
// 8.597