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linesearch.go
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linesearch.go
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// Copyright 2016 The Gosl Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package num
import (
"math"
"github.com/cpmech/gosl/chk"
"github.com/cpmech/gosl/fun"
)
// LineSearch finds a new point x along the direction dx, from x0, where the function
// has decreased sufficiently. The new function value is returned in fx
// Input:
// ffcn -- f(x) callback
// dx -- direction vector
// x0 -- initial x
// dφdx0 -- initial dφdx0 = fx * dfdx
// φ0 -- initial φ = 0.5 * dot(fx,fx)
// max_it -- max number of iterations
// dx_is_mdx -- whether dx is actually -dx ==> IMPORTANT: dx will then be changed dx := -dx
// Output:
// x -- updated x (along dx)
// fx -- updated f(x)
// φ0 -- updated φ = 0.5 * dot(fx,fx)
// dx -- changed to -dx if dx_is_mdx == true
// nFeval -- number of calls to f(x)
// Local constants:
// tol_gra_min -- tolerance to consider local minimum
// mul_dx_max -- multiplier to control maximum dx
// slope_max -- ~0 but < 0
// α -- Armijo coefficient
// ε -- machine epsilon
func LineSearch(x, fx []float64, ffcn fun.Vv, dx, x0, dφdx0 []float64, φ0 float64, max_it int, dx_is_mdx bool) (nFeval int, err error) {
// tolerances
tol_gra_min := 1e-12
mul_dx_max := 100.0
slope_max := -MACHEPS
slope_max = 0.0
// constants
α := 1e-4 // Armijo coefficient
ε := 1e-16 // machine epsilon
// scale dx if step is too big
n := len(x0)
var nrm_x0, nrm_dx float64
for i := 0; i < n; i++ {
if dx_is_mdx {
dx[i] = -dx[i]
}
nrm_x0 += x0[i] * x0[i]
nrm_dx += dx[i] * dx[i]
}
nrm_x0, nrm_dx = math.Sqrt(nrm_x0), math.Sqrt(nrm_dx)
nrm_dx_max := mul_dx_max * max(nrm_x0, float64(n))
if nrm_dx > nrm_dx_max {
for i := 0; i < n; i++ {
dx[i] *= nrm_dx_max / nrm_dx // scale if attempted step is to big
}
}
// descent slope and λ min
var slope, max_val, tmp float64
for i := 0; i < n; i++ {
slope += dφdx0[i] * dx[i]
tmp = math.Abs(dx[i]) / max(math.Abs(x0[i]), 1.0)
if tmp > max_val {
max_val = tmp
}
}
λ_min := ε / max_val
// check slope on the direction of dx
if slope > slope_max {
return nFeval, chk.Err(_linesearch_err1, slope)
}
// iterations
var λ, φ, λ2, φ2, gra, den, r1, r2, a, b, d float64
λ = 1.0 // always try full step first
var it int
for it = 0; it < max_it; it++ {
// update search
for i := 0; i < n; i++ {
x[i] = x0[i] + λ*dx[i]
}
err = ffcn(fx, x)
nFeval += 1
if err != nil {
return
}
// compute φ
φ = 0.0
for i := 0; i < n; i++ {
φ += fx[i] * fx[i]
}
φ *= 0.5
// dx is too small
if λ < λ_min {
// check for spurious convergence (local minimum)
gra = 0.0
den = max(φ, 0.5*float64(n))
for i := 0; i < n; i++ {
tmp = math.Abs(dφdx0[i]) * max(math.Abs(x[i]), 1.0) / den
if tmp > gra {
gra = tmp
}
}
if gra < tol_gra_min {
return nFeval, chk.Err(_linesearch_err2, λ, λ_min, gra)
}
return // converged
}
// converged? (sufficient function decrease)
if φ <= φ0+α*λ*slope {
return
}
// backtrack
if it == 0 {
tmp = -0.5 * slope / (φ - φ0 - slope)
} else {
r1 = φ - φ0 - λ*slope
r2 = φ2 - φ0 - λ2*slope
a = (r1/(λ*λ) - r2/(λ2*λ2)) / (λ - λ2)
b = (-λ2*r1/(λ*λ) + λ*r2/(λ2*λ2)) / (λ - λ2)
if math.Abs(a) < ε {
tmp = -0.5 * slope / b
} else {
d = b*b - 3.0*a*slope
if d < 0.0 {
tmp = 0.5 * λ
} else if b <= 0.0 {
tmp = (-b + math.Sqrt(d)) / (3.0 * a)
} else {
tmp = -slope / (b + math.Sqrt(d))
}
}
tmp = min(tmp, 0.5*λ) // make sure tmp is smaller than 0.5*λ
}
// save previous values
λ2, φ2 = λ, φ
// new λ
λ = max(tmp, 0.1*λ) // make sure λ is greater than 0.1*λ
}
// check convergence
if it == max_it {
return nFeval, chk.Err(_linesearch_err3, it+1)
}
return
}
// error messages
var (
_linesearch_err1 = "linesearch.go: LineSearch: slope must be negative (%g is invalid)"
_linesearch_err2 = "linesearch.go: LineSearch: local mininum reached? λ=%g, λ_min=%g, gra=%g"
_linesearch_err3 = "linesearch.go: LineSearch: failed to converge after %d iterations"
)