func (Q *Quadratic) Val(x mat.Vec) float64 {
val := 0.0
Q.temp.Apply(Q.A, x)
val += mat.Dot(x, Q.temp)
val += mat.Dot(x, Q.B)
val += Q.C
return val
}
func (Q *Quadratic) ValGrad(x, g mat.Vec) float64 {
At := Q.A.TrView()
Q.temp.Apply(Q.A, x)
g.Apply(At, x)
g.Add(g, Q.temp)
g.Add(g, Q.B)
val := 0.0
val += mat.Dot(x, Q.temp)
val += mat.Dot(x, Q.B)
val += Q.C
return val
}
func TestExampleModel(t *testing.T) {
//badly conditioned Hessian leads to zig-zagging of the steepest descent
//algorithm
condNo := 100.0
optSol := mat.Vec{1, 2}
A := mat.NewFromArray([]float64{condNo, 0, 0, 1}, true, 2, 2)
b := mat.Vec{-2 * optSol[0] * condNo, -2 * optSol[1]}
c := -0.5 * mat.Dot(b, optSol)
//define objective function
fun := opt.NewQuadratic(A, b, c)
//set inital solution estimate
sol := NewSolution(mat.NewVec(2))
//set termination parameters
p := NewParams()
p.IterMax = 5
//Use steepest descent solver to solve the model
result := NewSteepestDescent().Solve(fun, sol, p, NewDisplay(1))
fmt.Println("x =", result.X)
//should be [1,2], but because of the bad conditioning we made little
//progress in the second dimension
//Use a BFGS solver to refine the result:
result = NewLBFGS().Solve(fun, result.Solution, p, NewDisplay(1))
fmt.Println("x =", result.X)
}
func (lf *LineFuncDeriv) ValDeriv(x float64) (float64, float64) {
lf.xTemp.Copy(lf.x)
lf.xTemp.Axpy(x, lf.d)
val := lf.f.ValGrad(lf.xTemp, lf.g)
return val, mat.Dot(lf.d, lf.g)
}
func (sol LBFGS) Solve(o Grad, in *Solution, p *Params, u ...Updater) *Result {
r := NewResult(in)
obj := ObjGradWrapper{r: r, o: o}
r.initGrad(obj)
h := newHelper(r.Solution, u)
stepSize := 1.0
gLin := 0.0
n := len(r.X)
S := make([]mat.Vec, sol.Mem)
Y := make([]mat.Vec, sol.Mem)
for i := 0; i < sol.Mem; i++ {
S[i] = mat.NewVec(n)
Y[i] = mat.NewVec(n)
}
d := mat.NewVec(n)
xOld := mat.NewVec(n)
gOld := mat.NewVec(n)
sNew := mat.NewVec(n)
yNew := mat.NewVec(n)
alphas := mat.NewVec(sol.Mem)
betas := mat.NewVec(sol.Mem)
rhos := mat.NewVec(sol.Mem)
lineFunc := NewLineFuncDeriv(obj, r.X, d)
lsInit := uni.NewSolution()
lsParams := uni.NewParams()
for ; r.Status == NotTerminated; h.update(r, p) {
d.Copy(r.GradX)
if r.Iter > 0 {
yNew.Sub(r.GradX, gOld)
sNew.Sub(r.X, xOld)
temp := S[len(S)-1]
copy(S[1:], S)
S[0] = temp
S[0].Copy(sNew)
temp = Y[len(S)-1]
copy(Y[1:], Y)
Y[0] = temp
Y[0].Copy(yNew)
copy(rhos[1:], rhos)
rhos[0] = 1 / mat.Dot(sNew, yNew)
for i := 0; i < sol.Mem; i++ {
alphas[i] = rhos[i] * mat.Dot(S[i], d)
d.Axpy(-alphas[i], Y[i])
}
for i := sol.Mem - 1; i >= 0; i-- {
betas[i] = rhos[i] * mat.Dot(Y[i], d)
d.Axpy(alphas[i]-betas[i], S[i])
}
}
d.Scal(-1)
gLin = mat.Dot(d, r.GradX)
lsInit.SetX(stepSize)
lsInit.SetLB(0, r.ObjX, gLin)
lsRes := sol.LineSearch.Solve(lineFunc, lsInit, lsParams)
if lsRes.Status < 0 {
fmt.Println("Linesearch:", lsRes.Status)
d.Copy(r.GradX)
d.Scal(-1)
lsInit.SetLB(0, r.ObjX, -r.GradX.Nrm2Sq())
lsRes = sol.LineSearch.Solve(lineFunc, lsInit, lsParams)
if lsRes.Status < 0 {
fmt.Println("Linesearch:", lsRes.Status)
r.Status = Status(lsRes.Status)
break
}
}
stepSize, r.ObjX = lsRes.X, lsRes.ObjX
xOld.Copy(r.X)
gOld.Copy(r.GradX)
r.X.Axpy(stepSize, d)
obj.ValGrad(r.X, r.GradX)
}
return r
}
func (sol *Rosenbrock) Solve(o Function, in *Solution, p *Params, u ...Updater) *Result {
r := NewResult(in)
obj := ObjWrapper{r: r, o: o}
r.init(obj)
h := newHelper(r.Solution, u)
eps := 1.0
n := len(r.X)
d := make([]mat.Vec, n)
for i := range d {
d[i] = mat.NewVec(n)
d[i][i] = 1
}
lambda := make([]float64, n)
lf := make([]*LineFunc, n)
for i := range lf {
lf[i] = NewLineFunc(obj, r.X, d[i])
}
lsInit := uni.NewSolution()
lsParams := uni.NewParams()
lsParams.XTolAbs = p.XTolAbs
lsParams.XTolRel = p.XTolRel
lsParams.FunTolAbs = 0
lsParams.FunTolRel = 0
for ; r.Status == NotTerminated; h.update(r, p) {
//Search in all directions
for i := range d {
lf[i].Dir = 1
valNeg := 0.0
valPos := lf[i].Val(eps)
if valPos >= r.ObjX {
lf[i].Dir = -1
valNeg = lf[i].Val(eps)
if valNeg >= r.ObjX {
eps *= 0.5
lf[i].Dir = 1
lsInit.SetLB(-eps)
lsInit.SetUB(eps)
lsInit.SetX(0)
} else {
lsInit.SetUB()
lsInit.SetLB()
lsInit.SetX(eps)
}
} else {
lsInit.SetUB()
lsInit.SetLB()
lsInit.SetX(eps)
}
lsRes := sol.LineSearch.Solve(lf[i], lsInit, lsParams)
lambda[i] = lf[i].Dir * lsRes.X
r.X.Axpy(lambda[i], d[i])
r.ObjX = lsRes.ObjX
}
//Find new directions
for i := range d {
if math.Abs(lambda[i]) > p.XTolAbs {
d[i].Scal(lambda[i])
for j := i + 1; j < n; j++ {
d[i].Axpy(lambda[j], d[j])
}
}
}
//Gram-Schmidt, TODO:use QR factorization
for i := range d {
d[i].Scal(1 / d[i].Nrm2())
for j := i + 1; j < n; j++ {
d[j].Axpy(-mat.Dot(d[i], d[j]), d[i])
}
}
}
return r
}
//Predictor-Corrector Interior Point implementation
func (sol *PredCorr) Solve(prob *Problem, p *Params, u ...Updater) *Result {
res := NewResult(prob)
h := newHelper(u)
A := prob.A
m, n := A.Dims()
At := A.TrView()
var mu, sigma float64
res.X.AddSc(1)
res.S.AddSc(1)
res.Rd = mat.NewVec(n)
res.Rp = mat.NewVec(m)
res.Rs = mat.NewVec(n)
x := res.X
s := res.S
y := res.Y
dx := mat.NewVec(n)
ds := mat.NewVec(n)
dy := mat.NewVec(m)
dxAff := mat.NewVec(n)
dsAff := mat.NewVec(n)
dyAff := mat.NewVec(m)
dxCC := mat.NewVec(n)
dsCC := mat.NewVec(n)
dyCC := mat.NewVec(m)
xdivs := mat.NewVec(n)
temp := mat.New(m, n)
lhs := mat.New(m, m)
rhs := mat.NewVec(m)
soli := mat.NewVec(m)
triU := mat.New(m, m)
triUt := triU.TrView()
nTemp1 := mat.NewVec(n)
nTemp2 := mat.NewVec(n)
alpha := 0.0
for {
res.Rd.Sub(prob.C, res.S)
res.Rd.AddMul(At, res.Y, -1)
res.Rp.Apply(A, res.X)
res.Rp.Sub(prob.B, res.Rp)
res.Rs.Mul(res.X, res.S)
res.Rs.Neg(res.Rs)
if h.update(res, p); res.Status != 0 {
break
}
if checkKKT(res, p); res.Status != 0 {
break
}
mu = res.Rs.Asum() / float64(n)
//determining left hand side
temp.ScalCols(A, xdivs.Div(x, s))
lhs.Mul(temp, At)
//factorization
lhs.Chol(triU)
//right hand side
nTemp1.Add(res.Rd, s)
nTemp1.Mul(nTemp1, xdivs)
rhs.Apply(A, nTemp1)
rhs.Add(rhs, res.Rp)
//solving for dyAff
soli.Trsv(triUt, rhs)
dyAff.Trsv(triU, soli)
//calculating other steps (dxAff, dsAff)
nTemp1.Apply(At, dyAff)
dxAff.Sub(nTemp1, res.Rd)
dxAff.Sub(dxAff, s)
dxAff.Mul(dxAff, xdivs)
dsAff.Div(dxAff, xdivs)
dsAff.Add(dsAff, s)
dsAff.Neg(dsAff)
//determining step size
alpha = 1.0
for i := range dxAff {
if dxAff[i] < 0 {
alph := -x[i] / dxAff[i]
if alph < alpha {
alpha = alph
}
}
}
for i := range dsAff {
if dsAff[i] < 0 {
alph := -s[i] / dsAff[i]
if alph < alpha {
alpha = alph
}
}
}
alpha *= 0.99995
//calculating duality gap measure for affine case
nTemp1.Copy(x)
nTemp1.Axpy(alpha, dxAff)
nTemp2.Copy(s)
nTemp2.Axpy(alpha, dsAff)
mu_aff := mat.Dot(nTemp1, nTemp2) / float64(n)
//centering parameter
sigma = math.Pow(mu_aff/mu, 3)
//right hand side for predictor corrector step
res.Rs.Mul(dxAff, dsAff)
res.Rs.Neg(res.Rs)
res.Rs.AddSc(sigma * mu_aff)
nTemp1.Div(res.Rs, s)
nTemp1.Neg(nTemp1)
rhs.Apply(A, nTemp1)
//solving for dyCC
soli.Trsv(triUt, rhs)
dyCC.Trsv(triU, soli)
//calculating other steps (dxAff, dsAff)
nTemp1.Apply(At, dyCC)
dxCC.Mul(nTemp1, x)
dxCC.Add(res.Rs, dxCC)
dxCC.Div(dxCC, s)
dsCC.Mul(dxCC, s)
dsCC.Sub(res.Rs, dsCC)
dsCC.Div(dsCC, x)
dx.Add(dxAff, dxCC)
dy.Add(dyAff, dyCC)
ds.Add(dsAff, dsCC)
alpha = 1
for i := range dx {
if dx[i] < 0 {
alph := -x[i] / dx[i]
if alph < alpha {
alpha = alph
}
}
}
for i := range ds {
if ds[i] < 0 {
alph := -s[i] / ds[i]
if alph < alpha {
alpha = alph
}
}
}
alpha *= 0.99995
x.Axpy(alpha, dx)
y.Axpy(alpha, dy)
s.Axpy(alpha, ds)
}
return res
}