Beispiel #1
0
// Computes analytic center of A*x <= b with A m by n of rank n.
// We assume that b > 0 and the feasible set is bounded.
func Acent(A, b *matrix.FloatMatrix, niters int) (*matrix.FloatMatrix, []float64) {

	if niters <= 0 {
		niters = MAXITERS
	}
	ntdecrs := make([]float64, 0, niters)

	if A.Rows() != b.Rows() {
		return nil, nil
	}

	m, n := A.Size()
	x := matrix.FloatZeros(n, 1)
	H := matrix.FloatZeros(n, n)
	// Helper m*n matrix
	Dmn := matrix.FloatZeros(m, n)

	for i := 0; i < niters; i++ {

		// Gradient is g = A^T * (1.0/(b - A*x)). d = 1.0/(b - A*x)
		// d is m*1 matrix, g is n*1 matrix
		d := matrix.Minus(b, matrix.Times(A, x)).Inv()
		g := matrix.Times(A.Transpose(), d)

		// Hessian is H = A^T * diag(1./(b-A*x))^2 * A.
		// in the original python code expression d[:,n*[0]] creates
		// a m*n matrix where each column is copy of column 0.
		// We do it here manually.
		for i := 0; i < n; i++ {
			Dmn.SetColumn(i, d)
		}

		// Function mul creates element wise product of matrices.
		Asc := matrix.Mul(Dmn, A)
		blas.SyrkFloat(Asc, H, 1.0, 0.0, linalg.OptTrans)

		// Newton step is v = H^-1 * g.
		v := g.Copy().Scale(-1.0)
		lapack.PosvFloat(H, v)

		// Directional derivative and Newton decrement.
		lam := blas.DotFloat(g, v)
		ntdecrs = append(ntdecrs, math.Sqrt(-lam))
		if ntdecrs[len(ntdecrs)-1] < TOL {
			fmt.Printf("last Newton decrement < TOL(%v)\n", TOL)
			return x, ntdecrs
		}

		// Backtracking line search.
		// y = d .* A*v
		y := d.Mul(A.Times(v))
		step := 1.0
		for 1-step*y.Max() < 0 {
			step *= BETA
		}

	search:
		for {
			// t = -step*y
			t := y.Copy().Scale(-step)
			// t = (1 + t) [e.g. t = 1 - step*y]
			t.Add(1.0)

			// ts = sum(log(1-step*y))
			ts := t.Log().Sum()
			if -ts < ALPHA*step*lam {
				break search
			}
			step *= BETA
		}
		v.Scale(step)
		x = x.Plus(v)
	}
	// no solution !!
	fmt.Printf("Iteration %d exhausted\n", niters)
	return x, ntdecrs
}
Beispiel #2
0
func main() {
	m := 6
	Vdata := [][]float64{
		[]float64{1.0, -1.0, -2.0, -2.0, 0.0, 1.5, 1.0},
		[]float64{1.0, 2.0, 1.0, -1.0, -2.0, -1.0, 1.0}}

	V := matrix.FloatMatrixFromTable(Vdata, matrix.RowOrder)

	// V[1, :m] - V[1,1:]
	a0 := matrix.Minus(V.GetSubMatrix(1, 0, 1, m), V.GetSubMatrix(1, 1, 1))
	// V[0, :m] - V[0,1:]
	a1 := matrix.Minus(V.GetSubMatrix(0, 0, 1, m), V.GetSubMatrix(0, 1, 1))
	A0, _ := matrix.FloatMatrixStacked(matrix.StackDown, a0.Scale(-1.0), a1)
	A0 = A0.Transpose()
	b0 := matrix.Mul(A0, V.GetSubMatrix(0, 0, 2, m).Transpose())
	b0 = matrix.Times(b0, matrix.FloatWithValue(2, 1, 1.0))

	A := make([]*matrix.FloatMatrix, 0)
	b := make([]*matrix.FloatMatrix, 0)
	A = append(A, A0)
	b = append(b, b0)

	// List of symbols
	C := make([]*matrix.FloatMatrix, 0)
	C = append(C, matrix.FloatZeros(2, 1))
	var row *matrix.FloatMatrix = nil
	for k := 0; k < m; k++ {
		row = A0.GetRow(k, row)
		nrm := blas.Nrm2Float(row)
		row.Scale(2.0 * b0.GetIndex(k) / (nrm * nrm))
		C = append(C, row.Transpose())
	}

	// Voronoi set around C[1]
	A1 := matrix.FloatZeros(3, 2)
	A1.SetSubMatrix(0, 0, A0.GetSubMatrix(0, 0, 1).Scale(-1.0))
	A1.SetSubMatrix(1, 0, matrix.Minus(C[m], C[1]).Transpose())
	A1.SetSubMatrix(2, 0, matrix.Minus(C[2], C[1]).Transpose())
	b1 := matrix.FloatZeros(3, 1)
	b1.SetIndex(0, -b0.GetIndex(0))
	v := matrix.Times(A1.GetRow(1, nil), matrix.Plus(C[m], C[1])).Float() * 0.5
	b1.SetIndex(1, v)
	v = matrix.Times(A1.GetRow(2, nil), matrix.Plus(C[2], C[1])).Float() * 0.5
	b1.SetIndex(2, v)
	A = append(A, A1)
	b = append(b, b1)

	// Voronoi set around C[2] ... C[5]
	for k := 2; k < 6; k++ {
		A1 = matrix.FloatZeros(3, 2)
		A1.SetSubMatrix(0, 0, A0.GetSubMatrix(k-1, 0, 1).Scale(-1.0))
		A1.SetSubMatrix(1, 0, matrix.Minus(C[k-1], C[k]).Transpose())
		A1.SetSubMatrix(2, 0, matrix.Minus(C[k+1], C[k]).Transpose())
		b1 = matrix.FloatZeros(3, 1)
		b1.SetIndex(0, -b0.GetIndex(k-1))
		v := matrix.Times(A1.GetRow(1, nil), matrix.Plus(C[k-1], C[k])).Float() * 0.5
		b1.SetIndex(1, v)
		v = matrix.Times(A1.GetRow(2, nil), matrix.Plus(C[k+1], C[k])).Float() * 0.5
		b1.SetIndex(2, v)
		A = append(A, A1)
		b = append(b, b1)
	}

	// Voronoi set around C[6]
	A1 = matrix.FloatZeros(3, 2)
	A1.SetSubMatrix(0, 0, A0.GetSubMatrix(5, 0, 1).Scale(-1.0))
	A1.SetSubMatrix(1, 0, matrix.Minus(C[1], C[6]).Transpose())
	A1.SetSubMatrix(2, 0, matrix.Minus(C[5], C[6]).Transpose())
	b1 = matrix.FloatZeros(3, 1)
	b1.SetIndex(0, -b0.GetIndex(5))
	v = matrix.Times(A1.GetRow(1, nil), matrix.Plus(C[1], C[6])).Float() * 0.5
	b1.SetIndex(1, v)
	v = matrix.Times(A1.GetRow(2, nil), matrix.Plus(C[5], C[6])).Float() * 0.5
	b1.SetIndex(2, v)

	A = append(A, A1)
	b = append(b, b1)

	P := matrix.FloatIdentity(2)
	q := matrix.FloatZeros(2, 1)
	solopts := &cvx.SolverOptions{ShowProgress: false, MaxIter: 30}
	ovals := make([]float64, 0)
	for k := 1; k < 7; k++ {
		sol, err := cvx.Qp(P, q, A[k], b[k], nil, nil, solopts, nil)
		_ = err
		x := sol.Result.At("x")[0]
		ovals = append(ovals, math.Pow(blas.Nrm2Float(x), 2.0))
	}

	optvals := matrix.FloatVector(ovals)
	//fmt.Printf("optvals=\n%v\n", optvals)

	rangeFunc := func(n int) []float64 {
		r := make([]float64, 0)
		for i := 0; i < n; i++ {
			r = append(r, float64(i))
		}
		return r
	}

	nopts := 200
	sigmas := matrix.FloatVector(rangeFunc(nopts))
	sigmas.Scale((0.5 - 0.2) / float64(nopts)).Add(0.2)

	bndsVal := func(sigma float64) float64 {
		// 1.0 - sum(exp( -optvals/(2*sigma**2)))
		return 1.0 - matrix.Exp(matrix.Scale(optvals, -1.0/(2*sigma*sigma))).Sum()
	}
	bnds := matrix.FloatZeros(sigmas.NumElements(), 1)
	for j, v := range sigmas.FloatArray() {
		bnds.SetIndex(j, bndsVal(v))
	}
	plotData("plot.png", sigmas.FloatArray(), bnds.FloatArray())
}