Beispiel #1
0
func qcl1(A, b *matrix.FloatMatrix) (*cvx.Solution, error) {

	// Returns the solution u, z of
	//
	//   (primal)  minimize    || u ||_1
	//             subject to  || A * u - b ||_2  <= 1
	//
	//   (dual)    maximize    b^T z - ||z||_2
	//             subject to  || A'*z ||_inf <= 1.
	//
	// Exploits structure, assuming A is m by n with m >= n.

	m, n := A.Size()
	Fkkt := func(W *sets.FloatMatrixSet) (f cvx.KKTFunc, err error) {

		minor := 0
		if !checkpnt.MinorEmpty() {
			minor = checkpnt.MinorTop()
		}

		err = nil
		f = nil
		beta := W.At("beta")[0].GetIndex(0)
		v := W.At("v")[0]

		// As = 2 * v *(v[1:].T * A)
		//v_1 := matrix.FloatNew(1, v.NumElements()-1, v.FloatArray()[1:])
		v_1 := v.SubMatrix(1, 0).Transpose()

		As := matrix.Times(v, matrix.Times(v_1, A)).Scale(2.0)

		//As_1 := As.GetSubMatrix(1, 0, m, n)
		//As_1.Scale(-1.0)
		//As.SetSubMatrix(1, 0, matrix.Minus(As_1, A))
		As_1 := As.SubMatrix(1, 0, m, n)
		As_1.Scale(-1.0)
		As_1.Minus(A)
		As.Scale(1.0 / beta)

		S := matrix.Times(As.Transpose(), As)
		checkpnt.AddMatrixVar("S", S)

		d1 := W.At("d")[0].SubMatrix(0, 0, n, 1).Copy()
		d2 := W.At("d")[0].SubMatrix(n, 0).Copy()

		// D = 4.0 * (d1**2 + d2**2)**-1
		d := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2)).Inv().Scale(4.0)
		// S[::n+1] += d
		S.Diag().Plus(d.Transpose())

		err = lapack.Potrf(S)
		checkpnt.Check("00-Fkkt", minor)
		if err != nil {
			return
		}

		f = func(x, y, z *matrix.FloatMatrix) (err error) {

			minor := 0
			if !checkpnt.MinorEmpty() {
				minor = checkpnt.MinorTop()
			} else {
				loopf += 1
				minor = loopf
			}
			checkpnt.Check("00-f", minor)

			// -- z := - W**-T * z
			// z[:n] = -div( z[:n], d1 )
			z_val := z.SubMatrix(0, 0, n, 1)
			z_res := matrix.Div(z_val, d1).Scale(-1.0)
			z.SubMatrix(0, 0, n, 1).Set(z_res)

			// z[n:2*n] = -div( z[n:2*n], d2 )
			z_val = z.SubMatrix(n, 0, n, 1)
			z_res = matrix.Div(z_val, d2).Scale(-1.0)
			z.SubMatrix(n, 0, n, 1).Set(z_res)

			// z[2*n:] -= 2.0*v*( v[0]*z[2*n] - blas.dot(v[1:], z[2*n+1:]) )
			v0_z2n := v.GetIndex(0) * z.GetIndex(2*n)
			v1_z2n := blas.DotFloat(v, z, &linalg.IOpt{"offsetx", 1}, &linalg.IOpt{"offsety", 2*n + 1})
			z_res = matrix.Scale(v, -2.0*(v0_z2n-v1_z2n))
			z.SubMatrix(2*n, 0, z_res.NumElements(), 1).Plus(z_res)

			// z[2*n+1:] *= -1.0
			z.SubMatrix(2*n+1, 0).Scale(-1.0)

			// z[2*n:] /= beta
			z.SubMatrix(2*n, 0).Scale(1.0 / beta)

			// -- x := x - G' * W**-1 * z

			// z_n = z[:n], z_2n = z[n:2*n], z_m = z[-(m+1):],
			z_n := z.SubMatrix(0, 0, n, 1)
			z_2n := z.SubMatrix(n, 0, n, 1)
			z_m := z.SubMatrix(z.NumElements()-(m+1), 0)

			// x[:n] -= div(z[:n], d1) - div(z[n:2*n], d2) + As.T * z[-(m+1):]
			z_res = matrix.Minus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
			a_res := matrix.Times(As.Transpose(), z_m)
			z_res.Plus(a_res).Scale(-1.0)
			x.SubMatrix(0, 0, n, 1).Plus(z_res)

			// x[n:] += div(z[:n], d1) + div(z[n:2*n], d2)
			z_res = matrix.Plus(matrix.Div(z_n, d1), matrix.Div(z_2n, d2))
			x.SubMatrix(n, 0, z_res.NumElements(), 1).Plus(z_res)
			checkpnt.Check("15-f", minor)

			// Solve for x[:n]:
			//
			//    S*x[:n] = x[:n] - (W1**2 - W2**2)(W1**2 + W2**2)^-1 * x[n:]

			// w1 = (d1**2 - d2**2), w2 = (d1**2 + d2**2)
			w1 := matrix.Minus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))
			w2 := matrix.Plus(matrix.Mul(d1, d1), matrix.Mul(d2, d2))

			// x[:n] += -mul( div(w1, w2), x[n:])
			x_n := x.SubMatrix(n, 0)
			x_val := matrix.Mul(matrix.Div(w1, w2), x_n).Scale(-1.0)
			x.SubMatrix(0, 0, n, 1).Plus(x_val)
			checkpnt.Check("25-f", minor)

			// Solve for x[n:]:
			//
			//    (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]

			err = lapack.Potrs(S, x)
			if err != nil {
				fmt.Printf("Potrs error: %s\n", err)
			}
			checkpnt.Check("30-f", minor)

			// Solve for x[n:]:
			//
			//    (d1**-2 + d2**-2) * x[n:] = x[n:] + (d1**-2 - d2**-2)*x[:n]

			// w1 = (d1**-2 - d2**-2), w2 = (d1**-2 + d2**-2)
			w1 = matrix.Minus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
			w2 = matrix.Plus(matrix.Mul(d1, d1).Inv(), matrix.Mul(d2, d2).Inv())
			x_n = x.SubMatrix(0, 0, n, 1)

			// x[n:] += mul( d1**-2 - d2**-2, x[:n])
			x_val = matrix.Mul(w1, x_n)
			x.SubMatrix(n, 0, x_val.NumElements(), 1).Plus(x_val)
			checkpnt.Check("35-f", minor)

			// x[n:] = div( x[n:], d1**-2 + d2**-2)
			x_n = x.SubMatrix(n, 0)
			x_val = matrix.Div(x_n, w2)
			x.SubMatrix(n, 0, x_val.NumElements(), 1).Set(x_val)
			checkpnt.Check("40-f", minor)

			// x_n = x[:n], x-2n = x[n:2*n]
			x_n = x.SubMatrix(0, 0, n, 1)
			x_2n := x.SubMatrix(n, 0, n, 1)

			// z := z + W^-T * G*x
			// z[:n] += div( x[:n] - x[n:2*n], d1)
			x_val = matrix.Div(matrix.Minus(x_n, x_2n), d1)
			z.SubMatrix(0, 0, n, 1).Plus(x_val)
			checkpnt.Check("44-f", minor)

			// z[n:2*n] += div( -x[:n] - x[n:2*n], d2)
			x_val = matrix.Div(matrix.Plus(x_n, x_2n).Scale(-1.0), d2)
			z.SubMatrix(n, 0, n, 1).Plus(x_val)
			checkpnt.Check("48-f", minor)

			// z[2*n:] += As*x[:n]
			x_val = matrix.Times(As, x_n)
			z.SubMatrix(2*n, 0, x_val.NumElements(), 1).Plus(x_val)

			checkpnt.Check("50-f", minor)

			return nil
		}
		return
	}

	// matrix(n*[0.0] + n*[1.0])
	c := matrix.FloatZeros(2*n, 1)
	c.SubMatrix(n, 0).SetIndexes(1.0)

	h := matrix.FloatZeros(2*n+m+1, 1)
	h.SetIndexes(1.0, 2*n)
	// h[2*n+1:] = -b
	h.SubMatrix(2*n+1, 0).Plus(b).Scale(-1.0)
	G := &matrixFs{A}

	dims := sets.DSetNew("l", "q", "s")
	dims.Set("l", []int{2 * n})
	dims.Set("q", []int{m + 1})

	var solopts cvx.SolverOptions
	solopts.ShowProgress = true
	if maxIter > 0 {
		solopts.MaxIter = maxIter
	}
	if len(solver) > 0 {
		solopts.KKTSolverName = solver
	}
	return cvx.ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, nil, nil)
}
Beispiel #2
0
func mcsdp(w *matrix.FloatMatrix) (*cvx.Solution, error) {
	//
	// Returns solution x, z to
	//
	//    (primal)  minimize    sum(x)
	//              subject to  w + diag(x) >= 0
	//
	//    (dual)    maximize    -tr(w*z)
	//              subject to  diag(z) = 1
	//                          z >= 0.
	//
	n := w.Rows()
	G := &matrixFs{n}

	cngrnc := func(r, x *matrix.FloatMatrix, alpha float64) (err error) {
		// Congruence transformation
		//
		//    x := alpha * r'*x*r.
		//
		// r and x are square matrices.
		//
		err = nil

		// tx = matrix(x, (n,n)) is copying and reshaping
		// scale diagonal of x by 1/2, (x is (n,n))
		tx := x.Copy()
		matrix.Reshape(tx, n, n)
		tx.Diag().Scale(0.5)

		// a := tril(x)*r
		// (python: a = +r is really making a copy of r)
		a := r.Copy()

		err = blas.TrmmFloat(tx, a, 1.0, linalg.OptLeft)

		// x := alpha*(a*r' + r*a')
		err = blas.Syr2kFloat(r, a, tx, alpha, 0.0, linalg.OptTrans)

		// x[:] = tx[:]
		tx.CopyTo(x)
		return
	}

	Fkkt := func(W *sets.FloatMatrixSet) (cvx.KKTFunc, error) {

		//    Solve
		//                  -diag(z)                           = bx
		//        -diag(x) - inv(rti*rti') * z * inv(rti*rti') = bs
		//
		//    On entry, x and z contain bx and bs.
		//    On exit, they contain the solution, with z scaled
		//    (inv(rti)'*z*inv(rti) is returned instead of z).
		//
		//    We first solve
		//
		//        ((rti*rti') .* (rti*rti')) * x = bx - diag(t*bs*t)
		//
		//    and take z  = -rti' * (diag(x) + bs) * rti.

		var err error = nil
		rti := W.At("rti")[0]

		// t = rti*rti' as a nonsymmetric matrix.
		t := matrix.FloatZeros(n, n)
		err = blas.GemmFloat(rti, rti, t, 1.0, 0.0, linalg.OptTransB)
		if err != nil {
			return nil, err
		}

		// Cholesky factorization of tsq = t.*t.
		tsq := matrix.Mul(t, t)
		err = lapack.Potrf(tsq)
		if err != nil {
			return nil, err
		}

		f := func(x, y, z *matrix.FloatMatrix) (err error) {
			// tbst := t * zs * t = t * bs * t
			tbst := z.Copy()
			matrix.Reshape(tbst, n, n)
			cngrnc(t, tbst, 1.0)

			// x := x - diag(tbst) = bx - diag(rti*rti' * bs * rti*rti')
			diag := tbst.Diag().Transpose()
			x.Minus(diag)

			// x := (t.*t)^{-1} * x = (t.*t)^{-1} * (bx - diag(t*bs*t))
			err = lapack.Potrs(tsq, x)
			if err != nil {
				fmt.Printf("Fkkt.f.Potrs: %v\n", err)
			}

			// z := z + diag(x) = bs + diag(x)
			// z, x are really column vectors here
			z.AddIndexes(matrix.MakeIndexSet(0, n*n, n+1), x.FloatArray())

			// z := -rti' * z * rti = -rti' * (diag(x) + bs) * rti
			cngrnc(rti, z, -1.0)
			return nil
		}
		return f, nil
	}

	c := matrix.FloatWithValue(n, 1, 1.0)

	// initial feasible x: x = 1.0 - min(lmbda(w))
	lmbda := matrix.FloatZeros(n, 1)
	wp := w.Copy()
	lapack.Syevx(wp, lmbda, nil, 0.0, nil, []int{1, 1}, linalg.OptRangeInt)
	x0 := matrix.FloatZeros(n, 1).Add(-lmbda.GetAt(0, 0) + 1.0)
	s0 := w.Copy()
	// Diag() return a row vector, x0 is column vector
	s0.Diag().Plus(x0.Transpose())
	matrix.Reshape(s0, n*n, 1)

	// initial feasible z is identity
	z0 := matrix.FloatIdentity(n)
	matrix.Reshape(z0, n*n, 1)

	dims := sets.DSetNew("l", "q", "s")
	dims.Set("s", []int{n})

	primalstart := sets.FloatSetNew("x", "s")
	dualstart := sets.FloatSetNew("z")
	primalstart.Set("x", x0)
	primalstart.Set("s", s0)
	dualstart.Set("z", z0)

	var solopts cvx.SolverOptions
	solopts.ShowProgress = true
	if maxIter > 0 {
		solopts.MaxIter = maxIter
	}
	if len(solver) > 0 {
		solopts.KKTSolverName = solver
	}
	h := w.Copy()
	matrix.Reshape(h, h.NumElements(), 1)
	return cvx.ConeLpCustomMatrix(c, G, h, nil, nil, dims, Fkkt, &solopts, primalstart, dualstart)
}