10.4 Inner and outer Löwner-John Ellipsoids

In this section we show how to compute the Löwner-John inner and outer ellipsoidal approximations of a polytope. They are defined as, respectively, the largest volume ellipsoid contained inside the polytope and the smallest volume ellipsoid containing the polytope, as seen in Fig. 10.5.

_images/ellipses_polygon.png

Fig. 10.5 The inner and outer Löwner-John ellipse of a polygon.

For further mathematical details, such as uniqueness of the two ellipsoids, consult [BenTalN01]. Our solution is a mix of conic quadratic and semidefinite programming. Among other things, in Section 10.4.3 we show how to implement bounds involving the determinant of a PSD matrix.

10.4.1 Inner Löwner-John Ellipsoids

Suppose we have a polytope given by an h-representation

\[\mathcal{P} = \{ x \in \real^n \mid Ax \leq b \}\]

and we wish to find the inscribed ellipsoid with maximal volume. It will be convenient to parametrize the ellipsoid as an affine transformation of the standard disk:

\[\mathcal{E} = \{ x \mid x = Cu + d,\ u\in\real^n,\ \| u \|_2 \leq 1 \}.\]

Every non-degenerate ellipsoid has a parametrization such that \(C\) is a positive definite symmetric \(n\times n\) matrix. Now the volume of \(\mathcal{E}\) is proportional to \(\mbox{det}(C)^{1/n}\). The condition \(\mathcal{E}\subseteq\mathcal{P}\) is equivalent to the inequality \(A(Cu+d)\leq b\) for all \(u\) with \(\|u\|_2\leq 1\). After a short computation we obtain the formulation:

(1)\[\begin{split}\begin{array}{lll} \maximize & t & \\ \st & t \leq \mbox{det}(C)^{1/n}, & \\ & (b-Ad)_i\geq \|(AC)_i\|_2, & i=1,\ldots,m,\\ & C \succeq 0, & \end{array}\end{split}\]

where \(X_i\) denotes the \(i\)-th row of the matrix \(X\). This can easily be implemented using Fusion, where the sequence of conic inequalities can be realized at once by feeding in the matrices \(b-Ad\) and \(AC\).

Listing 10.7 Fusion implementation of model (1). Click here to download.
    public static Tuple<double[], double[]> lownerjohn_inner(double[][] A, double[] b)
    {
      using( Model M = new Model("lownerjohn_inner"))
      {
        int m = A.Length;
        int n = A[0].Length;

        // Setup variables
        Variable t = M.Variable("t", 1, Domain.GreaterThan(0.0));
        Variable C = M.Variable("C", new int[] {n, n}, Domain.Unbounded());
        Variable d = M.Variable("d", n, Domain.Unbounded());

        // (bi - ai^T*d, C*ai) \in Q
        for (int i = 0; i < m; ++i)
          M.Constraint("qc" + i, Expr.Vstack(Expr.Sub(b[i], Expr.Dot(A[i], d)), Expr.Mul(C, A[i])),
                       Domain.InQCone() );

        // t <= det(C)^{1/n}
        det_rootn(M, C, t);

        // Objective: Maximize t
        M.Objective(ObjectiveSense.Maximize, t);
        M.Solve();

        return Tuple.Create(C.Level(), d.Level());
      }
    }

The only black box is the method det_rootn which implements the constraint \(t\leq \mbox{det}(C)^{1/n}\). It will be described in Section 10.4.3.

10.4.2 Outer Löwner-John Ellipsoids

To compute the outer ellipsoidal approximation to a polytope, let us now start with a v-representation

\[\mathcal{P} = \mbox{conv}\{ x_1, x_2, \ldots , x_m \} \subseteq \real^n,\]

of the polytope as a convex hull of a set of points. We are looking for an ellipsoid given by a quadratic inequality

\[\mathcal{E} = \{ x\in\real^n \mid \| Px-c \|_2 \leq 1 \},\]

whose volume is proportional to \(\mbox{det}(P)^{-1/n}\), so we are after maximizing \(\mbox{det}(P)^{1/n}\). Again, there is always such a representation with a symmetric, positive definite matrix \(P\). The inclusion conditions \(x_i\in\mathcal{E}\) translate into a straightforward problem formulation:

(2)\[\begin{split}\begin{array}{lll} \maximize & t &\\ \st & t \leq \mbox{det}(P)^{1/n}, &\\ & \|Px_i - c\|_2 \leq 1, &i=1,\ldots,m,\\ & P \succeq 0, & \end{array}\end{split}\]

and then directly into Fusion code:

Listing 10.8 Fusion implementation of model (2). Click here to download.
    public static Tuple<double[], double[]> lownerjohn_outer(double[][] x)
    {
      using( Model M = new Model("lownerjohn_outer") )
      {
        int m = x.Length;
        int n = x[0].Length;

        // Setup variables
        Variable t = M.Variable("t", 1, Domain.GreaterThan(0.0));
        Variable P = M.Variable("P", new int[] {n, n}, Domain.Unbounded());
        Variable c = M.Variable("c", n, Domain.Unbounded());

        // (1, P(*xi+c)) \in Q
        for (int i = 0; i < m; ++i)
          M.Constraint("qc" + i, Expr.Vstack(Expr.Ones(1), Expr.Sub(Expr.Mul(P, x[i]), c)),
                       Domain.InQCone() );

        // t <= det(P)^{1/n}
        det_rootn(M, P, t);

        // Objective: Maximize t
        M.Objective(ObjectiveSense.Maximize, t);
        M.Solve();

        return Tuple.Create(P.Level(), c.Level());
      }
    }

10.4.3 Bound on the Determinant Root

It remains to show how to express the bounds on \(\mbox{det}(X)^{1/n}\) for a symmetric positive definite \(n\times n\) matrix \(X\) using PSD and conic quadratic variables. We want to model the set

(3)\[C = \lbrace (X, t) \in \PSD^n \times \real \mid t \leq \mbox{det}(X)^{1/n} \rbrace.\]

A standard approach when working with the determinant of a PSD matrix is to consider a semidefinite cone

(4)\[\begin{split}\left( {\begin{array}{cc}X & Z \\ Z^T & \mbox{Diag}(Z) \\ \end{array} } \right) \succeq 0\end{split}\]

where \(Z\) is a matrix of additional variables and where we intuitively identify \(\mbox{Diag}(Z)=\{\lambda_1,\ldots,\lambda_n\}\) with the eigenvalues of \(X\). With this in mind, we are left with expressing the constraint

(5)\[t \leq (\lambda_1\cdot\ldots\cdot\lambda_n)^{1/n}.\]

This is easy to implement recursively using rotated quadratic cones when \(n\) is a power of \(2\); otherwise we need to round \(n\) up to the nearest power of \(2\) as in Listing 10.10. For example, \(t\leq (\lambda_1\lambda_2\lambda_3\lambda_4)^{1/4}\) is equivalent to

\[\lambda_1\lambda_2\geq y_1^2,\ \lambda_3\lambda_4\geq y_2^2,\ y_1y_2\geq t^2\]

while \(t\leq (\lambda_1\lambda_2\lambda_3)^{1/3}\) can be achieved by writing \(t\leq (t\lambda_1\lambda_2\lambda_3)^{1/4}\).

For further details and proofs see [BenTalN01] or [MOSEKApS12].

Listing 10.9 Approaching the determinant, see (4). Click here to download.
    public static void det_rootn(Model M, Variable X, Variable t)
    {
      int n = X.GetShape().Dim(0);

      // Setup variables
      Variable Y = M.Variable(Domain.InPSDCone(2 * n));

      // Setup Y = [X, Z; Z^T diag(Z)]
      Variable Y11 = Y.Slice(new int[] {0, 0}, new int[] {n, n});
      Variable Y21 = Y.Slice(new int[] {n, 0}, new int[] {2 * n, n});
      Variable Y22 = Y.Slice(new int[] {n, n}, new int[] {2 * n, 2 * n});

      M.Constraint( Expr.Sub(Expr.MulElm(Matrix.Eye(n), Y21), Y22), Domain.EqualsTo(0.0));
      M.Constraint( Expr.Sub(X, Y11), Domain.EqualsTo(0.0) );

      // t^n <= (Z11*Z22*...*Znn)
      Variable[] tmpv = new Variable[n];
      for (int i = 0; i < n; ++i) tmpv[i] = Y22.Index(i, i);
      Variable z = Var.Reshape(Var.Vstack(tmpv), n);
      geometric_mean(M, z, t);
    }
Listing 10.10 Bounding the geometric mean, see (5). Click here to download.
    public static void geometric_mean(Model M, Variable x, Variable t)
    {
      int n = (int)x.Size();
      int l = (int)System.Math.Ceiling(log2(n));
      int m = pow2(l) - n;

      Variable x0;

      if (m == 0)
        x0 = x;
      else
        x0 = Var.Vstack(x, M.Variable(m, Domain.GreaterThan(0.0)));

      Variable z = x0;

      for (int i = 0; i < l - 1; ++i)
      {
        Variable xi = M.Variable(pow2(l - i - 1), Domain.GreaterThan(0.0));
        for (int k = 0; k < pow2(l - i - 1); ++k)
          M.Constraint(Var.Vstack(z.Index(2 * k), z.Index(2 * k + 1), xi.Index(k)),
                       Domain.InRotatedQCone());
        z = xi;
      }

      Variable t0 = M.Variable(1, Domain.GreaterThan(0.0));
      M.Constraint(Var.Vstack(z, t0), Domain.InRotatedQCone());

      M.Constraint(Expr.Sub(Expr.Mul(System.Math.Pow(2, 0.5 * l), t), t0), Domain.EqualsTo(0.0));

      for (int i = pow2(l - m); i < pow2(l); ++i)
        M.Constraint(Expr.Sub(x0.Index(i), t), Domain.EqualsTo(0.0));
    }