11.6 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. 11.7.

_images/ellipses_polygon.png

Fig. 11.7 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 and semidefinite programming. Among other things, in Sec. 11.6.3 (Bound on the Determinant Root) we show how to implement bounds involving the determinant of a PSD matrix.

11.6.1 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:

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

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 11.12 Fusion implementation of model (11.22). 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 = det_rootn(M, t, n);
        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().Axis(0) );

        // 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 Sec. 11.6.3 (Bound on the Determinant Root).

11.6.2 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:

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

and then directly into Fusion code:

Listing 11.13 Fusion implementation of model (11.23). Click here to download.
    public static Tuple<double[], double[]> lownerjohn_outer(double[,] x)
    {
      using( Model M = new Model("lownerjohn_outer") )
      {
        int m = x.GetLength(0);
        int n = x.GetLength(1);

        // Setup variables
        Variable t = M.Variable("t", 1, Domain.GreaterThan(0.0));
        Variable P = det_rootn(M, t, n);
        Variable c = M.Variable("c", Domain.Unbounded().WithShape(1,n));

        // (1, P(*xi+c)) \in Q
        M.Constraint("qc",
                     Expr.Hstack(Expr.Ones(m), Expr.Sub(Expr.Mul(x,P), Expr.Repeat(c,m,0))),
                     Domain.InQCone().Axis(1));

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

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

11.6.3 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

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

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

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

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

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

but this is exactly the geometric mean cone Domain.InPGeoMeanCone. We obtain the following model:

Listing 11.14 Bounding the n-th root of the determinant, see (11.25). Click here to download.
    public static Variable det_rootn(Model M, Variable t, int n)
    {
      // Setup variables
      Variable Y = M.Variable(Domain.InPSDCone(2 * n));

      Variable X   = Y.Slice(new int[]{0, 0}, new int[]{n, n});
      Variable Z   = Y.Slice(new int[]{0, n}, new int[]{n, 2 * n});
      Variable DZ  = Y.Slice(new int[]{n, n}, new int[]{2 * n, 2 * n});

      // Z is lower-triangular
      int[,] low_tri = new int[n*(n-1)/2,2];
      int k = 0;
      for(int i = 0; i < n; i++)
        for(int j = i+1; j < n; j++)
          { low_tri[k,0] = i; low_tri[k,1] = j; ++k; }
      M.Constraint(Z.Pick(low_tri), Domain.EqualsTo(0.0));
      // DZ = Diag(Z)
      M.Constraint(Expr.Sub(DZ, Expr.MulElm(Z, Matrix.Eye(n))), Domain.EqualsTo(0.0));

      // t^n <= (Z11*Z22*...*Znn)
      M.Constraint(Expr.Vstack(DZ.Diag(), t), Domain.InPGeoMeanCone());

      // Return an n x n PSD variable which satisfies t <= det(X)^(1/n)
      return X;
    }