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. 7.

_images/ellipses_polygon.png

Fig. 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 quadratic and semidefinite programming. Among other things, in Sec. 10.4.3 (Bound on the Determinant Root) 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 22 Fusion implementation of model (1). Click here to download.
  public static Object[] lownerjohn_inner(double[][] A, double[] b)
  throws SolutionError {
    Model M = new Model("lownerjohn_inner");
    try {
      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", Domain.inPSDCone(n));
      Variable d = M.variable("d", n, Domain.unbounded());

      // (b-Ad, AC) generate cones
      M.constraint("qc", Expr.hstack(Expr.sub(b, Expr.mul(A, d)), Expr.mul(A, C)),
                   Domain.inQCone());
      //t <= det(C)^{1/n}
      det_rootn(M, C, t);

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

      return new Object[] {C.level(), d.level()};
    } finally {
      M.dispose();
    }
  }

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. 10.4.3 (Bound on the Determinant Root).

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 23 Fusion implementation of model (2). Click here to download.
  public static Object[] lownerjohn_outer(double[][] x)
  throws SolutionError {
    Model M = new Model("lownerjohn_outer");
    try {
      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", Domain.inPSDCone(n));
      Variable c = M.variable("c", n, Domain.unbounded());

      //(1, Px-c) in cone
      M.constraint("qc",
                   Expr.hstack(Expr.ones(m),
                               Expr.sub(Expr.mul(x, P),
                                        Var.reshape(Var.repeat(c, m), new int[] {m, n})
                                       )
                              ),
                   Domain.inQCone());

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

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

      return new Object[] {P.level(), c.level()};
    } finally {
      M.dispose();
    }
  }

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 25. 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 24 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(Y21.diag(), Y22.diag()), Domain.equalsTo(0.0) );
    M.constraint( Expr.sub(X, Y11), Domain.equalsTo(0.0) );

    // t^n <= (Z11*Z22*...*Znn)
    geometric_mean(M, Y22.diag(), t);
  }
Listing 25 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)Math.ceil(Math.log(n) / Math.log(2));
    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(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));
  }