# 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.

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 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 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 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 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(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 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)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));
}