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.
std::pair<std::shared_ptr<ndarray<double, 1>>, std::shared_ptr<ndarray<double, 1>>>
    lownerjohn_inner
    ( std::shared_ptr<ndarray<double, 2>> A,
      std::shared_ptr<ndarray<double, 1>> b)
{
  Model::t M = new Model("lownerjohn_inner"); auto _M = finally([&]() { M->dispose(); });
  int m = A->size(0);
  int n = A->size(1);

  // Setup variables
  Variable::t t = M->variable("t", 1, Domain::greaterThan(0.0));
  Variable::t C = M->variable("C", Domain::inPSDCone(n));
  Variable::t d = M->variable("d", n, Domain::unbounded());

  // quadratic cones
  M->constraint(Expr::hstack(Expr::sub(b, Expr::mul(A, d)), Expr::mul(A, C)),
                Domain::inQCone());

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

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

  return std::make_pair(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.
std::pair<std::shared_ptr<ndarray<double, 1>>, std::shared_ptr<ndarray<double, 1>>>
    lownerjohn_outer(std::shared_ptr<ndarray<double, 2>> x)
{
  Model::t M = new Model("lownerjohn_outer");
  int m = x->size(0);
  int n = x->size(1);

  // Setup variables
  Variable::t t = M->variable("t", 1, Domain::greaterThan(0.0));
  Variable::t P = M->variable("P", Domain::inPSDCone(n));
  Variable::t c = M->variable("c", n, Domain::unbounded());

  // (1, Px-c) \in Q
  M->constraint(Expr::hstack(
                  Expr::ones(m), Expr::sub(Expr::mul(x, P),
                      Var::reshape(Var::repeat(c, m), new_array_ptr<int, 1>({m, n}))) ),
                Domain::inQCone());

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

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

  return std::make_pair(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.
void det_rootn(Model::t M, int n, Variable::t X, Variable::t t)
{
  // Setup variables
  Variable::t Y = M->variable(Domain::inPSDCone(2 * n));

  // Setup Y = [X, Z; Z^T diag(Z)]
  Variable::t Y11 = Y->slice(new_array_ptr<int, 1>({0, 0}), new_array_ptr<int, 1>({n, n}));
  Variable::t Y21 = Y->slice(new_array_ptr<int, 1>({n, 0}), new_array_ptr<int, 1>({2 * n, n}));
  Variable::t Y22 = Y->slice(new_array_ptr<int, 1>({n, n}), new_array_ptr<int, 1>({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 10.10 Bounding the geometric mean, see (5). Click here to download.
void geometric_mean(Model::t M, Variable::t x, Variable::t t)
{
  int n = (int) x->size();
  int l = (int)std::ceil(std::log(n) / std::log(2));
  int m = pow2(l) - n;

  Variable::t x0 =
    m == 0 ? x : Var::vstack(x, M->variable(m, Domain::greaterThan(0.0)));

  Variable::t z = x0;

  for (int i = 0; i < l - 1; ++i)
  {
    Variable::t 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::t t0 = M->variable(1, Domain::greaterThan(0.0));
  M->constraint(Var::vstack(z, t0), Domain::inRotatedQCone());

  M->constraint(Expr::sub(Expr::mul(std::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));
}