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International audience; We study balls of homogeneous cubics on Rn, n=2,3, which are bounded by unity on the unit sphere. For n=2 we completely describe the facial structure of this norm ball, while for n=3 we classify all extremal points and describe some families of faces.

We provide a complete classification of the extreme rays of the $$6 \times 6$$ copositive cone $$\mathcal {COP}^{6}$$ . We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $$A \in \mathcal {COP}^{6}$$ . To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric $$6 \times 6$$ matrices $${\mathcal {S}}^{6}$$ , parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices $$A \in \mathcal {COP}^{6}$$ having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros u as well as the supports of the corresponding matrix-vector products Au. The analysis uses recently and newly developed methods that are applicable to copositive matrices of arbitrary order.

We consider non-degenerate graph immersions into affine space $\mathbb A^{n+1}$ whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs $(J,\gamma)$, where $J$ is an $n$-dimensional real Jordan algebra and $\gamma$ is a non-degenerate trace form on $J$. Every graph immersion with parallel cubic form can be extended to an affine complete symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra $J$. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form., Comment: some proofs have been simplified with respect to the first version

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter $\alpha \leq \frac12$ one can define a projectively invariant $2\pi$-periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, which we call balanced, is unique up to a shift of the parameter. The curve is an ellipse if and only if $\alpha = \frac12$, and the value of $\alpha$ is a global projective invariant of the curve. The parametrization is equivariant with respect to duality., Comment: submitted to Journal of Geometry

International audience; The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of adapted martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the pathwise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

For every proper convex cone $K \subset \mathbb R^3$ there exists a unique complete hyperbolic affine 2-sphere with mean curvature $-1$ which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups $SL(3,\mathbb R)$ and $S^1$, where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of $S^1$ generalizes conic duality, which acts by multiplication of the cubic differential by $-1$. We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of $S^1$ induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parametrizations of the corresponding affine spheres. The solutions can be expressed in terms of degenerate Painlev\'e III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also cones with polyhedral boundary parts., Comment: The third version contains simplified proofs

International audience; Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that ${\rm supp} v \subset {\rm supp} u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$.

International audience

Let $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal. Then the minimal zeros of $A$ all have supports of cardinality two if and only if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal $\{-1,0,1\}$ unit diagonal matrices characterized by Hoffman and Pereira in 1973., Comment: 4 pages

Let $n \geq 5$ and let $u^1,\dots,u^n$ be nonnegative real $n$-vectors such that the indices of their positive elements form the sets $\{1,2,\dots,n-2\},\{2,3,\dots,n-1\},\dots,\{n,1,\dots,n-3\}$, respectively. Here each index set is obtained from the previous one by a circular shift. The set of copositive forms which vanish on the vectors $u^1,\dots,u^n$ is a face of the copositive cone ${\cal C}^n$. We give an explicit semi-definite description of this face and of its subface consisting of positive semi-definite forms, and study their properties. If the vectors $u^1,\dots,u^n$ and their positive multiples exhaust the zero set of an exceptional copositive form belonging to this face, then we say it has minimal circulant zero support set, and otherwise non-minimal circulant zero support set. We show that forms with non-minimal circulant zero support set are always extremal, and forms with minimal circulant zero support sets can be extremal only if $n$ is odd. We construct explicit examples of extremal forms with non-minimal circulant zero support set for any order $n \geq 5$, and examples of extremal forms with minimal circulant zero support set for any odd order $n \geq 5$. The set of all forms with non-minimal circulant zero support set, i.e., defined by different collections $u^1,\dots,u^n$ of zeros, is a submanifold of codimension $2n$, the set of all forms with minimal circulant zero support set a submanifold of codimension $n$., 32 pages

International audience

Let $A$ be an element of the copositive cone $coposn$. A zero $vu$ of $A$ is a nonnegative vector whose elements sum up to one and such that $vu^TAvu = 0$. The support of $vu$ is the index set $Suppvu subset 1,dots,n$ corresponding to the nonzero entries of $vu$. A zero $vu$ of $A$ is called minimal if there does not exist another zero $vv$ of $A$ such that its support $Suppvv$ is a strict subset of $Suppvu$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $VarKA$ of real symmetric $n times n$ matrices $B$ such that there exists $delta > 0$ satisfying $A + delta B in coposn$. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in $coposn$, and of the minimal exposed face of $A$ in $coposn$, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of $coposn$ by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $delta > 0$ we have $A - delta C notin coposn$.

Let $${M_n \subset \mathbb R^{n+1}}$$ be a complete hyperbolic affine hypersphere with mean curvature H, $${H < 0}$$ , and let C be its cubic form. We derive a differential inequality and an upper bound on the scalar function $${||C||_{\infty}}$$ defined by the fiber-wise maximum of the value of C on the unit sphere bundle of M. The bounds are attained for the affine hyperspheres which are asymptotic to a simplicial cone. The results have applications in conic optimization.

Special Issue Dedicated to Brian Anderson on the Occasion of His 70th Birthday: Part II; International audience; This article reviews the development of experiment design in the field of identification of dynamical systems, from the early work of the seventies on input design for open loop identification to the developments of the last decade that were spurred by the research on identification for control. While the early work focused entirely on criteria based on the asymptotic parameter covariance, the results of the last decade aim at minimizing a wide range of possible criteria, including measures of the estimated transfer function, or of functions of this estimated transfer function. Two important recent developments are the solution of the experiment design problem for closed loop identification, and the formulation and solution of the dual optimal design problem in which the cost of identification is minimized subject to a quality constraint on the estimated model. We shall conclude this survey with new results on the optimal closed loop experiment design problem, where the optimization is performed jointly with respect to the controller and the spectrum of the external excitation.

Let K ⊂ E , K ′ ⊂ E ′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E ⊗ E ′ is K ⊗ K ′ -separable if it can be represented as finite sum y = ∑ l x l ⊗ x l ′ , where x l ∈ K and x l ′ ∈ K ′ for all l . Let S ( n ) , H ( n ) , Q ( n ) be the spaces of n × n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S + ( n ) , H + ( n ) , Q + ( n ) be the cones of positive semidefinite matrices in these spaces. If a matrix A ∈ H ( mn ) = H ( m ) ⊗ H ( n ) is H + ( m ) ⊗ H + ( n ) -separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S ( m ) ⊗ S ( n ) , H ( m ) ⊗ S ( n ) , and for m ⩽ 2 in the space Q ( m ) ⊗ S ( n ) . We provide a complete enumeration of all pairs ( n , m ) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q ( n ) ⊗ S ( 2 ) is Q + ( n ) ⊗ S + ( 2 ) - separable if and only if it is positive semidefinite.

International audience; Let $n \in \mathbb N$ and let $L_n \subset \mathbb R^n$ be the $n$-dimensional second order cone, or Lorentz cone. A linear map $M$ from $\mathbb R^m$ to $\mathbb R^n$ is called positive if $M[L_m] \subset L_n$. The set of positive maps forms a convex cone, the positive cone. Its dual cone, the separable cone, is the convex hull of tensor products $x \otimes y$, where $x \in L_n$, $y \in L_m$. The structure of the positive and the separable cones is the subject of the present contribution. We study the action of the automorphism group of the positive cone and provide a complete classification of its orbits by constructing explicitly canonical forms. This enables us to identify and to classify the extreme rays of the positive cone, as well as their dual counterparts, the largest faces of the separable cone. We also characterize positive maps which are scalable to doubly stochastic ones.