Level sets and non Gaussian integrals of positively homogeneous functions

We investigate various properties of the sublevel set $\{x \,:\,g(x)\leq 1\}$ and the integration of $h$ on this sublevel set when $g$ and $h$are positively homogeneous functions. For instance, the latter integral reduces to integrating $h\exp(-g)$ on the whole space $R^n$ (a non Gaussian integral) and when $g$ is a polynomial, then the volume of the sublevel set is a convex function of the coefficients of $g$. In fact, whenever $h$ is nonnegative, the functional $\int \phi(g(x))h(x)dx$ is a convex function of $g$ for a large class of functions $\phi:R_+\to R$. We also provide a numerical approximation scheme to compute the volume or integrate $h$ (or, equivalently to approximate the associated non Gaussian integral). We also show that finding the sublevel set $\{x \,:\,g(x)\leq 1\}$ of minimum volume that contains some given subset $K$ is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function

Inverse polynomial optimization

We consider the inverse optimization problem associated with the polynomial program f^*=\min \{f(x): x\in K\}$and a given current feasible solution$y\in K$. We provide a systematic numerical scheme to compute an inverse optimal solution. That is, we compute a polynomial$\tilde{f}$(which may be of same degree as$f$if desired) with the following properties: (a)$y$is a global minimizer of$\tilde{f}$on$K$with a Putinar's certificate with an a priori degree bound$d$fixed, and (b),$\tilde{f}$minimizes$\Vert f-\tilde{f}\Vert$(which can be the$\ell_1$,$\ell_2$or$\ell_\infty$-norm of the coefficients) over all polynomials with such properties. Computing$\tilde{f}_d$reduces to solving a semidefinite program whose optimal value also provides a bound on how far is$f(\y)$from the unknown optimal value$f^*$. The size of the semidefinite program can be adapted to the computational capabilities available. Moreover, if one uses the$\ell_1$-norm, then$\tilde{f}$ takes a simple and explicit canonical form. Some variations are also discussed.Comment: 25 pages; to appear in Math. Oper. Res; Rapport LAAS no. 1114

Reconstruction of algebraic-exponential data from moments

Let $G$ be a bounded open subset of Euclidean space with real algebraic boundary $\Gamma$. Under the assumption that the degree $d$ of $\Gamma$ is given, and the power moments of the Lebesgue measure on $G$ are known up to order $3d$, we describe an algorithmic procedure for obtaining a polynomial vanishing on $\Gamma$. The particular case of semi-algebraic sets defined by a single polynomial inequality raises an intriguing question related to the finite determinateness of the full moment sequence. The more general case of a measure with density equal to the exponential of a polynomial is treated in parallel. Our approach relies on Stokes theorem and simple Hankel-type matrix identities

GloptiPoly 3: moments, optimization and semidefinite programming

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming