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Jean B. Lasserre

Publications and source records attributed to Jean B. Lasserre.

28 records · Page 2Linked to original sources
arXiv2008

Nonlinear optimal control synthesis via occupation measures

We consider nonlinear optimal control problems (OCPs) for which all problem data are polynomial. In the first part of the paper, we review how occupation measures can be used to approximate pointwise the optimal value function of a given OCP, using a hierarchy of linear matrix inequality (LMI) relaxations. In the second part, we extend the methodology to approximate the optimal value function on a given set and we use such a function to constructively and computationally derive an almost optimal control law. Numerical examples show the effectiveness of the approach.

math.OC↗
arXiv2007

The moment problem with bounded density

Let $μ$ be a given Borel measure on $\K\subseteq\R^n$ and let $y=(y_α)$, $α\in\N^n$, be a given sequence. We provide several conditions linking $y$ and the moment sequence $z=(z_α)$ of $μ$, for $y$ to be the moment sequence of a Borel measure $ν$ on $\K$ which is absolutely continuous with respect to $μ$ and such that its density is in $L_\infty(\K,μ)$. The conditions are necessary and sufficient if $\K$ is a compact basic semi-algebraic set, and sufficient if $\K\equiv\R^n$. Moreover, arbitrary finitely many of these conditions can be checked by solving either a semidefinite program or a linear program with a single variable

math.FA↗
arXiv2007

Explicit Formula for Counting Lattice Points of Polyhedra

Given $z\in C^n$ and $A\in Z^{m\times n}$, we consider the problem of evaluating the counting function $h(y;z):=\sum\{z^x : x\in Z^n; Ax=y, x\geq 0\}$. We provide an explicit expression for $h(y;z)$ as well as an algorithm with possibly numerous but very simple calculations. In addition, we exhibit finitely many fixed convex cones, explicitly and exclusively defined by $A$, such that for any $y\in Z^m$, the sum $h(y;z)$ can be obtained by a simple formula involving the evaluation of $\sum z^x$ over the integral points of those cones only. At last, we also provide an alternative (and different) formula from a decomposition of the generating function into simpler rational fractions, easy to invert.

math.AG↗
arXiv2007

Sufficient conditions for a real polynomial to be a sum of squares

We provide explicit conditions for a real polynomial $f$ of degree 2d to be a sum of squares (s.o.s.), stated only in terms of the coefficients of $f$, i.e. with no lifting. All conditions are simple and provide an explicit description of a convex polyhedral subcone of the cone of s.o.s. polynomials of degree at most 2d. We also provide a simple condition to ensure that $f$ is s.o.s., possibly modulo a constant.

math.AG↗
arXiv2006

A Positivstellensatz which Preserves the Coupling Pattern of Variables

We specialize Schmüdgen's Positivstellensatz and its Putinar and Jacobi and Prestel refinement, to the case of a polynomial $f\in R[X,Y]+R[Y,Z]$, positive on a compact basic semi algebraic set $K$ described by polynomials in $R[X,Y]$ and $R[Y,Z]$ only, or in $R[X]$ and $R[Y,Z]$ only (i.e. $K$ is a cartesian product). In particular, we show that the preordering $P(g,h)$ (resp. quadratic module $Q(g,h)$) generated by the polynomials $\{g_j\}\subset R[X,Y]$ and $\{h_k\}\subset R[Y,Z]$ that describe $K$, is replaced with $P(g)+P(h)$ (resp. $Q(g)+Q(h)$), so that the absence of coupling between $X$ and $Z$ is also preserved in the representation. A similar result applies with Krivine's Positivstellensatz involving the cone generated by $\{g_j,h_k\}$.

math.AC↗
arXiv2005

SOS approximations of nonnegative polynomials via simple high degree perturbation

We show that every real polynomial $f$ nonnegative on $[-1,1]^{n}$ can be approximated in the $l_{1}$-norm of coefficients, by a sequence of polynomials $\{f_{\ep r}\}$ that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and \textit{explicit} approximation sequence. Then we show that if the Moment Problem holds for a basic closed semi-algebraic set $K_S\subset\R^n$ with nonempty interior, then every polynomial nonnegative on $K_S$ can be approximated in a similar fashion by elements from the corresponding preordering. Finally, we show that the degree of the perturbation in the approximating sequence depends on $ε$ as well as the degree and the size of coefficients of the nonnegative polynomial $f$, but not on the specific values of its coefficients.

math.AG↗
arXiv2004

A sum of squares approximation of nonnegative polynomials

We show that every real nonnegative polynomial $f$ can be approximated as closely as desired by a sequence of polynomials $\{f_ε\}$ that are sums of squares. Each $f_ε$ has a simple et explicit form in terms of $f$ and $ε$. A special representation is also obtained for convex polynomials, nonnegative on a convex semi-algebraic set.

math.AG↗
arXiv2004

S.o.s. approximation of polynomials nonnegative on a real algebraic set

With every real polynomial $f$, we associate a family $\{f_{εr}\}_{ε, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $ε>0,r\in N$, and such that $\Vert f-f_{εr}\Vert_1\to 0$ as $ε\to 0$. Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $ε>0$, there exist nonnegative scalars $\{λ_j(ε)\}_{j\in J}$ such that, for all $r$ sufficiently large, $$f_{εr}+\sum_{j\in J} λ_j(ε) g_j^2,\quad is a sum of squares.$$ This representation is an obvious certificate of nonnegativity of $f_{εr}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {\it no} assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and again, with {\it no} assumption on $V$ or $K$.

math.AG↗
arXiv2003

A discrete Farkas lemma

Given $A\in \Z^{m\times n}$ and $b\in\Z^m$, we consider the issue of existence of a nonnegative integral solution $x\in \N^n$ to the system of linear equations $Ax=b$. We provide a discrete and explicit analogue of the celebrated Farkas lemma for linear systems in $\R^n$ and prove that checking existence of integral solutions reduces to solving an explicit linear programming problem of fixed dimension, known in advance.

math.CO↗