Abstract:
We consider the general linear programming problem over the cone of positive semi-definite matrices. We first provide a simple sufficient condition for existence of optimal solutions and absence of a duality gap without requiring existence of a strictly feasible solution. We then simply characterize the analogues of the standard concepts of linear programming, i.e., extreme points, basis, reduced cost, degeneracy, pivoting step as well as a Simplex-like algorithm.

Abstract:
For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety $V_\mathbb{R}(I)$ as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gr\"obner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

Abstract:
We provide a real algebraic symbolic-numeric algorithm for computing the real variety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may not be. Our approach uses sets of linear functionals on $R[X]$, vanishing on a given set of polynomials generating $I$ and their prolongations up to a given degree, as well as on polynomials of the real radical ideal of $I$, obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm, based on standard numerical linear algebra routines and semidefinite optimization, combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases.

Abstract:
We provide two algorithms for computing the volume of a convex polytope with half-space representation {x>=0; Ax <=b} for some (m,n) matrix A and some m-vector b. Both algorithms have a O(n^m) computational complexity which makes them especially attractive for large n and relatively small m when the other methods with O(m^n) complexity fail. The methodology which differs from previous existing methods uses a Laplace transform technique that is well-suited to the half-space representation of the polytope.

Abstract:
We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure $\mu$ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with $\mu$.

Abstract:
We investigate the multi-dimensional Super Resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the 1 -minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal 1 -minimization exact recovery problem of Super Resolution that unleashes standard techniques (such as moment-sum-of-squares hier-archies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the Super Resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets.

Abstract:
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and global minimum using a sequence of semidefinite programming (SDP) relaxations and provide convergence results for the methods. Our scheme for problems with a convex lower-level problem involves solving a transformed equivalent single-level problem by a sequence of SDP relaxations; whereas our approach for general problems involving a non-convex polynomial lower-level problem solves a sequence of approximation problems via another sequential SDP relaxations.

Abstract:
Plasma levels of glucose, insulin and glucagon were measured at various time intervals after pancreatic duct ligation (PDL) in rabbits. Two hyperglycemic periods were observed: one between 15–90 days (peak at 30 days of 15.1 ± 1.2mmol/l, p < 0.01), and the other at 450 days (11.2 ± 0.5 mmol/l, p < 0.02). The first hyperglycemic episode was significantly correlated with both hypoinsulinemia (41.8 ± 8pmol/l, r= –0.94, p < 0.01) and hyperglucagonemia (232 ± 21ng/l, r=0.95, p < 0.01). However, the late hyperglycemic phase (450 days), which was not accompanied by hypoinsulinemia, was observed after the hyperglucagonemia (390 days) produced by abundant immunostained A-cells giving rise to a 3-fold increase in pancreatic glucagon stores. The insulin and glucagon responses to glucose loading at 180, 270 and 450 days reflected the insensitivity of B- and A-cells to glucose. The PDL rabbit model with chronic and severe glycemic disorders due to the predominant role of glucagon mimicked key features of the NIDDM syndrome secondary to exocrine disease.

Abstract:
We consider the class of Markov kernels for which the weak or strong Feller property fails to hold at some discontinuity set. We provide a simple necessary and sufficient condition for existence of an invariant probability measure as well as a Foster-Lyapunov sufficient condition. We also characterize a subclass, the quasi (weak or strong) Feller kernels, for which the sequences of expected occupation measures share the same asymptotic properties as for (weak or strong) Feller kernels. In particular, it is shown that the sequences of expected occupation measures of strong and quasi strong-Feller kernels with an invariant probability measure converge setwise to an invariant measure.