The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets dynamical by using semidefinite programming solve sum-of-squares (SOS) programmes. With the emergence SOS programmes, it is possible efficiently search for Lyapunov functions that guarantee st...

Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due the expansive list geometric primitives , e.g., splines, triangles, and hexahedra, joined with an ever-expanding objectives one might want achieve them. With recent increase in attention toward higher-order surfaces we can expect challenges porti...

This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the Sparse Principal Component Analysis (Sparse PCA) problem, and the family of Sum-of-Squares (SoS, aka Lasserre/Parillo) convex relaxations. It was well known that in large dimension p, a planted k-spa...

PURPOSE To develop a novel approach for calculating the accurate sensitivity profiles of phased-array coils, resulting in correction of nonuniform intensity in parallel MRI. MATERIALS AND METHODS The proposed intensity-correction method estimates the accurate sensitivity profile of each channel of the phased-array coil. The sensitivity profile is estimated by fitting a nonlinear curve to ever...

A wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities (LMIs). These techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares (SOS) programming. In this paper we further extend the class of systems that c...

This paper studies the problem of minimizing a homogeneous polynomial (form) f(x) over the unit sphere Sn−1 = {x ∈ R : ‖x‖2 = 1}. The problem is NP-hard when f(x) has degree 3 or higher. Denote by fmin (resp., fmax) the minimum (resp., maximum) value of f(x) on Sn−1. First, when f(x) is an even form of degree 2d, we study the standard sum of squares (SOS) relaxation for finding a lower bound of...

Maximum A posteriori Probability (MAP) inference in graphical models amounts to solving a graph-structured combinatorial optimization problem. Popular inference algorithms such as belief propagation (BP) and generalized belief propagation (GBP) are intimately related to linear programming (LP) relaxation within the Sherali-Adams hierarchy. Despite the popularity of these algorithms, it is well ...

This work is concerned with the proof-complexity of certifying that optimization problems do not have good solutions. Specifically we consider bounded-degree “Sum of Squares” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning ...

this paper presents a gain scheduled autopilot for pitch channel of a flying vehicle. the selected method is based on polynomial fuzzy systems. the method does not involve linearization about operating point. first the polynomial fuzzy model of pitch channel of the flight body is derived. next, using polynomial fuzzy system methodology the controller is design such that the outputs of the nonli...

To prove that a polynomial is nonnegative on Rn one can try to show that it is a sum of squares of polynomials. The latter problem is now known to be reducible to a semidefinite programming computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials ...