We present a faster interior-point method for optimizing sum-of-squares (SOS) polynomials, which are central tool in polynomial optimization and capture convex programming the Lasserre hierarchy. Let $$p = \sum _i q^2_i$$ be an n-variate SOS of degree 2d. Denoting by $$L:= \left( {\begin{array}{c}n+d\\ d\end{array}}\right) $$ $$U:= {\begin{array}{c}n+2d\\ 2d\end{array}}\right) dimensions vector...

Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials via the definition of convexity, its first order characterization, and its second order characterization are equivalent. These three equivalent algebraic conditions, henceforth referred to as sos-convexity, can be checked by semidefinite programming w...

The contributions of the first half of this thesis are on the computational and algebraic aspects of convexity in polynomial optimization. We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1...

We provide a new degree bound on the weighted sum-of-squares (SOS) polynomials for Putinar–Vasilescu's Positivstellensatz. This leads to another Positivstellensatz saying that if f is polynomial of at most 2 d nonnegative semialgebraic set having nonempty interior defined by finitely many inequalities g j ( x ) ≥ 0 , = 1 … m with : L − ‖ some > then there exist positive constants c ¯ and depend...

This study addresses the issue of robust dynamic output feedback control (DOF) for polynomial Takagi–Sugeno (T–S) fuzzy systems in Finite Frequency (FF) domain. Sufficient conditions designing DOF are derived terms sum squares (SOS). The proposed strategy is built FF domain to reduce conservation-generated by techniques established whole frequency In addition, there no transformation matrices o...

Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for two-player games. By contrast, a simple quasi-polynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than by changing their strategy. The LMM algorithm ca...

Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather some invariant subset of the state space called attractor set. For a given ODE, in general, existence, shape and structure sets ODE unknown. Fortunately, sublevel Lyapunov functions can provide bounds on ODEs. In this paper we propose new character...

This paper presents an observer-based sensorless drive system for the position control of permanent magnet linear synchronous motor (PMLSM). To achieve purpose and maintain stability global system, is designed using sum-of-squares (SOS) method. With help SOSTOOLS toolbox, algorithm can be established without need complicated mathematical derivation. First, to apply SOS technique, model treated ...

In this letter, we investigate the verification of dissipativity properties for polynomial systems without an explicitly identified model but directly from noise-corrupted measurements. Contrary to most data-driven approaches nonlinear systems, determine over all finite time horizons using noisy input-state data. To end, propose two noise characterizations deduce data-based set-membership repre...

Awidely usedmethod for determiningwhether amultivariate polynomial is a sum of squares of polynomials (SOS), called SOS decomposition, is to decide the feasibility of corresponding semi-definite programming (SDP) problem which can be efficiently solved in theory. In practice, although existing SDP solvers can work out some problems of big scale, the efficiency and reliability of such method dec...