More Like this

1. A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

   https://doi.org/10.3934/jcd.2022019  

   Jones, Morgan; Peet, Matthew M. (January 2022, Journal of Computational Dynamics)

   Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose \begin{document}$ 1 $$\end{document}$-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator. 

   more » « less
2. High-Resolution Modeling of the Fastest First-Order Optimization Method for Strongly Convex Functions

   https://doi.org/10.1109/CDC42340.2020.9304444  

   Sun, Boya; George, Jemin; Kia, Solmaz (December 2020, 2020 59th IEEE Conference on Decision and Control (CDC))

   null (Ed.)

   Motivated by the fact that the gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs), here we derive an ODE representation of the accelerated triple momentum (TM) algorithm. For unconstrained optimization problems with strongly convex cost, the TM algorithm has a proven faster convergence rate than the Nesterov's accelerated gradient (NAG) method but with the same computational complexity. We show that similar to the NAG method, in order to accurately capture the characteristics of the TM method, we need to use a high-resolution modeling to obtain the ODE representation of the TM algorithm. We propose a Lyapunov analysis to investigate the stability and convergence behavior of the proposed high-resolution ODE representation of the TM algorithm. We compare the rate of the ODE representation of the TM method with that of the NAG method to confirm its faster convergence. Our study also leads to a tighter bound on the worst rate of convergence for the ODE model of the NAG method. In this paper, we also discuss the use of the integral quadratic constraint (IQC) method to establish an estimate on the rate of convergence of the TM algorithm. A numerical example verifies our results. 

   more » « less
3. Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood

   Jin, Qiujiang; Koppel, Alec; Rajawat, Ketan; Mokhtari, Aryan. (January 2022, International Conference on Machine Learning)

   Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of O((1/sqrt{t})^t) for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more updates compared to the number of iterations that BFGS requires for a local superlinear rate. This is due to the fact that in Greedy-BFGS the Hessian is directly approximated and the Newton direction approximation may not be as accurate as the one for BFGS. In this paper, we close this gap and present a novel BFGS method that has the best of both worlds in that it leverages the approximation ideas of both BFGS and GreedyBFGS to properly approximate the Newton direction and the Hessian matrix simultaneously. Our theoretical results show that our method outperforms both BFGS and Greedy-BFGS in terms of convergence rate, while it reaches its quadratic convergence rate with fewer steps compared to Greedy-BFGS. Numerical experiments on various datasets also confirm our theoretical findings. 

   more » « less
4. Positivstellensätze and Moment Problems with Universal Quantifiers

   https://doi.org/10.1287/moor.2024.0402  

   Hu, Xiaomeng; Klep, Igor; Nie, Jiawang (May 2025, Mathematics of Operations Research)

   This paper studies Positivstellensätze and moment problems for sets K that are given by universal quantifiers. Let Q be the closed set of universal quantifiers. Fix a finite nonnegative Borel measure whose support is Q and assume it satisfies the multivariate Carleman condition. First, we prove a Positivstellensatz with universal quantifiers: if a polynomial f is positive on K, then f belongs to the associated quadratic module, under the archimedeanness assumption. Second, we prove some necessary and sufficient conditions for a full (or truncated) multisequence to admit a representing measure supported in K. In particular, the classical flat extension theorem of Curto and Fialkow is generalized to truncated moment problems on such a set K. Third, we present applications of the above Positivstellensatz and moment problems in semi-infinite optimization, where feasible sets are given by infinitely many constraints with universal quantifiers. This results in a new hierarchy of Moment-SOS relaxations. Its convergence is shown under some usual assumptions. The quantifier set Q is allowed to be non-semialgebraic, which makes it possible to solve some optimization problems with non-semialgebraic constraints. Funding: X. Hu and J. Nie are partially supported by the NSF [Grant DMS-2110780]. I. Klep is supported by the Slovenian Research Agency program P1-0222 [also Grants J1-50002, J1-60011, J1-50001, J1-2453, N1-0217, and J1-3004] and was partially supported by the Marsden Fund Council of the Royal Society of New Zealand. I. Klep’s work was partly performed within the project COMPUTE, funded within the QuantERA II program that has received funding from the EU’s H2020 research and innovation program under the GA No 101017733. 

   more » « less
5. The Kloosterman circle method and weighted representation numbers of quadratic forms

   https://doi.org/10.1007/s40687-025-00544-4  

   Jones, Edna (August 2025, Research in the Mathematical Sciences)

   Abstract We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local–global principle for certain Kleinian sphere packings. 

   more » « less