Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

Abstract

In this paper, the fixed-time stability of a novel proximal dynamical system is investigated for solving mixed variational inequality problems. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that the solution of the proposed proximal dynamical system exists in the classical sense, is uniquely determined and converges to the unique solution of the associated mixed variational inequality problem in a fixed time. As a special case, the proposed proximal dynamical system reduces to a novel fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumptions of strong monotonicity are relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature are shown as special cases of the proposed proximal dynamical system considered in this paper. Finally, several numerical examples are presented that corroborate the fixed-time convergent behavior of the proposed proximal dynamical system.

Fixed-Time Extremum Seeking

“…Thus, we analyze again the stability properties of system (35) by studying the properties of system (36), and using standard robustness results for perturbed ODEs with a continuous righthand side, e.g., [40,Thm. 7.21].…”

“…From here, the stability analysis follows the exact same steps of the proof of Theorem 1, using [36, Thm.2] and [40,Thm. 7.21] successively to link the stability properties of the original dynamics (10) and its average perturbed dynamics (35). Local existence of complete solutions from the neighborhood N follows directly by the local stability properties of (36) and the closeness of solutions between (36) and (31).…”

“…Remark 2. Unlike the model-based fixed-time gradient dynamics of [3] and [35], the FTGES dynamics are model-free and only need measurements of the cost function φ. Because of this, their stability properties are highly dependent on an appropriate ordered tuning of the parameters (ε 2 a, 1 ).…”

“…The structure of the NFTES is similar to the multi-variable Newton-based extremum seeking controller considered in [38], where, on average, the state ξ 2 serves as an estimation of the gradient, and the state ξ 1 approximates the inverse of the Hessian ∇ 2 φ(z). However, the NFTES dynamics have two main differences: a) for admissible parameters (q 1 , q 2 ), the dynamics of u are continuous but not Lipschitz continuous, and they aim to approximate a Newton-based flow with fixedtime convergence properties [3], [35] instead of the standard Newton-based flow ẋ = −∇ 2 φ(x) −1 ∇φ(x) considered in [38]; b) the excitation signals N (•) and M (•) are both generated by a time-invariant linear oscillator, which facilitates the analysis of the algorithm via averaging theory for non-smooth φ(z) time-invariant dynamical systems [36]. A scheme illustrating the FTNES dynamics is shown in Figure 5.…”

“…Algorithms with fixed time stability properties have been recently studied in [29], [30], [31], [32], [33], [3] and [34]. As shown in [31] and [33], this type of convergence property can be established via Lyapunov functions for a class of continuous-time dynamical systems, which has opened the door to novel optimization algorithms characterized by continuous vector fields with non-asymptotic convergence properties; e.g., [3] and [35]. However, to the best of our knowledge, all existing optimization algorithms with fixed-time stability properties are model-based and require access to the first or second derivatives of the cost functions.…”