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Second-order dynamical systems are important tools for solving optimization problems, and most of existing works in this field have focused on unconstrained optimization problems. In this paper, we propose an inertial primal-dual dynamical…
The convergence rate of various first-order optimization algorithms is a pivotal concern within the numerical optimization community, as it directly reflects the efficiency of these algorithms across different optimization problems. Our…
In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our…
In this paper, we present a novel framework for deriving the evolution equation of the level set function in topology optimization, departing from conventional Hamilton-Jacobi based formulations. The key idea is the introduction of an…
In this paper we study a second order dynamical system with variable coefficients in connection to the minimization problem of a smooth nonconvex function. The convergence of the trajectories generated by the dynamical system to a critical…
This paper is devoted to a new modification of a recently proposed adaptive stochastic mirror descent algorithm for constrained convex optimization problems in the case of several convex functional constraints. Algorithms, standard and its…
In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the…
We develop a new Riemannian descent algorithm that relies on momentum to improve over existing first-order methods for geodesically convex optimization. In contrast, accelerated convergence rates proved in prior work have only been shown to…
For the general problem of minimizing a convex function over a compact convex domain, we will investigate a simple iterative approximation algorithm based on the method by Frank & Wolfe 1956, that does not need projection steps in order to…
This paper proposes novel primal-dual dynamical systems for solving linear equality constrained convex optimization. First, we introduce a primal-dual dynamical system with implicit Hessian damping, which can neutralize the transversal…
Topology optimization of natural convection problems is computationally expensive, due to the large number of degrees of freedom (DOFs) in the model and its two-way coupled nature. Herein, a method is presented to reduce the computational…
We consider the minimization of submodular functions subject to ordering constraints. We show that this optimization problem can be cast as a convex optimization problem on a space of uni-dimensional measures, with ordering constraints…
In this article, we consider the problem of unconstrained time-varying convex optimization, where the cost function changes with time. We provide an in-depth technical analysis of the problem and argue why freezing the cost at each time…
Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…
We study the homogenization of first-order Hamilton-Jacobi equations on an infinite-dimensional Hilbert space, motivated by systems of infinitely many indistinguishable particles on the torus. A central difficulty is that the analysis takes…
This paper develops a primal-dual dynamical system where the coefficients are designed in closed-loop way for solving a convex optimization problem with linear equality constraints. We first introduce a ``second-order primal" +…
In this paper, we establish some second order necessary/sufficient optimality conditions for optimal control problems of stochastic evolution equations in infinite dimensions. The control acts on both the drift and diffusion terms and the…
We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential…
Optimizing problems in a distributed manner is critical for systems involving multiple agents with private data. Despite substantial interest, a unified method for analyzing the convergence rates of distributed optimization algorithms is…
In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. We demonstrate that when the monotone operator flow is augmented with a Tikhonov regularization term, the…