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In multiobjective optimization, inertial gradient systems accelerate convergence toward weakly Pareto optimal solutions. To achieve even faster convergence, we introduce a multiobjective inertial gradient system with time scaling (MITS),…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
Discrete updates of numerical partial differential equations (PDEs) rely on two branches of temporal integration. The first branch is the widely-adopted, traditionally popular approach of the method-of-lines (MOL) formulation, in which…
This paper proposes a new algorithm for solving constrained global optimization problems where both the objective function and constraints are one-dimensional non-differentiable multiextremal Lipschitz functions. Multiextremal constraints…
Recent studies have shown that fractional calculus is an effective alternative mathematical tool in various scientific fields. However, some investigations indicate that results established in differential and integral calculus do not…
The so-called fast inertial relaxation engine is a first-order method for unconstrained smooth optimization problems. It updates the search direction by a linear combination of the past search direction, the current gradient and the…
Classical global convergence results for first-order methods rely on uniform smoothness and the \L{}ojasiewicz inequality. Motivated by properties of objective functions that arise in machine learning, we propose a non-uniform refinement of…
We propose a family of optimization methods that achieve linear convergence using first-order gradient information and constant step sizes on a class of convex functions much larger than the smooth and strongly convex ones. This larger…
We consider the application of the type-I Anderson acceleration to solving general non-smooth fixed-point problems. By interleaving with safe-guarding steps, and employing a Powell-type regularization and a re-start checking for strong…
We study stochastic second-order methods for solving general non-convex optimization problems. We propose using a special version of momentum to stabilize the stochastic gradient and Hessian estimates in Newton's method. We show that…
We are concerned with a class of nonconvex and nonsmooth composite optimization problems, comprising a twice differentiable function and a prox-regular function. We establish a sufficient condition for the proximal mapping of a prox-regular…
This work proposes and analyzes a generalized acceleration technique for decreasing the computational complexity of using stochastic collocation (SC) methods to solve partial differential equations (PDEs) with random input data. The SC…
This paper introduces a class of model-free feedback methods for solving generic constrained optimization problems where the specific mathematical forms of the objective and constraint functions are not available. The proposed methods,…
Recently, there has been great interest in connections between continuous-time dynamical systems and optimization methods, notably in the context of accelerated methods for smooth and unconstrained problems. In this paper we extend this…
We propose a new first-order method for minimizing nonconvex functions with a Lipschitz continuous gradient and Hessian. The proposed method is an accelerated gradient descent with two restart mechanisms and finds a solution where the…
Treating optimization methods as dynamical systems can be traced back centuries ago in order to comprehend the notions and behaviors of optimization methods. Lately, this mind set has become the driving force to design new optimization…
This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for…
In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large…
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of…
In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many…