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We propose a communication and computation efficient second-order method for distributed optimization. For each iteration, our method only requires $\mathcal{O}(d)$ communication complexity, where $d$ is the problem dimension. We also…
A very simple first-order algorithm is proposed for solving nonlinear optimization problems with deterministic nonlinear equality constraints. This algorithm adaptively selects steps in the plane tangent to the constraints or steps that…
We present iterative solvers to approximate the solution of numerical schemes for stochastic Stefan problems. After briefly talking about the convergence results, we tackle the question of efficient strategies for solving the nonlinear…
Solving complex optimization problems in engineering and the physical sciences requires repetitive computation of multi-dimensional function derivatives. Commonly, this requires computationally-demanding numerical differentiation such as…
We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to…
In this paper new descent line search iterative schemes for unconstrained as well as constrained optimization problems are developed using q-derivative. At every iteration of the scheme, a positive definite matrix is provided which is…
The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified…
We are concerned with the tensor equations whose coefficient tensor is an M-tensor. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend…
We propose a new type of multilevel method for solving eigenvalue problems based on Newton iteration. With the proposed iteration method, solving eigenvalue problem on the finest finite element space is replaced by solving a small scale…
Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary…
The recently introduced Gradient Methods with Memory use a subset of the past oracle information to create an accurate model of the objective function that enables them to surpass the Gradient Method in practical performance. The model…
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of…
An inexact Newton type method for numerical minimization of convex piecewise quadratic functions is considered and its convergence is analyzed. Earlier, a similar method was successfully applied to optimizaton problems arising in numerical…
We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the…
In this paper, we study Newton-conjugate gradient (Newton-CG) methods for minimizing a nonconvex function $f$ whose Hessian is $(H_f,\nu)$-H\"older continuous with modulus $H_f>0$ and exponent $\nu\in(0,1]$. Recently proposed Newton-CG…
We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould and Toint…
In this paper, we propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. The local convergence theorems are established…
A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation…
This paper proposes a novel approach for solving linear programs. We reformulate a primal-dual linear program as an unconstrained minimization of a convex and twice continuously differentiable merit function. When the optimal set of the…
The unsupervised task of aligning two or more distributions in a shared latent space has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. Existing flow-based approaches estimate…