Related papers: Convergence of Newton's method in shape optimisati…
This paper proposes several novel optimization algorithms for minimizing a nonlinear objective function. The algorithms are enlightened by the optimal state trajectory of an optimal control problem closely related to the minimized objective…
In this paper we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with H\"older continuous Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG) method…
In this paper, we propose a descent method for composite optimization problems with linear operators. Specifically, we first design a structure-exploiting preconditioner tailored to the linear operator so that the resulting preconditioned…
This study proposes an efficient Newton-type method for the optimal control of switched systems under a given mode sequence. A mesh-refinement-based approach is utilized to discretize continuous-time optimal control problems (OCPs) and…
We analyze the convergence rate of the randomized Newton-like method introduced by Qu et. al. (2016) for smooth and convex objectives, which uses random coordinate blocks of a Hessian-over-approximation matrix $\bM$ instead of the true…
This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape…
Stochastic optimisation problems minimise expectations of random cost functions. We use 'optimise then discretise' method to solve stochastic optimisation. In our approach, accurate quadrature methods are required to calculate the…
Quasi-Newton methods are widely used for solving convex optimization problems due to their ease of implementation, practical efficiency, and strong local convergence guarantees. However, their global convergence is typically established…
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…
This paper aims to develop a Newton-type method to solve a class of nonconvex composite programs. In particular, the nonsmooth part is possibly nonconvex. To tackle the nonconvexity, we develop a notion of strong prox-regularity which is…
We study a Newton-like method for the minimization of an objective function that is the sum of a smooth convex function and an l-1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton…
Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values…
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…
We propose a second-order method for unconditional minimization of functions $f(z)$ of complex arguments. We call it the Mixed Newton Method due to the use of the mixed Wirtinger derivative $\frac{\partial^2f}{\partial\bar z\partial z}$ for…
The cubic regularized Newton method of Nesterov and Polyak has become increasingly popular for non-convex optimization because of its capability of finding an approximate local solution with second-order guarantee. Several recent works…
For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…
In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape…
In this work, we investigate stochastic quasi-Newton methods for minimizing a finite sum of cost functions over a decentralized network. In Part I, we develop a general algorithmic framework that incorporates stochastic quasi-Newton…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…