The Douglas-Rachford algorithm is a classical and a successful method for solving the feasibility problems. Here, we provide a region for global convergence of the algorithm for the feasibility problem involving a disc and a circle in the Euclidean space of dimension two. Citation 1. Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence … Read more
To improve the performance of the limited-memory variable metric L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed in [1, 2]. But the suitable extra updates need to be selected carefully, since the repeating process can be time consuming. We show that for the limited-memory variable metric BNS method, matrix … Read more
In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a … Read more
Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity … Read more
This paper deals with the minimization of large sum of convex functions by Inexact Newton (IN) methods employing subsampled Hessian approximations. The Conjugate Gradient method is used to compute the inexact Newton step and global convergence is enforced by a nonmonotone line search procedure. The aim is to obtain methods with affordable costs and fast … Read more
Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are … Read more
This work is devoted to studying a family of Middle Proximal Alternating Direction Method of Multipliers (MP-ADM) for solving multi-block constrained separable convex optimization. Such one-parameter family of MP-ADM combines both Jacobian and Gauss-Seidel types of alternating direction method, and proximal point techniques are only applied to the middle subproblems to promote the convergence. We … Read more
We present DFO-GN, a derivative-free version of the Gauss-Newton method for solving nonlinear least-squares problems. As is common in derivative-free optimization, DFO-GN uses interpolation of function values to build a model of the objective, which is then used within a trust-region framework to give a globally-convergent algorithm requiring $O(\epsilon^{-2})$ iterations to reach approximate first-order criticality … Read more
In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. For this purpose, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual … Read more
A block version of the BFGS variable metric update formula and its modifications are investigated. In spite of the fact that this formula satisfies the quasi-Newton conditions with all used difference vectors and that the improvement of convergence is the best one in some sense for quadratic objective functions, for general functions it does not … Read more