We solve saddle point problems by accelerated versions of the classical Cimmino’s method. For that, saddle point problems are reviewed and reformulated as best approximation problems. In that setting, low-cost optimization techniques are used for accelerating Cimmino’s method. Encouraging numerical results coming from KKT systems, Stokes problems and domain decomposition problems are presented....

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate KKT and Approximate Gradient Projection conditions are analyzed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed.

We prove local and global convergence results for an interior-point methodapplied to mathematical programs with equilibrium constraints. The globalresult shows the algorithm minimizes infeasibility regardless of startingpoint, while one result proves local convergence when penalty functions areexact; another local result proves convergence when the solution is not evena KKT point.

We consider the application of the globalized semismooth Newton method to the solution of (the KKT conditions of) quasi variational inequalities. We show that the method is globally and locally superlinearly convergent for some important classes of quasi variational inequality problems. We report numerical results to illustrate the practical behavior of the method.

Homogeneous programming is an important class of optimization problems. The purpose of this note is to give a truly equivalent characterization of KKT-points of homogeneous programming problems, which corrects a result given by Lasserre and Hiriart-Urruty in Ref. 9.

The paper presents a reformulation of the Karush-KuhnTucker (KKT) system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by proposed method, is proved. The preliminary numerical results with a problems test set are presented.

This article points out an important source of inefficiency in Platt’s sequential minimal optimization (SMO) algorithm that is caused by the use of a single threshold value. Using clues from the KKT conditions for the dual problem, two threshold parameters are employed to derive modifications of SMO. These modified algorithms perform significantly faster than the original SMO on all benchmark d...