Recent developments in nonlinear optimization algorithms

This article provides a condensed overview of some of the major today's features (both classical or recently developed), used in the design and development of algorithms to solve nonlinear continuous optimization problems. We first consider the unconstrained optimization case to introduce the line-search and trust-region approaches as globalization techniques to force an algorithm to convergence from any starting point. We then focus on constrained optimization and give the main ideas of two classes of methods, the Sequential Quadratic Programming (SQP) methods and the interior-point methods. We briefly discuss why interior-point methods are now so popular, in their primal-dual version, while they have been abandoned about twenty years ago. We also introduce a newly emerging alternative, called filter method, to the use of a merit function as a tool to measure progress from one iteration to the next in constrained optimization. We relate some of the most widely used nonlinear optimization solvers to the algorithmic features presented, and we finally give some useful tools for an easy and comprehensive access to recent developments in nonlinear optimization algorithms and to practical solvers and their performance.