Steepest descent method using novel adaptive stepsizes for unconstrained nonlinear multiobjective programming

We propose new adaptive strategies to compute stepsizes for the steepest descent method to solve unconstrained nonlinear multiobjective optimization problems without employing any linesearch procedure. The resulting algorithms can be applied to a wide class of nonconvex unconstrained multi-criteria optimization problems satisfying a global Lipschitz continuity condition imposed on the gradients of all objectives. In … Read more

Nonmonotone line searches for unconstrained multiobjective optimization problems

In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the … Read more

Analysis of the Gradient Method with an Armijo-Wolfe Line Search on a Class of Nonsmooth Convex Functions

It has long been known that the gradient (steepest descent) method may fail on nonsmooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point. We give an … Read more

An implementation of the steepest descent method using retractions on riemannian manifolds

In 2008 Absil et al. published a book with optimization methods in Riemannian manifolds. The authors developed steepest descent, Newton, trust-region and conjugate gradients methods using an approximation of the geodesic called retraction. In this paper we present implementations of the of steepest descent method of Absil et al. using Matlab software. We show the … Read more

On the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming

We estimate the worst-case complexity of minimizing an unconstrained, nonconvex composite objective with a structured nonsmooth term by means of some first-order methods. We find that it is unaffected by the nonsmoothness of the objective in that a first-order trust-region or quadratic regularization method applied to it takes at most O($\epsilon^{-2}$) function-evaluations to reduce the … Read more