We are concerned with Lipschitzian error bounds and Lipschitzian stability properties for solutions of a complementarity system. For this purpose, we deal with a nonsmooth slack-variable reformulation of the complementarity system, and study conditions under which the reformulation serves as a local error bound for the solution set of the complementarity system. We also discuss … Read more
Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension increases. Recent advances in developing randomized derivative-free techniques have tackled this issue by working in low-dimensional subspaces that are … Read more
Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been … Read more
Bilevel optimization problems are hierarchical problems with a constraint set which is a subset of the graph of the solution set mapping of a second optimization problem. To investigate their properties and derive solution algorithms, their transformation into single-level ones is necessary. For this, various approaches have been developed. The rst and most often used … Read more
In this work, we introduce new direct search schemes for the solution of bilevel optimization (BO) problems. Our methods rely on a fixed accuracy black box oracle for the lower-level problem, and deal both with smooth and potentially nonsmooth true objectives. We thus analyze for the first time in the literature direct search schemes in … Read more
Restricted normal cones are of interest, for instance, in the theory of local error bounds, where they have recently been used to characterize the exis- tence of a constrained Lipschitzian error bound. In this paper, we establish rela- tions between two concepts for restricted normals. The first of these concepts was introduced in the late … Read more
We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of this class is that the inner function can be merely lower semicontinuous instead of continuously differentiable. It covers a range of … Read more
In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent developments from nonlinear programming to the context of optimization on Riemannian manifolds, including equality and inequality constraints. Many research have … Read more
\(\) The main computational cost per iteration of adaptive cubic regularization methods for solving large-scale nonconvex problems is the computation of the step \(s_k\), which requires an approximate minimizer of the cubic model. We propose a new approach in which this minimizer is sought in a low dimensional subspace that, in contrast to classical approaches, … Read more