Exact penalties for variational inequalities with applications to nonlinear complementarity problems

In this paper, we present a new reformulation of the KKT system associated to a variational inequality as a semismooth equation. The reformulation is derived from the concept of differentiable exact penalties for nonlinear programming. The best theoretical results are presented for nonlinear complementarity problems, where simple, verifiable, conditions ensure that the penalty is exact. We close the paper with some preliminary computational tests on the use of a semismooth Newton method to solve the equation derived from the new reformulation. We also compare its performance with the Newton method applied to classical reformulations based on the Fischer-Burmeister function and on the minimum. The new reformulation combines the best features of the classical ones, being as easy to solve as the reformulation that uses the Fischer-Burmeister function while requiring as few Newton steps as the one that is based on the minimum.

On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.

Short fiction of Thomas Hardy: an exploration and assessment

Hardy's novels and poetry have received extensive criticism, in due proportion to their merit. Hardy's short stories, however, have been virtually excluded from the annals of Hardy criticism, even though Hardy wrote over forty short stories, several of which are truly outstanding. In part, the reason for this neglect is because of the neglected state of the short story in Victorian England. Short fiction, published mainly only in periodicals and never collected in volume form, was obscured to a large extent by the highly popular serial novel. This thesis examines Thomas Hardy's short stories in the context of both the Victorian period and the Victorian short story genre, and explores the ways in which Thomas Hardy improved upon and deviated from some of the common types of short fiction being written in his day.