Andrew Eberhard

Applying Generalised Convexity Notions to Jets

By viewing jets as an inductive extension of the first order proximal analysis of loffe and Mordukhovich one is able to extend many constructs appearing in the first order theories to the second order case. We find that the second order Dini derivative is the rank one support of a set of matrices whose rank one representer coincides with the subjets. The rank one support plays a key role in this development as does the infimal convolution (or negative the Φ2-conjugate). Both of these concepts interact in unexpected ways. In particular we find that the infimal convolution smoothing of a function makes the subjet set more rotund (in a rank one sense). We also demonstrate a number of approximation theorems for subjets involving the infimal convolution.

Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.

A geometrical insight on pseudoconvexity and pseudomonotonicity

Generalised convexity is revisited from a geometric point of view. A substitute to the subdifferential is proposed. Then generalised monotonicity is considered. A representation of generalised monotone maps allows us to obtain a symmetry between maps and their inverses. Finally, maximality of generalised monotone maps is analysed.

Prox-Regularity and Subjets

Here we study the subjets of the class of prox-regular functions as introduced by Rockafellar and Poliquin. The infimal convolution turns out be an invaluable tool in characterizing these second order non-smooth derivatives for this class of functions. A number of relationships with previously defined second order derivatives are explored. Some unconstrained necessary and sufficient optimality conditions utilizing subjets are derived. Then via a calculus these are applied to the constrained case using the composite programming formulation.

Some Sufficient Optimality Conditions in Nonsmooth Analysis

In this paper we study some second order conditions that may be added to the first order nessessary optimality condition

A new approach to the feasibility pump in mixed integer programming

0\in\partial_{p}f(\bar{x})

A Numerical Method for a Class of Mixed Switching and Impulsive Optimal Control Problems

(with

On the augmented Lagrangian dual for integer programming

\partial_{p}f

Maximal quasimonotonicity and dense single-directional properties of quasimonotone operators

denoting the proximal subdifferential) in order to obtain a sufficient condition for a strict local minimum for extended-real-valued, nonsmooth functions. Three different types of second order conditions are investigated, all based on a different second order subdifferential. Namely, the subhessian, the graphical derivative, and the (contingent) coderivative to the proximal subdifferential.

First-Order and Second-Order Optimality Conditions for Nonsmooth Constrained Problems via Convolution Smoothing

The feasibility pump is a recent, highly successful heuristic for general mixed integer linear programming problems. We show that the feasibility pump heuristic can be interpreted as a discrete ver...