LANCS Workshop on Modelling and Solving Complex Optimisation Problems

Towards optimal Newton-type methods for nonconvex smooth optimization Coralia Cartis Coralia.Cartis (at) ed.ac.uk School of Mathematics, Edinburgh University We show that the steepest-descent and Newton methods for unconstrained non-convex optimization, under standard assumptions, may both require a number of iterations and function evaluations arbitrarily close to the steepest-descent’s global worst-case complexity bound. This shows that the latter upper bound is essentially tight for steepest descent and that the Newton method may be as slow as the steepest-descent method in the worst case. Next, the cubic regularization of the Newton method (Griewank (1981), Nesterov & Polyak (2006)) is considered and extended to large-scale problems, while preserving the same order of its improved worst-case complexity (by comparison to that of steepest-descent). This improved worst-case bound is also shown to be essentially tight. We further address the optimality of cubic regularization from a worst-case complexity point of view amongst a class of secondorder methods. An extension of cubic regularization to bound-constrained problems will be presented that satisfies the unconstrained function-evaluation complexity bound of cubic regularization. Joint work with Nick Gould (Rutherford Appleton Laboratory, UK) and Philippe Toint (Namur University, Belgium). Convexification of mixed-integer quadratically constrained quadratic programs Laura Galli l.galli (at) unibo.it DEIS, Bologna University In this study we explore when and how one can obtain convex reformulations of mixedinteger quadratically constrained quadratic programs (MIQCQP). In particular, we show that semidefinite programming (SDP) can be used both to guarantee convexity of MIQCQP functions, and to perturb them in such a way that the lower bound obtained from the continuous relaxation of the problem is equal to the SDP bound (sometimes even better). A serious problem is however posed by continuous variables that have quadratic terms in one or more constraints. Such variables can cause convex reformulations to be weak, or even prevent them from existing at all. This is not surprising, however, given that non-convex QCQP, a purely continuous problem, is an NP-hard global optimisation problem. Joint work with Adam Letchford (Lancaster University).