# EPSRC Symposium Capstone Conference

**Mini Symposium on Optimization**

Organiser: Coralia Cartis

Thursday 2 July 2009

Speakers

**Pierre-Antoine Absil** (Louvain) *Accelerated optimization methods and their convergence*

Abstract: Let $T$ be a descent mapping for a differentiable real-valued function $f$. Let $(x_k)$ be a ``$T$-accelerated" sequence, namely, there is $c>0$ such that, for all $k$, $f(x_k) - f(x_{k+1}) \geq c (f(x_k) - f(T(x_k)))$. We propose weak conditions on $T$ that ensure that every accumulation point of $(x_k)$ is a critical point of $f$. This theory has applications, e.g., in the analysis of eigenvalue algorithms and of certain types of switched discrete-time systems.

**Frank E. Curtis** (Courant) * Inexact Newton Methods for Nonlinear Constrained Optimization*

Inexact Newton methods play a fundamental role in the solution of large-scale unconstrained optimization problems and nonlinear equations. The key advantage of these approaches is that they can be made to emulate the properties of Newton's method while allowing flexibility in the computational cost per iteration. Due to the multi-objective nature of *constrained* optimization problems, however, that require an algorithm to find both a feasible and optimal point, it has not been known how to successfully apply an inexact Newton method within a globally convergent framework. In this talk, we present a new methodology for applying inexactness to the most fundamental iteration in constrained optimization: a line-search primal-dual Newton algorithm. We illustrate that the choice of merit function is crucial for ensuring global convergence and discuss novel techniques for handling non-convexity, ill-conditioning, and the presence of inequality constraints in such an environment. Preliminary numerical results are presented for PDE-constrained optimization problems.

**Raphael Hauser **(Oxford) *Adversarial smoothed analysis*

Coauthors: Martin Lotz, Felipe Cucker.

We present smoothed analysis bounds for conic condition numbers under a general class of radially symmetric perturbation distributions around an input data point of a numerical problem. The generality of our theory is such that it applies to the conditioning of a family of important problems in numerical analysis and to perturbation distributions whose density can have a singularity at the center of the perturbation, a class of distributions we call {\em adversarial}.

**Michal Kocvara** (Birmingham) *On the solution of large-scale semidefinite problems in structural optimization*

We propose a new approach for solving the structural optimization problems where stability of the optimal structure is enforced either by vibration control, that is control of the fundamental eigenfrequency, or by control of the critical buckling load. The problem leads to a large-scale nonlinear programming problem. When stability control in enforced, a large and sparse semidefinite constraint is added to this problem; in the case of the critical buckling load, this constraint is nonlinear. Such problems are demanding from the viewpoint of computational cost, due to the dimension of real life instances. In particular, the problems are characterized by a large number of variables and, at the same time, a high dimension of the matrix constraint. This kind of problems is impossible to solve by recent semidefinite programming software based on interior point methods. Our approach is based on a nonlinear reformulation of the positive semidefinite constraint which was first proposed by Fletcher. It applies to linear as well as nonlinear semidefinite constraints. The resulting nonlinear program is solved by a variant of the Method of Moving Asymptotes, which is popular and provably efficient in Structural Optimisation. A working set strategy is also introduced in order to save computational effort. This is a joint Work with Claudio Bogani (University of Birmingham) and Michael Stingl (University of Erlangen).

*Further details will be added as they become available *