On barrier and modified barrier multigrid methods for 3d topology optimization

One of the challenges encountered in optimization of mechanical structures, in particular in what is known as topology optimization, is the size of the problems, which can easily involve millions of variables. A basic example is the minimum compliance formulation of the variable thickness sheet (VTS) problem, which is equivalent to a convex problem. We propose to solve the VTS problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\ Polyak and later studied by Ben-Tal and Zibulevsky and others. The most computationally expensive part of the algorithm is the solution of linear systems arising from the Newton method used to minimize a generalized augmented Lagrangian. We use a special structure of the Hessian of this Lagrangian to reduce the size of the linear system and to convert it to a form suitable for a standard multigrid method. This converted system is solved approximately by a multigrid preconditioned MINRES method. The proposed PBM algorithm is compared with the optimality criteria (OC) method and an interior point (IP) method, both using a similar iterative solver setup. We apply all three methods to different loading scenarios. In our experiments, the PBM method clearly outperforms the other methods in terms of computation time required to achieve a certain degree of accuracy

Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g. the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: Firstly, we derive a dual representation of the considered problem and prove that strong duality holds. Secondly, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real world instance.Comment: Revised version. Accepted for publication in "Mathematical Finance

Multitask learning without label correspondences

We propose an algorithm to perform multitask learning where each task has potentially distinct label sets and label correspondences are not readily available. This is in contrast with existing methods which either assume that the label sets shared by different tasks are the same or that there exists a label mapping oracle. Our method directly maximizes the mutual information among the labels, and we show that the resulting objective function can be efficiently optimized using existing algorithms. Our proposed approach has a direct application for data integration with different label spaces for the purpose of classification, such as integrating Yahoo! and DMOZ web directories

Local Linear Convergence Analysis of Primal-Dual Splitting Methods

In this paper, we study the local linear convergence properties of a versatile class of Primal-Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework we first show that (i) the sequences generated by Primal-Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal-Dual splitting can be specialized to cover existing ones on Forward-Backward splitting and Douglas-Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning, etc. The demonstration not only verifies the local linear convergence behaviour of Primal-Dual splitting methods, but also the insights on how to accelerate them in practice