Continuous optimization methods for convex mixed-integer nonlinear programming

The topic of this dissertation is the design of fast branch-and-bound algorithms that use intelligently adapted approaches from continuous optimization for solving convex mixed-integer nonlinear programming problems. This class of optimization problems is NP-hard and polynomial-time algorithms for these problems are therefore unlikely to exist (unless P=NP). The importance of this class is highlighted by the fact that many real-world applications can be modeled as a (convex) mixed-integer nonlinear programming problem. Currently, there are several standard techniques such as outer approximation that are used within recent state-of-the-art software. Although all these algorithms include sophisticated improvements such as primal heuristics and effective preprocessing, they do not take into account the large gap between the algorithmic performance of NLP and IP solvers. While NLP solvers are well-engineered for large-scale problems, MIP problems of similar sizes are by far harder to solve in practice. Therefore, when using NLP techniques within MIP solvers, these NLP algorithms have to be adjusted to handle small-size instances effectively. Taking this problem into account, we present three branch-and-bound algorithms, based on a former work by Buchheim et al. (2012) on unconstrained convex quadratic integer programming problems. The main strategies used within this branch-andbound framework include extensive preprocessing and fast incremental computations, aiming at a very fast enumeration of the nodes. The first algorithm we present is designed to solve convex quadratic mixed-integer programming problems with linear inequality constraints and is based on a new feasible active set algorithm applied to the dual of the continuous relaxation. This active set algorithm is tailored for the continuous problem and fully exploits its structure. Furthermore, a warmstarting procedure is used to reduce the number of active set iterations per node. The second algorithm we introduce is an approach called quadratic outer approximation for solving box-constrained convex mixed-integer nonlinear programming problems. It extends the classical outer approximation by using quadratic underestimators leading to a faster convergence in practice. Finally, the last algorithm we devise is aimed at a class of mean-risk portfolio optimization problems that can be modeled as convex mixed-integer programming problems with a single linear budget constraint. For this application we propose a branch-and-bound scheme using a modified Frank-Wolfe type algorithm to solve the node relaxations. Similarly to the branch-and-bound algorithms mentionded above we exploit the simplicity of the relaxations to enumerate the nodes as quickly as possible rather than focussing on strong dual bounds. We implemented all three algorithms and compared their performance with several state-of-the art approaches. Our extensive computational studies show that all new approaches presented in this thesis are able to effectively solve large classes of real-world instances

Failure Processes in Elastic Fiber Bundles

The fiber bundle model describes a collection of elastic fibers under load. the fibers fail successively and for each failure, the load distribution among the surviving fibers change. Even though very simple, the model captures the essentials of failure processes in a large number of materials and settings. We present here a review of fiber bundle model with different load redistribution mechanism from the point of view of statistics and statistical physics rather than materials science, with a focus on concepts such as criticality, universality and fluctuations. We discuss the fiber bundle model as a tool for understanding phenomena such as creep, and fatigue, how it is used to describe the behavior of fiber reinforced composites as well as modelling e.g. network failure, traffic jams and earthquake dynamics.Comment: This article has been Editorially approved for publication in Reviews of Modern Physic

A Lattice Basis Reduction Approach for the Design of Finite Wordlength FIR Filters

International audienceMany applications of finite impulse response (FIR) digital filters impose strict format constraints for the filter coefficients. Such requirements increase the complexity of determining optimal designs for the problem at hand. We introduce a fast and efficient method, based on the computation of good nodes for polynomial interpolation and Euclidean lattice basis reduction. Experiments show that it returns quasi-optimal finite wordlength FIR filters; compared to previous approaches it also scales remarkably well (length 125 filters are treated in < 9s). It also proves useful for accelerating the determination of optimal finite wordlength FIR filters