Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Convex relaxation of mixture regression with efficient algorithms

We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semidefinite matrix variable, we reformulate the problem to eliminate the need for general semidefinite programming. In particular, we provide two reformulations that admit fast algorithms. The first is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

Techniques for Optimum Design of Actively Controlled Structures Including Topological Considerations

The design and performance of complex engineering systems often depends on several conflicting objectives which, in many cases, cannot be represented as a single measure of performance. This thesis presents a multi-objective formulation for a comprehensive treatment of the structural and topological considerations in the design of actively controlled structures. The dissertation addresses three main problems. The first problem deals with optimum placement of actuators in actively controlled structures. The purpose of control is to reduce the vibrations when the structure is subjected to a disturbance. In order to mitigate the structural vibrations as quickly as possible, it is necessary to place the actuators at locations such that their ability to control the vibrations is maximized. Since the actuator locations are discrete (0-1) variables, a genetic algorithm based approach is used to solve the resulting optimization problem. The second problem this dissertation addresses is the multi-objective design of actively controlled structures. Structural weight, controller performance index and energy dissipated by the actuators are considered as the objective functions. It is assumed that a hierarchical structure exist between the actuator placement and structural-control design objective functions with the actuator placement problem considered being more important. The resulting multi-objective optimization problem is solved using Stackelberg game and cooperative game theory approaches. The exchange of information between different levels of the multi-level problem is done by constructing the rational reaction set of follower solution using design of experiments and response surface methods. The third problem addressed in this dissertation is the optimization of structural topology in the context of structural/control system design. Despite the recognition that an optimization of topology can significantly improve structural performance, most of the work in design of actively controlled structures has been done with structures of a known topology. The combined topology and sizing optimization of actively controlled structures is also considered in this thesis. The approach presented involves the determination of optimum topology followed by a sizing and control system optimization of the optimum topology. Using two numerical examples, it is shown that a simultaneous consideration of topological, control and structural aspects yields solutions that outperform designs when topological considerations are neglected