Physical Sciences and Mathematics Commons^™

Optimization Methods For Learning Graph-Structured Sparse Models, Baojian Zhou

Legacy Theses & Dissertations (2009 - 2024)

Learning graph-structured sparse models has recently received significant attention thanks to their broad applicability to many important real-world problems. However, such models, of more effective and stronger interpretability compared with their counterparts, are difficult to learn due to optimization challenges. This thesis presents optimization algorithms for learning graph-structured sparse models under three different problem settings. Firstly, under the batch learning setting, we develop methods that can be applied to different objective functions that enjoy linear convergence guarantees up to constant errors. They can effectively optimize the statistical score functions in the task of subgraph detection; Secondly, under stochastic learning setting, …

Variational Geometric Approach To Generalized Differential And Conjugate Calculi In Convex Analysis, Boris S. Mordukhovich, Nguyen Mau Nam, R. Blake Rector, T. Tran

Mathematics and Statistics Faculty Publications and Presentations

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes. Key words. Convex and variational analysis, Fenchel conjugates, normals and subgradients, coderivatives, convex calculus, optimal value functions.

The Dc Algorithm & The Constrained Fermat-Torricelli Problem, Nathan Peron Lawrence, George Blikas

Student Research Symposium

The theory of functions expressible as the Difference of Convex (DC) functions has led to the development of a rich field in applied mathematics known as DC Programming.We survey the work of Pham Dinh Tao and Le Thi Hoai An in order to understand the DC Algorithm (DCA) and its use in solving clustering problems. Further, we present several other methods that generalize the DCA for any norm. These powerful tools enable researchers to reformulate objective functions, not necessarily convex, into DC Programs.

The Fermat-Torricelli problem is visited in light of convex analysis and various norms. Pierre de Fermat proposed …

Geometric Approach To Convex Subdifferential Calculus, Boris S. Mordukhovich, Nguyen Mau Nam

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.

Transients Of Platoons With Asymmetric And Different Laplacians, Ivo Herman, Dan Martinec, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

We consider an asymmetric control of platoons of identical vehicles with nearest-neighbor interaction. Recent results show that if the vehicle uses different asymmetries for position and velocity errors, the platoon has a short transient and low overshoots. In this paper we investigate the properties of vehicles with friction. To achieve consensus, an integral part is added to the controller, making the vehicle a third-order system. We show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities. Simulations suggest that our system has a better performance than other nearest-neighbor scenarios. Moreover, …

Nonsmooth Algorithms And Nesterov's Smoothing Technique For Generalized Fermat-Torricelli Problems, Nguyen Mau Nam, Nguyen Thai An, R. Blake Rector, Jie Sun

Mathematics and Statistics Faculty Publications and Presentations

We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.

Screening And Sufficiency In Multiobjective Decision Problems With Large Alternative Sets, Michael D. Cote

Theses and Dissertations

Portfolio selection problems with combinatorially-large alternative sets can be impossible to evaluate precisely on a reasonable timescale. When portfolios require complex modeling for performance assessment, prohibitive computational processing times can result. Eliminating a small number of alternatives through an intelligent screening process can greatly reduce the number of alternative combinations, thereby decreasing a problem's evaluation time and cost. A methodology was developed for the class of hierarchical portfolio selection problems in which multiple objectives are all judged on the same sub-objectives. First, a novel capability-based alternative screening process was devised to identify and remove poor alternatives, thereby reducing the number …

Finite Sample Properties Of Minimum Kolmogorov-Smirnov Estimator And Maximum Likelihood Estimator For Right-Censored Data, Jerzy Wieczorek

Dissertations and Theses

MKSFitter computes minimum Kolmogorov-Smirnov estimators (MKSEs) for several different continuous univariate distributions, using an evolutionary optimization algorithm, and recommends the distribution and parameter estimates that best minimize the Kolmogorov-Smirnov (K-S) test statistic. We modify this tool by extending it to use the Kaplan-Meier estimate of the cumulative distribution function (CDF) for right-censored data. Using simulated data from the most commonly-used survival distributions, we demonstrate the tool's inability to consistently select the correct distribution type with right-censored data, even for large sample sizes and low censoring rates. We also compare this tool's estimates with the right-censored maximum likelihood estimator (MLE). While …

Determining The Orbit Locations Of Turkish Airborne Early Warning And Control Aircraft Over The Turkish Air Space, Nebi Sarikaya

Theses and Dissertations

The technology improvement affects the military needs of individual countries. The new doctrine of defense for many countries emphasizes detecting threats as far away as you can from your homeland. Today, the military uses both ground RADAR and Airborne Early Warning and Control (AEW&C) Aircraft. AEW&C aircraft has become vital to detect low altitude threats that a ground RADAR cannot detect because of obstacles on the earth. Turkey has ordered four AEW&C aircraft for her air defense system because of the lack of complete coverage by ground RADAR. This research provides optimal orbit locations that can be updated according to …

The Derivation Of Hybridizable Discontinuous Galerkin Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.

Optimizing The Replication Of Multi-Quality Web Applications Using Aco And Wolf, Judson C. Dressler

Theses and Dissertations

This thesis presents the adaptation of Ant Colony Optimization to a new NP-hard problem involving the replication of multi-quality database-driven web applications (DAs) by a large application service provider (ASP). The ASP must assign DA replicas to its network of heterogeneous servers so that user demand is satisfied and replica update loads are minimized. The algorithm proposed, AntDA, for solving this problem is novel in several respects: ants traverse a bipartite graph in both directions as they construct solutions, pheromone is used for traversing from one side of the bipartite graph to the other and back again, heuristic edge values …

Multipoint Quadratic Approximation For Numerical Optimization, Michael A. Blaylock

Theses and Dissertations

A quadratic approximation for nonlinear functions is developed in order to realize computational savings in solving numerical optimization problems. Function and gradient information accumulated from multiple design points during the iteration history is used in estimating the Hessian matrix. The approximate Hessian matrix is the available for a second order Taylor series approximation to the functions of interest. Several truss and frame models will be used to demonstrate the effectiveness of the new Multipoint Quadratic Approximation (MQA) in solving structural optimization problems.