Digital Commons Network^™

Revisiting Rockafellar’S Theorem On Relative Interiors Of Convex Graphs With Applications To Convex Generalized Differentiation, Boris S. Mordukhovich, Nguyen Mau Nam, Dang Van Cuong, G. Sandine

Mathematics and Statistics Faculty Publications and Presentations

In this paper we revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex set-valued mapping in terms of the relative interior of its domain and function values. Then we apply this theorem to provide a simple way to prove many calculus rules of generalized differentiation for set-valued mappings and nonsmooth functions in finite dimensions. Using this important theorem by Rockafellar allows us to improve some results on generalized differentiation of set-valued mappings in [13] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.

Development Of A Configurable Real-Time Event Detection Framework For Power Systems Using Swarm Intelligence Optimization, Umar Farooq

Dissertations and Theses

Modern power systems characterized by complex topologies require accurate situational awareness to maintain an adequate level of reliability. Since they are large and spread over wide geographical areas, occurrence of failures is inevitable in power systems. Various generation and transmission disturbances give rise to a mismatch between generation and demand, which manifest as frequency events. These events can take the form of negligible frequency deviations or more severe emergencies that can precipitate cascading outages, depending on the severity of the disturbance and efficacy of remedial action schema. The impacts of such events have become more critical with recent decline in …

Making Curry With Rice: An Optimizing Curry Compiler, Steven Libby

Dissertations and Theses

In this dissertation we present the RICE optimizing compiler for the functional logic language Curry. This is the first general optimizing compiler for a functional logic language. Our work is based on the idea of compiling through program transformations, which we have adapted from the functional language compiler community. We also present the GAS system for generating new program transformations, which uses the power of functional logic programming to provide a flexible framework for describing transformations. This allows us to describe and implement a wide range of optimizations including inlining, shortcut deforestation, unboxing, and case shortcutting, a new optimization we …

Fenchel-Rockafellar Theorem In Infinite Dimensions Via Generalized Relative Interiors, Dang Van Cuong, Mau Nam Nguyen, B. S. Mordukhovich, G. Sandine

Mathematics and Statistics Faculty Publications and Presentations

In this paper we provide further studies of the Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

Using Intrinsically-Typed Definitional Interpreters To Verify Compiler Optimizations In A Monadic Intermediate Language, Dani Barrack

Dissertations and Theses

Compiler optimizations are critical to the efficiency of modern functional programs. At the same time, optimizations that unintentionally change the semantics of programs can systematically introduce errors into programs that pass through them. The question of how to best verify that optimizations and other program transformations preserve semantics is an important one, given the potential for error introduction. Dependent types allow us to prove that properties about our programs are correct, as well as to design data types and interpreters in such a way that they are correct-by-construction. In this thesis, we explore the use of dependent types and intrinsically-typed …

Quantum Grover's Oracles With Symmetry Boolean Functions, Peng Gao

Dissertations and Theses

Quantum computing has become an important research field of computer science and engineering. Among many quantum algorithms, Grover's algorithm is one of the most famous ones. Designing an effective quantum oracle poses a challenging conundrum in circuit and system-level design for practical application realization of Grover's algorithm.

In this dissertation, we present a new method to build quantum oracles for Grover's algorithm to solve graph theory problems. We explore generalized Boolean symmetric functions with lattice diagrams to develop a low quantum cost and area efficient quantum oracle. We study two graph theory problems: cycle detection of undirected graphs and generalized …

Proximal Policy Optimization For Radiation Source Search, Philippe Erol Proctor

Dissertations and Theses

Rapid localization and search for lost nuclear sources in a given area of interest is an important task for the safety of society and the reduction of human harm. Detection, localization and identification are based upon the measured gamma radiation spectrum from a radiation detector. The nonlinear relationship of electromagnetic wave propagation paired with the probabilistic nature of gamma ray emission and background radiation from the environment leads to ambiguity in the estimation of a source's location. In the case of a single mobile detector, there are numerous challenges to overcome such as weak source activity, multiple sources, or the …

Forecasting Optimal Parameters Of The Broken Wing Butterfly Option Strategy Using Differential Evolution, David Munoz Constantine

Dissertations and Theses

Obtaining an edge in financial markets has been the objective of many hedge funds, investors, and market participants. Even with today's abundance of data and computing power, few individuals achieve a consistent edge over an extended time. To obtain this edge, investors usually use options strategies. The Broken Wing Butterfly (BWB) is an options strategy that has increased in popularity among traders. Profit is generated primarily by exploiting option value time decay. In this thesis, the selection of entry and exit BWB parameters, such as profit and loss targets, are optimized for an in-sample period. Afterward, they are used to …

The Optimization Of Machining Parameters For Milling Operations By Using The Nelder–Mead Simplex Method, Yubin Lee, Alin Resiga, Sung Yi, Chien Wern

Mechanical and Materials Engineering Faculty Publications and Presentations

The purpose of machining operations is to make specific shapes or surface characteristics for a product. Conditions for machining operations were traditionally selected based on geometry and surface finish requirements. However, nowadays, many researchers are optimizing machining parameters since high-quality products can be produced using more expensive and advanced machines and tools. There are a few methods to optimize the machining process, such as minimizing unit production time or cost or maximizing profit. This research focused on maximizing the profit of computer numerical control (CNC) milling operations by optimizing machining parameters. Cutting speeds and feed are considered as the main …

Convex And Nonconvex Optimization Techniques For Multifacility Location And Clustering, Tuyen Dang Thanh Tran

Dissertations and Theses

This thesis contains contributions in two main areas: calculus rules for generalized differentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Banach spaces. These calculus rules are useful for further applications to nonsmooth optimization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class …

On Dc And Local Dc Functions, Liam Jemison

University Honors Theses

In this project we investigate the class of functions which can be represented by a difference of convex functions, hereafter referred to simply as 'DC' functions. DC functions are of interest in optimization because they allow the use of convex optimization techniques in certain non-convex problems. We present known results about DC and locally DC functions, including detailed proofs of important theorems by Hartman and Vesely.

We also investigate the DCA algorithm for optimizing DC functions and implement it to solve the support vector machine problem.

Aggregated Water Heater System (Awhs) Optimization For Ancillary Services, Manasseh Obi

Dissertations and Theses

In this dissertation, I present a two-stage optimization routine that schedules an Aggregated Water Heater System (AWHS) to concurrently provide three utility ancillary services, namely, frequency regulation, frequency response, and peak demand mitigation.

Water heaters can be controlled to manage their energy take, the amount of energy a water heater can absorb upon command. The AWHS is a model aggregation of thousands of water heaters, the energy take and power characteristics of which are based on U.S Census household data and usage behavior patterns. The aggregate energy take available in the AWHS may be dispatched en masse for participation in …

Dictionary Learning For Image Reconstruction Via Numerical Non-Convex Optimization Methods, Lewis M. Hicks

University Honors Theses

This thesis explores image dictionary learning via non-convex (difference of convex, DC) programming and its applications to image reconstruction. First, the image reconstruction problem is detailed and solutions are presented. Each such solution requires an image dictionary to be specified directly or to be learned via non-convex programming. The solutions explored are the DCA (DC algorithm) and the boosted DCA. These various forms of dictionary learning are then compared on the basis of both image reconstruction accuracy and number of iterations required to converge.

The Optimization Of Machining Parameters For Milling Operations By Using The Nelder Mead Simplex Method, Yubin Lee

Dissertations and Theses

Machining operations need to be optimized to maximize profit for computer numerical control (CNC) machines. Although minimum production time could mean high productivity, it can not guarantee maximum profit rate in CNC milling operations. The possible range of machining parameters is limited by several constraints, such as maximum machine power, surface finish requirements, and maximum cutting force for the stability of milling operations. Among CNC machining parameters, cutting speed and feed have the greatest effect on machining operations. Therefore, cutting speed and feed are considered as main process variables to maximize the profit rate of CNC milling operations.

A variety …

Paints-R-Us Term Project, Tyler Campbell, Skye Gilbreth, Michael Oluwole, Elijah Raffo, Brad Unruh

Engineering and Technology Management Student Projects

This project will consider a linear product mix optimization problem for a fictional paint company, Paints-R-Us. Paints-R-Us is a wholesale paint manufacturer located in the Pacific Northwest. The Global Production Manager, Steve Brush, has been tasked with maximizing Paints-R-Us’s profit in the upcoming quarter. Steve Brush oversees the global production plan, and in collaboration with the production planners will develop a production plan which optimizes the profits that Paints-R-Us can create in the quarter accounting for the following criteria:

• Demand in the given quarter for each of the 5 paint types that Paints-R-Us produces • The warehousing storage capacity …

Utilizing Parallelism In The Conjugate Gradient Algorithm, Adam James Craig

University Honors Theses

This paper develops the original conjugate gradient method and the idea of preconditioning a system. I also propose a unique type of additive-Schwarz preconditioner that can be solved in parallel, which creates a speed increase for large systems. To show this, I developed a C++11 linear algebra library used in conjunction with the OpenMP parallel computing library to empirically show a speed increase.

Efficient Methods For Robust Circuit Design And Performance Optimization For Carbon Nanotube Field Effect Transistors, Muhammad Ali

Dissertations and Theses

Carbon nanotube field-effect transistors (CNFETs) are considered to be promising candidate beyond the conventional CMOSFET due to their higher current drive capability, ballistic transport, lesser power delay product and higher thermal stability. CNFETs show great potential to build digital systems on advanced technology nodes with big benefits in terms of power, performance and area (PPA). Hence, there is a great need to develop proven models and CAD tools for performance evaluation of CNFET-based circuits. CNFETs specific parameters, such as number of tubes, pitch (spacing between the tubes) and diameter of CNTs determine current driving capability, speed, power consumption and area …

Design Optimization For A Cnc Machine, Alin Resiga

Dissertations and Theses

Minimizing cost and optimization of nonlinear problems are important for industries in order to be competitive. The need of optimization strategies provides significant benefits for companies when providing quotes for products. Accurate and easily attained estimates allow for less waste, tighter tolerances, and better productivity. The Nelder-Mead Simplex method with exterior penalty functions was employed to solve optimum machining parameters. Two case studies were presented for optimizing cost and time for a multiple tools scenario. In this study, the optimum machining parameters for milling operations were investigated. Cutting speed and feed rate are considered as the most impactful design variables …

Clustering And Multifacility Location With Constraints Via Distance Function Penalty Methods And Dc Programming, Mau Nam Nguyen, Thai An Nguyen, Sam Reynolds, Tuyen Tran

Mathematics and Statistics Faculty Publications and Presentations

This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide different numerical examples to test our method.

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.

Generalized Differential Calculus And Applications To Optimization, R. Blake Rector

Dissertations and Theses

This thesis contains contributions in three areas: the theory of generalized calculus, numerical algorithms for operations research, and applications of optimization to problems in modern electric power systems. A geometric approach is used to advance the theory and tools used for studying generalized notions of derivatives for nonsmooth functions. These advances specifically pertain to methods for calculating subdifferentials and to expanding our understanding of a certain notion of derivative of set-valued maps, called the coderivative, in infinite dimensions. A strong understanding of the subdifferential is essential for numerical optimization algorithms, which are developed and applied to nonsmooth problems in operations …

Subgradients Of Minimal Time Functions Without Calmness, Nguyen Mau Nam, Dang Van Cuong

Mathematics and Statistics Faculty Publications and Presentations

In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we continue this line of research by providing new results on generalized differentiation of this class of functions, relaxing assumptions imposed on the functions and sets involved for the results. In particular, we focus on the singular subdifferential and the limiting subdifferential of this class of functions.

Nesterov's Smoothing Technique And Minimizing Differences Of Convex Functions For Hierarchical Clustering, Mau Nam Nguyen, Wondi Geremew, Sam Raynolds, Tuyen Tran

Mathematics and Statistics Faculty Publications and Presentations

A bilevel hierarchical clustering model is commonly used in designing optimal multicast networks. In this paper we will consider two different formulations of the bilevel hierarchical clustering problem -- a discrete optimization problem which can be shown to be NP-hard. Our approach is to reformulate the problem as a continuous optimization problem by making some relaxations on the discreteness conditions. This approach was considered by other researchers earlier, but their proposed methods depend on the square of the Euclidian norm because of its differentiability. By applying the Nesterov smoothing technique and the DCA -- a numerical algorithm for minimizing differences …

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 …

Convex And Nonconvex Optimization Techniques For The Constrained Fermat-Torricelli Problem, Nathan Lawrence

University Honors Theses

The Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to three given points in the plane. This problem was introduced by the French mathematician Fermat in the 17th century and was solved by the Italian mathematician and physicist Torricelli. In this thesis we introduce a constrained version of the Fermat-Torricelli problem in high dimensions that involves distances to a finite number of points with both positive and negative weights. Based on the distance penalty method, Nesterov’s smoothing technique, and optimization techniques for minimizing differences of convex functions, we provide effective algorithms to solve the problem. …

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, …

The Smallest Intersecting Ball Problem, Daniel J. Giles, Mau Nam Nguyen

Student Research Symposium

The smallest intersecting ball problem involves finding the minimal radius necessary to intersect a collection of closed convex sets. This poster discusses relevant tools of convex optimization and explores three methods of finding the optimal solution: the subgradient method, log-exponential smoothing, and an original approach using target set expansion. A fourth algorithm based on weighted projections is given, but its convergence is yet unproven. Numerical tests and comparison between methods are also presented.

The Majorization Minimization Principle And Some Applications In Convex Optimization, Daniel Giles

University Honors Theses

The majorization-minimization (MM) principle is an important tool for developing algorithms to solve optimization problems. This thesis is devoted to the study of the MM principle and applications to convex optimization. Based on some recent research articles, we present a survey on the principle that includes the geometric ideas behind the principle as well as its convergence results. Then we demonstrate some applications of the MM principle in solving the feasible point, closest point, support vector machine, and smallest intersecting ball problems, along with sample MATLAB code to implement each solution. The thesis also contains new results on effective algorithms …

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.