Mathematics Commons^™

√(X2 + Μ) Is The Most Computationally Efficient Smooth Approximation To |X|: A Proof, Carlos Ramirez, Reinaldo Sanchez, Vladik Kreinovich, Miguel Argaez

Departmental Technical Reports (CS)

In many practical situations, we need to minimize an expression of the type |c₁| + ... + |c_n|. The problem is that most efficient optimization techniques use the derivative of the objective function, but the function |x| is not differentiable at 0. To make optimization efficient, it is therefore reasonable to approximate |x| by a smooth function. We show that in some reasonable sense, the most computationally efficient smooth approximation to |x| is the function √(x² + μ), a function which has indeed been successfully used in such optimization.

Near-Optimal Compressed Sensing Guarantees For Total Variation Minimization, Deanna Needell, R. Ward

CMC Faculty Publications and Research

Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images x ∈ ℂN2 to signals x ∈ ℂNd of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this …

Generalized Local Test For Local Extrema In Single-Variable Functions, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test is local, in the sense that it is only necessary to evaluate all relevant derivatives at the critical point itself, and it is reasonably robust. We illustrate an application of the generalized test on a trigonometric function where the second derivative test fails to classify some of the critical …

Optimization In Non-Parametric Survival Analysis And Climate Change Modeling, Iuliana Teodorescu

USF Tampa Graduate Theses and Dissertations

Many of the open problems of current interest in probability and statistics involve complicated data

sets that do not satisfy the strong assumptions of being independent and identically distributed. Often,

the samples are known only empirically, and making assumptions about underlying parametric

distributions is not warranted by the insufficient information available. Under such circumstances,

the usual Fisher or parametric Bayes approaches cannot be used to model the data or make predictions.

However, this situation is quite often encountered in some of the main challenges facing statistical,

data-driven studies of climate change, clinical studies, or financial markets, to name a few. …

An Adaptive Total Variation Algorithm For Computing The Balanced Cut Of A Graph, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht

Mathematics, Statistics and Data Science Faculty Works

We propose an adaptive version of the total variation algorithm proposed in [3] for computing the balanced cut of a graph. The algorithm from [3] used a sequence of inner total variation minimizations to guarantee descent of the balanced cut energy as well as convergence of the algorithm. In practice the total variation minimization step is never solved exactly. Instead, an accuracy parameter is specified and the total variation minimization terminates once this level of accuracy is reached. The choice of this parameter can vastly impact both the computational time of the overall algorithm as well as the accuracy of …

Integer Solutions To Optimization Problems And Modular Sequences Of Nexus Numbers, Jeremy T. Davis

College of Graduate Studies: Theses & Dissertations

In this thesis, we examine the use of integers through two ideas. As mathematics teachers, we prefer students not use calculators on assessments. In order to require this, students compute the problems by hand. We take a look at the classic Calculus I optimization box problem while restricting values to integers. In addition, sticking with the integer theme, we take a new look at the nexus numbers. Nexus numbers are extensions of the hex and rhombic dodecahedral numbers. We put these numbers into a sequence, and through a few computations of modular arithmetic, we analyze the sequences and their patterns …

Generating Minimal T-Wise Test Suites, Luis C. Gutierrez, Carlos Nieto, Francisco Zapata, Martine Ceberio

COURI Symposium Abstracts, Summer 2012

As the use of computing devices increases every day, users rely on the adequate functioning of software. When software is not tested properly, it can yield erroneous information or a complete failure of the system. The NIST estimates that defective software cost the United States economy close to $60 billion a year. Therefore, there is a need to develop software testing techniques that are time and cost effective. Fully testing software under all possible combinations of parameters values cannot be reduced. However, testing can focus on covering all combinations of subsets of parameters and empirical data shows that doing so …

Generating Minimal Pair-Wise Covering Test Suites, Luis C. Gutierrez ^, Martine Ceberio *

COURI Symposium Abstracts, Spring 2012

Software is ubiquitous and needs to be reliable. Software testing therefore plays an important role in software development. Proper testing a software system informs about its quality and reliability so as to prevent unexpected behavior during system execution. One of the methods to prevent failures consists in testing a system under different input values, but when all possible input values are tested, an impractical number of test cases might result. In software testing, pair-wise testing is a combinatorial technique which uses combination of pair input values to generate test cases. Using pair-wise testing dramatically reduces the number of test cases, …

Proof-Of-Concept For A Green Energy Linear Program For Optimizing Deployments, James M. Taylor, Betty Love

Mathematics Faculty Proceedings & Presentations

The US military has spent billions of dollars and sacrificed many lives in the effort to bring electrical power services and the fuel that drives the generators to forward-deployed bases in Afghanistan and Iraq over the past 10 years. In an effort to reduce some of these tremendous costs, the US military has considered using alternative energy sources to generate electricity and reduce costs and exposure of fuel truck convoys. While some research [10] has used detailed software packages to model the electrical demand and renewable energy production tradeoffs in this environment, the impact of operational constraints is not readily …

A Multimodal Freight Collaborative Hub Location And Network Design Problem, Jiri Tylich

Open Access Theses & Dissertations

The study presents an analytical framework to explore the rail-road collaborative paradigm.

New collaborative technologies have been developed in recent years and they offer a potential solutions and opportunities for collaboration among all modes of transportation. The most progressive technologies that could fulfill the gap in rail-road collaborative paradigm are identified and presented in this research.

The research deals with current state and possible development of collaboration of rail and highway modes of transportation, referred to as rail-road collaboration. Multimodal transportation is the shipment of goods in a single transportation unit. The longest part of the route takes place by …

Convergence Of A Steepest Descent Algorithm For Ratio Cut Clustering, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht

Mathematics, Statistics and Data Science Faculty Works

Unsupervised clustering of scattered, noisy and high-dimensional data points is an important and difficult problem. Tight continuous relaxations of balanced cut problems have recently been shown to provide excellent clustering results. In this paper, we present an explicit-implicit gradient flow scheme for the relaxed ratio cut problem, and prove that the algorithm converges to a critical point of the energy. We also show the efficiency of the proposed algorithm on the two moons dataset.

Optimization Of A Pressure-Treating Process, Josean Velez

Undergraduate Journal of Mathematical Modeling: One + Two

A company that pressure-treats wood wants to minimize its annual cost without using more than 250 days of operation per year. In addition, they want to find the corresponding value of time, batches and cost for each category. We develop an expression in terms of boards per batch to model the total cost of the treatment process. We then take the derivative and use Newton's Method to find the number of boards per batch that minimizes total cost.

Optimal Summer Camp Layout, Anthony Bonifonte

Honors Papers

Convex optimization is an important branch of operations research. It generalizes linear programming and offers powerful tools for modelling problems and discovering optimal solutions to real world problems. Mathematically it is an interesting topic because it ties together many branches: linear algebra, multivariable calculus, and numerical analysis, to name a few. Modelling a problem as a convex optimization problem can be challenging but offers many benefits. Algorithm design is critically important to ensure precision of solutions that solve with minimal computation power. From an engineering perspective it is also incredibly useful, since many more situations can be modeled than with …

Optimization Of A Chemical Reaction Train, Bahar Sansar

Undergraduate Journal of Mathematical Modeling: One + Two

This project consists of the optimization of a chemical reactor train. The reactor considered here is the continuous stirred tank reactor (CSTR), one of the reactor models used in engineering. Given the design equation for the CSTR and the cost function for a reactor, the following values are determined; the optimum number of reactors in the reaction train, the volume of each reactor and the total cost.

Radiotherapy Optimal Design: An Academic Radiotherapy Treatment Design System, Ryan Acosta, William Brick, A Hanna, Allen G. Holder, D Lara, G Mcquilen, D Nevin, P Uhlig, B Salter

Mathematics Faculty Research

Optimally designing radiotherapy and radiosurgery treatments to increase the likelihood of a successful recovery from cancer is an important application of operations research. Researchers have been hindered by the lack of academic software that supports head-to-head comparisons of different techniques, and this article addresses the inherent difficulties of designing and implementing an academic treatment planning system. In particular, this article details the algorithms and the software design of Radiotherapy optimAl Design (RAD).

Diversity Graphs, P Blain, C Davis, Allen G. Holder, J Silva, C Vinzant

Mathematics Faculty Research

Bipartite graphs have long been used to study and model matching problems, and in this paper we introduce the bipartite graphs that explain a recent matching problem in computational biology. The problem is to match haplotypes to genotypes in a way that minimizes the number of haplotypes, a problem called the Pure Parsimony problem. The goal of this work is not to address the computational or biological issues but rather to explore the mathematical structure through a study of the underlying graph theory.

Radiotherapy Optimal Design: An Academic Radiotherapy Treatment Design System, R Acosta, W Brick, A Hanna, Allen Holder, D Lara, G Mcquillen, D Nevin, P Uhlig, B Salter

Mathematical Sciences Technical Reports (MSTR)

Optimally designing radiotherapy and radiosurgery treatments to increase the likelihood of a successful recovery from cancer is an important application of operations research. Researchers have been hindered by the lack of academic software that supports head-to-head comparisons of different techniques, and this article addresses the inherent difficulties of designing and implementing an academic treatment planning system. In particular, this article details the algorithms and the software design of Radiotherapy optimAl Design (RAD).

Optimal Treatments For Photodynamic Therapy, Allen G. Holder, D Llagostera

Mathematics Faculty Research

Photodynamic therapy is a complex treatment for neoplastic diseases that uses the light-harvesting properties of a photosensitizer. The treatment depends on the amount of photosensitizer in the tissue and on the amount of light that is focused on the targeted area. We use a pharmacokinetic model to represent a photosensitizer's movement through the anatomy and design treatments with a linear program. This technique allows us to investigate how a treatment's success varies over time.

Beam Selection In Radiotherapy Design, M Ehrgott, Allen G. Holder, Josh Reese

Mathematics Faculty Research

The optimal design of a radiotherapy treatment depends on the collection of directions from which radiation is focused on the patient. These directions are manually selected by a physician and are typically based on the physician's previous experiences. Once the angles are chosen, there are numerous optimization models that decide a fluency pattern (exposure times) that best treats a patient. So, while optimization techniques are often used to decide the length of time a patient is exposed to a high-energy particle beam, the directions themselves are not optimized. The problem with optimally selecting directions is that the underlying mixed integer …

The Influence Of Dose Grid Resolution On Beam Selection Strategies In Radiotherapy Treatment Design, Ryan Acosta, Matthias Ehrgott, Allen G. Holder, Daniel Nevin, Josh Reese, Bill Salter

Mathematics Faculty Research

The design of a radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of a fluence pattern for each selected beam angle (intensity problem), and finding a sequence of configurations of a multilef collimator to deliver the treatment (realization problem). While many mathematical optimization models and algorithms have been proposed for the intensity problem and (to a lesser extent) the realization problem, this is not the case for the geometry problem. In clinical practice, beam directions are manually selected by a clinician and are typically based on the clinician’s experience. Solving the beam selection problem optimally is …

An Introduction To Systems Biology For Mathematical Programmers, Evind Almaas, Allen G. Holder, Kevin D. Livingstone

Mathematics Faculty Research

Many recent advances in biology, medicine and health care are due to computational efforts that rely on new mathematical results. These mathematical tools lie in discrete mathematics, statistics & probability, and optimization, and when combined with savvy computational tools and an understanding of cellular biology they are capable of remarkable results. One of the most significant areas of growth is in the field of systems biology, where we are using detailed biological information to construct models that describe larger entities. This chapter is designed to be an introduction to systems biology for individuals in Operations Research (OR) and mathematical programming …

Mathematically Modeling Pcr: An Asymptotic Approximation With Potential For Optimization, Martha J. Garlick

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

A mathematical model for PCR (Polymerase Chain Reaction) is developed using the law of mass action. Differential equations are written from the chemical equations, preserving the detail of the complementary DNA single strand being extended one bas e pair at a time. The equations for the annealing stage are solved analytically. The method of multiple scales is used to approximate solutions for the extension stage. A map is then developed from the solutions to simulate PCR. The advantage of this model is the ability to use the map to optimize the process. Our results suggest that dynamically optimizing the extension …

The Relationship Between Discrete Vector Quantization And The P-Median Problem, Allen G. Holder, G Lim, J Reese

Mathematics Faculty Research

We show that a well studied problem in the engineering community is the same as a problem studied by mathematical combinatorialists. Specifically, we show that the question of optimally designing a vector quantizer, which is an important problem in coding theory, is the same as the p-median problem, which is a classic graph theory problem with important applications in operations research. The importance of the relationship lies in the fact that both communities have spent years developing solution methodologies, and this connection permits each community to glean new ideas from the other. We show that two of the most popular …

Some Geometrical Aspects Of The Cone Linear Complementarity Problem., Madhur Malik Dr.

Doctoral Theses

Cone Linear Complementarity ProblemLet V be a finite dimensional real inner product space and K be a closed convex cone in V. Given a linear transformation L : V → V and a vector q ∈ V the cone linear complementarity problem or linear complementarity problem over K, denoted as LCP(K, L, q), is to find a vector x ∈ K such thatL(x) + q ∈ K+ and hx, L(x) + qi = 0,where h., .i denotes an inner product on V and K is the dual cone of K defined as:K∗ := {y ∈ V : hx, yi ≥ …

A Tutorial On Radiation Oncology And Optimization, Allen G. Holder, Bill Salter

Mathematics Faculty Research

Designing radiotherapy treatments is a complicated and important task that affects patient care, and modern delivery systems enable a physician more flexibility than can be considered. Consequently, treatment design is increasingly automated by techniques of optimization, and many of the advances in the design process are accomplished by a collaboration among medical physicists, radiation oncologists, and experts in optimization. This tutorial is meant to aid those with a background in optimization in learning about treatment design. Besides discussing several optimization models, we include a clinical perspective so that readers understand the clinical issues that are often ignored in the optimization …

Haplotyping And Minimum Diversity Graphs, C Davis, Allen G. Holder

Mathematics Faculty Research

Haplotyping is the process of reconstructing the genetic information donated by a prior generation to form a current population. Haplotyping is important because it allows us to study how traits are passed from one generation to another, which in turn allows us to find genetic markers that describe a current population's susceptibility to diseases. Our goal is to study the underlying graph theory problem, and we study the bipartite graphs, called diversity graphs, that describe haplotyping. In particular, we investigate the problem of finding the minimum number of haplotypes that can reconstruct a population, called the Pure Parsimony problem. The …

Computer Simulation And Homogenization In Heating Design Optimization, Daniel K. Balls

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The ability to ensure uniformity of temperature within a given finite physical region is an essential element in the success of many scientific processes, especially those that involve extreme fluctuation in temperature. Such a process is performed in an instrument called the LightTyper developed by Idaho Technology, Inc. of Salt Lake City Utah. This paper details the development and results of a scheme intended to obtain a heating design that ensures a high degree of temperature uniformity within the Idaho Technology instrument. Due to the experiments performed during this project, we were able to answer many questions that concerned finding …

Test Suite For Multiobjective Optimization And Results Using Normal Boundary Intersection (Nbi) In Design Explorer, Siva Kumar Natarajan

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Several methods have been developed to solve multiobjective optimization problems (MOP's). One of these, Normal Boundary Intersection (NBI), is a method developed by John Dennis and Indraneel Das. NBI is used at The Boeing Company as a tool to solve MOP's. This report presents a test suite of MOP's that I developed for Boeing during me internship in summer 2003.

The problems in the test suite were chosen to represent the different types of multiobjective optimization problems that could arise in practice and the complexities involved in solving them. These problems range from those that have nice convex Pareto surfaces …

On The Approximability Of Linear Ordering And Related Np-Optimization Problems., Sounaka Mishra Dr.

Doctoral Theses

We investigate approximability of both maximum and minimum linear ordering problems (MAX-LOP and MIN-LOP) and several related problems such as the well known feedback set problems, acyclie subdigraph problem and several others and their variants.We show that both MAX-LOP and MIN-LOP are strongly NP-complete, and MIN- LOP, MIN-QAP(S) (a special case of minimum quadratic assignment problem) and MIN-W-FAS are equivalent with respect to strict-reduction. The strict-equivalence is also established among these problems as well as MIN-W-FVS, with weights on arcs/vertices bounded by a polynomial, and the unweighted versions of the feedback set. problems. We also show that MAX-LOP is strict-equivalent …

Engineering Production Function And Choice Of Technique., N. S. S. Narayana Dr.

Doctoral Theses

This study deala wi th the selection of technological al ternaid ves of production, a problom referred to as choice of techninues" in ecommie theory. Selection of an appropriate technique (or a technolo gical al ternative) haa to be Bede with ro spe et to speci fied objectives and avail phle reDurces. Selection of proper techniques from smong many avall ablo el ternatives ha boen one of the min tasks whilu planni ng for econonio development. Generally develooping co untries like India are handi capped by problema of BCerci ty of on-lebour reaources, abundant labour force, lini ted foreign-aid, 1inited …