Physical Sciences and Mathematics Commons^™

Equilibrium Problems With Equilibrium Constraints Via Multiobjective Optimization, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns a new class of optimization-related problems called Equilibrium Problems with Equilibrium Constraints (EPECs). One may treat them as two level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of Mathematical Programs with Equilibrium Constraints (MPECs). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This paper presents necessary …

Necessary Conditions In Nonsmooth Minimization Via Lower And Upper Subgradients, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Frechetjregular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with …

Optimal Control Of Delayed Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

This paper concerns constrained dynamic optimization problems governed by delayed differential-algebraic systems. Dynamic constraints in such systems, which are particularly important for engineering applications, are described by interconnected delay-differential inclusions and algebraic equations. We pursue a two-hold goal: to study variational stability of such control systems with respect to discrete approximations and to derive necessary optimality conditions for both delayed differential-algebraic systems and their finite-difference counterparts using modern tools of variational analysis and generalized differentiation. We are not familiar with any results in these directions for differential-algebraic inclusions even in the delay-free case. In the first part of the paper …

The Approximate Maxium Principle In Constrained Optimal Control, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

The paper concerns optimal control problems for dynamic systems governed by a parametric family of discrete approximations of control systems with continuous time. Discrete approximations play an important role in both qualitative and numerical aspects of optimal control and occupy an intermediate position between discrete-time and continuous-time control systems. The central result in optimal control of discrete approximations is the Approximate Maximum Principle (AMP), which is justified for smooth control problems with endpoint constraints under certain assumptions without imposing any convexity, in contrast to discrete systems with a fixed step. We show that these assumptions are essential for the validity …

Kernel Estimation Of Rate Function For Recurrent Event Data, Chin-Tsang Chiang, Mei-Cheng Wang, Chiung-Yu Huang

Johns Hopkins University, Dept. of Biostatistics Working Papers

Recurrent event data are largely characterized by the rate function but smoothing techniques for estimating the rate function have never been rigorously developed or studied in statistical literature. This paper considers the moment and least squares methods for estimating the rate function from recurrent event data. With an independent censoring assumption on the recurrent event process, we study statistical properties of the proposed estimators and propose bootstrap procedures for the bandwidth selection and for the approximation of confidence intervals in the estimation of the occurrence rate function. It is identified that the moment method without resmoothing via a smaller bandwidth …

Optimization And Feedback Control Of Constrained Parabolic Systems Under Uncertain Perturbations, Boris S. Mordukhovich, Ilya Shvartsman

Mathematics Research Reports

This paper concerns a minimax control design problem for a class of parabolic systems with nonregular boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We deal with boundary controllers acting through Dirichlet boundary conditions that are the most challenging for the parabolic dynamics.

Global Solutions To The Lake Equations With Isolated Vortex Regions, Chaocheng Huang

Mathematics and Statistics Faculty Publications

The vorticity formulation for the lake equations in R² is studied.

Normal Forms For Nonlinear Discrete Time Control Systems, Boumediene Hamzi, Issa Amadou Tall

Miscellaneous (presentations, translations, interviews, etc)

We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation.

Unified Cross-Validation Methodology For Selection Among Estimators And A General Cross-Validated Adaptive Epsilon-Net Estimator: Finite Sample Oracle Inequalities And Examples, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

In Part I of this article we propose a general cross-validation criterian for selecting among a collection of estimators of a particular parameter of interest based on n i.i.d. observations. It is assumed that the parameter of interest minimizes the expectation (w.r.t. to the distribution of the observed data structure) of a particular loss function of a candidate parameter value and the observed data structure, possibly indexed by a nuisance parameter. The proposed cross-validation criterian is defined as the empirical mean over the validation sample of the loss function at the parameter estimate based on the training sample, averaged over …

Neumann Boundary Control Of Hyperbolic Equations With Pointwise State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.

Semi-Parametric Box-Cox Power Transformation Models For Censored Survival Observations, Tianxi Cai, Lu Tian, L. J. Wei

Harvard University Biostatistics Working Paper Series

No abstract provided.

Statistical Inferences Based On Non-Smooth Estimating Functions, Lu Tian, Jun S. Liu, Mary Zhao, L. J. Wei

Harvard University Biostatistics Working Paper Series

No abstract provided.

Maximum Likelihood Estimation Of Ordered Multinomial Parameters , Nicholas P. Jewell, Jack Kalbfleisch

The University of Michigan Department of Biostatistics Working Paper Series

The pool-adjacent violator-algorithm (Ayer et al., 1955) has long been known to give the maximum likelihood estimator of a series of ordered binomial parameters, based on an independent observation from each distribution (see, Barlow et al., 1972). This result has immediate application to estimation of a survival distribution based on current survival status at a set of monitoring times. This paper considers an extended problem of maximum likelihood estimation of a series of ‘ordered’ multinomial parameters pi = (p1i, p2i, . . . , pmi) for 1 < = I < = k, where ordered means that pj1 < = pj2 < = .. . < = pjk for each j with 1 < = j < = m-1. The data consist of k independent observations X1, . . . ,Xk where Xi has a multinomial distribution with probability parameter pi and known index ni > = 1. By making use of variants of the pool adjacent violator algorithm, …

A Population Pharmacokinetic Model With Time-Dependent Covariates Measured With Errors, Lang Lil, Xihong Lin, Mort B. Brown, Suneel Gupta, Kyung-Hoon Lee

The University of Michigan Department of Biostatistics Working Paper Series

We propose a population pharmacokinetic (PK) model with time-dependent covariates measured with errors. This model is used to model S-oxybutynin's kinetics following an oral administration of Ditropan, and allows the distribution rate to depend on time-dependent covariates blood pressure and heart rate, which are measured with errors. We propose two two-step estimation methods: the second order two-step method with numerical solutions of differential equations (2orderND), and the second order two-step method with closed form approximate solutions of differential equations (2orderAD). The proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear …

Controllability And Local Accessibility—A Normal Form Approach, Wei Kang, Mingqing Xiao, Issa Amadou Tall

Articles and Preprints

Given a system with an uncontrollable linearization at the origin, we study the controllability of the system at equilibria around the origin. If the uncontrollable mode is nonzero, we prove that the system always has other equilibria around the origin. We also prove that these equilibria are linearly controllable provided a coefficient in the normal form is nonzero. Thus, the system is qualitatively changed from being linearly uncontrollable to linearly controllable when the equilibrium point is moved from the origin to a different one. This is called a bifurcation of controllability. As an application of the bifurcation, systems with a …

Dirichlet Boundary Control Of Hyperbolic Equations In The Presence Of State Constraints, Boris S. Mordukhovich, Jean-Pierre Raymond

Mathematics Research Reports

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.

Exact Multiplicity For Periodic Solutions Of Duffing Type, Hongbin Chen, Yi Li, Xiaojie Hou

Yi Li

In this paper, we study the following Duffing-type equation: x″+cx′+g(t,x)=h(t),

where g(t,x) is a 2π-periodic continuous function in t and concave–convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

Exact Multiplicity For Periodic Solutions Of Duffing Type, Hongbin Chen, Yi Li, Xiaojie Hou

Mathematics and Statistics Faculty Publications

In this paper, we study the following Duffing-type equation:

x″+cx′+g(t,x)=h(t),

where g(t,x) is a 2π-periodic continuous function in t and concave–convex in x, and h(t) is a small continuous 2π-periodic function. The exact multiplicity and stability of periodic solutions are obtained.

Optimization And Equilibrium Problems With Equilibrium Constraints, Boris S. Mordukhovich

Mathematics Research Reports

The paper concerns optimization and equilibrium problems with the so-called equilibrium constraints (MPEC and EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special …

Cross-Calibration Of Stroke Disability Measures: Bayesian Analysis Of Longitudinal Ordinal Categorical Data Using Negative Dependence, Giovanni Parmigiani, Heidi W. Ashih, Gregory P. Samsa, Pamela W. Duncan, Sue Min Lai, David B. Matchar

Johns Hopkins University, Dept. of Biostatistics Working Papers

It is common to assess disability of stroke patients using standardized scales, such as the Rankin Stroke Outcome Scale (RS) and the Barthel Index (BI). The Rankin Scale, which was designed for applications to stroke, is based on assessing directly the global conditions of a patient. The Barthel Index, which was designed for general applications, is based on a series of questions about the patient’s ability to carry out 10 basis activities of daily living. As both scales are commonly used, but few studies use both, translating between scales is important in gaining an overall understanding of the efficacy of …

Optimal Control Of Differential-Algebraic Inclusions, Boris S. Mordukhovich, Lianwen Wang

Mathematics Research Reports

No abstract provided.

When Abelian Groups Split, Rachel M. Thomas, Robert C. Rhoades

Mathematical Sciences Technical Reports (MSTR)

Let S be a hyperbolic surface tiled by kaleidoscopic triangles. Let R_e denote the set of fixed points by the reflection in an edge, e, of a triangle. We say that R_e is separating if S-R_e has two components. Once we have a tiling, we can define a group of orientation preserving transformations, G. We develop a method for determining when a reflection is separating using the group algebra of G. Using this method we give necessary and sufficient conditions for a mirror to be separating when G is abelian. We also conjecture, that …

An Extended General Location Model For Causal Inference From Data Subject To Noncompliance And Missing Values, Yahong Peng, Rod Little, Trivellore E. Raghuanthan

The University of Michigan Department of Biostatistics Working Paper Series

Noncompliance is a common problem in experiments involving randomized assignment of treatments, and standard analyses based on intention-to treat or treatment received have limitations. An attractive alternative is to estimate the Complier-Average Causal Effect (CACE), which is the average treatment effect for the subpopulation of subjects who would comply under either treatment (Angrist, Imbens and Rubin, 1996, henceforth AIR). We propose an Extended General Location Model to estimate the CACE from data with non-compliance and missing data in the outcome and in baseline covariates. Models for both continuous and categorical outcomes and ignorable and latent ignorable (Frangakis and Rubin, 1999) …

Computational Models For Diffusion Of Second Messengers In Visual Transduction, Harihar Khanal

Publications

The process of phototransduction, whereby light is converted into an electrical response in retinal rod and cone photoreceptors, involves, as a crucial step, the diffusion of cytoplasmic signaling molecules, termed second messengers. A barrier to mathematical and computational modeling is the complex geometry of the rod outer segment which contains about 1000 thin discs. Most current investigations on the subject assume a well-stirred bulk aqueous environment thereby avoiding such geometrical complexity. We present theoretical and computational spatio-temporal models for phototransduction in vertebrate rod photoreceptors, which are pointwise in nature and thus take into account the complex geometry of the …

Network Security Risk Assessment Modeling Tools For Critical Infrastructure Assessment, George H. Baker, Samuel Redwine, Joseph Blandino

George H Baker

The James Madison University (JMU) CIPP research team is developing Network Security Risk Assessment Modeling (NSRAM) tools that will enable the assessment of both cyber and physical infrastructure security risks. The effort is driven by the need to predict and compute the probability of adverse effects stemming from system attacks and malfunctions, to understand their consequences, and to improve existing systems to minimize these consequences.

The tools are targeted at systems supporting critical infrastructures varying from individual systems to organization-wide systems, to systems covering entire geographical regions. Early work emphasizes computing systems, but systems sharing the network nature of computing …

Locally Efficient Estimation Of Nonparametric Causal Effects On Mean Outcomes In Longitudinal Studies, Romain Neugebauer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Marginal Structural Models (MSM) have been introduced by Robins (1998a) as a powerful tool for causal inference as they directly model causal curves of interest, i.e. mean treatment-specific outcomes possibly adjusted for baseline covariates. Two estimators of the corresponding MSM parameters of interest have been proposed, see van der Laan and Robins (2002): the Inverse Probability of Treatment Weighted (IPTW) and the Double Robust (DR) estimators. A parametric MSM approach to causal inference has been favored since the introduction of MSM. It relies on correct specification of a parametric MSM to consistently estimate the parameter of interest using the IPTW …

Natural Superconvergent Points Of Triangular Finite Elements, Zhimin Zhang, Runchang Lin

Mathematics Research Reports

In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of [2], especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory [14] are the only superconvergent points for the Poisson equation. Finally, we …

On The Stability Of The Positive Radial Steady States For A Semilinear Cauchy Problem, Yinbin Deng, Yi Li, Yi Liu

Mathematics and Statistics Faculty Publications

No abstract provided.

On Adaptive Estimation In Orthogonal Saturated Designs, Weizhen Wang, Daniel T. Voss

Mathematics and Statistics Faculty Publications

A simple method is provided to construct a general class of individual and simultaneous confidence intervals for the effects in orthogonal saturated designs. These intervals use the data adaptively, maintain the confidence levels sharply at 1 - α at the least favorable parameter configuration, work effectively under effect sparsity, and include the intervals by Wang and Voss (2001) as a special case.

A Forward-Backward Fluence Model For The Low-Energy Neutron Boltzmann Equation, Gary Alan Feldman

Mathematics & Statistics Theses & Dissertations

In this research work, the neutron Boltzmann equation was separated into two coupled integro-differential equations describing forward and backward neutron fluence in selected materials. Linear B-splines were used to change the integro-differential equations into a coupled system of ordinary differential equations (O.D.E.'s). Difference approximations were then used to recast the O.D.E.'s into a coupled system of linear equations that were solved for forward and backward neutron fluences. Adding forward and backward fluences gave the total fluence at selected energies and depths in the material. Neutron fluences were computed in single material shields and in a shield followed by a target …