Large Scale Computational Problems in Numerical Optimization

Our work under this support broadly falls into five categories: automatic differentiation, sparsity, constraints, parallel computation, and applications. Automatic Differentiation (AD): We developed strong practical methods for computing sparse Jacobian and Hessian matrices which arise frequently in large scale optimization problems [10,35]. In addition, we developed a novel view of "structure" in applied problems along with AD techniques that allowed for the efficient application of sparse AD techniques to dense, but structured, problems. Our AD work included development of freely available MATLAB AD software. Sparsity: We developed new effective and practical techniques for exploiting sparsity when solving a variety of optimization problems. These problems include: bound constrained problems, robust regression problems, the null space problem, and sparse orthogonal factorization. Our sparsity work included development of freely available and published software [38,39]. Constraints: Effectively handling constraints in large scale optimization remains a challenge. We developed a number of new approaches to constrained problems with emphasis on trust region methodologies. Parallel Computation: Our work included the development of specifically parallel techniques for the linear algebra tasks underpinning optimization algorithms. Our work contributed to the nonlinear least-squares problem, nonlinear equations, triangular systems, orthogonalization, and linear programming. Applications: Our optimization work is broadly applicable across numerous application domains. Nevertheless we have specifically worked in several application areas including molecular conformation, molecular energy minimization, computational finance, and bone remodeling

On affine scaling inexact dogleg methods for bound-constrained nonlinear systems

Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given

A Two-Process View of Trust and Distrust Building in Recommendation Agents: A Process-Tracing Study

Prior literature focuses on trust, while largely ignoring distrust, partly because of the assumption that an Information Technology (IT) design that builds trust in the IT will also prevent distrust-building. However, this assumption may not be true if trust-building processes and distrust-building processes in the context of IT usage are different. This paper proposes a two-process view of trust and distrust building, i.e., that trust-building and distrust-building processes are distinct and separate. In the context of recommendation agent (RA) usage in electronic commerce, a trust (distrust) process is defined as a customer’s favorable (unfavorable) interpretation of his or her interactions with an RA, resulting in a positive (negative) expectation that the RA can be relied upon for his or her shopping decisions. This study empirically tests a process theory rather than a variance theory. Variance theory research relies on logical arguments to explain and test the causality relationships among variables. Process theory research complements variance theory research by revealing and testing the mechanisms that constitute the processes by which certain variables influence others. In this process-tracing study, we collected and analyzed the concurrent verbal protocols from 49 participants using two RAs. The results of our protocol analysis support the proposed two-process view. The pattern of trust-building processes in RA usage is systematically different from that of distrust-building processes, which may suggest that some RA features should be designed to increase trust, and others to decrease distrust. The findings also suggest that distrust deserves research attention on its own merit. In a complex relationship involving both trust building and distrust building, understanding both trust and distrust processes, rather than focusing on trust alone, can lead to a more accurate representation and improved management of that complex relationship

Two Affine Scaling Methods for Solving Optimization Problems Regularized with an L1-norm

In finance, the implied volatility surface is plotted against strike price and time to maturity. The shape of this volatility surface can be identified by fitting the model to what is actually observed in the market. The metric that is used to measure the discrepancy between the model and the market is usually defined by a mean squares of error of the model prices to the market prices. A regularization term can be added to this error metric to make the solution possess some desired properties. The discrepancy that we want to minimize is usually a highly nonlinear function of a set of model parameters with the regularization term. Typically monotonic decreasing algorithm is adopted to solve this minimization problem. Steepest descent or Newton type algorithms are two iterative methods but they are local, i.e., they use derivative information around the current iterate to find the next iterate. In order to ensure convergence, line search and trust region methods are two widely used globalization techniques. Motivated by the simplicity of Barzilai-Borwein method and the convergence properties brought by globalization techniques, we propose a new Scaled Gradient (SG) method for minimizing a differentiable function plus an L1-norm. This non-monotone iterative method only requires gradient information and safeguarded Barzilai-Borwein steplength is used in each iteration. An adaptive line search with the Armijo-type condition check is performed in each iteration to ensure convergence. Coleman, Li and Wang proposed another trust region approach in solving the same problem. We give a theoretical proof of the convergence of their algorithm. The objective of this thesis is to numerically investigate the performance of the SG method and establish global and local convergence properties of Coleman, Li and Wang’s trust region method proposed in [26]. Some future research directions are also given at the end of this thesis

Traffic matrix estimation in IP networks

An Origin-Destination (OD) traffic matrix provides a major input to the design, planning and management of a telecommunications network. Since the Internet is being proposed as the principal delivery mechanism for telecommunications traffic at the present time, and this network is not owned or managed by a single entity, there are significant challenges for network planners and managers needing to determine equipment and topology configurations for the various sections of the Internet that are currently the responsibility of ISPs and traditional telcos. Planning of these sub-networks typically requires a traffic matrix of demands that is then used to infer the flows on the administrator's network. Unfortunately, computation of the traffic matrix from measurements of individual flows is extremely difficult due to the fact that the problem formulation generally leads to the need to solve an under-determined system of equations. Thus, there has been a major effort from among researchers to obtain the traffic matrix using various inference techniques. The major contribution of this thesis is the development of inference techniques for traffic matrix estimation problem according to three different approaches, viz: (1) deterministic, (2) statistical, and (3) dynamic approaches. Firstly, for the deterministic approach, the traffic matrix estimation problem is formulated as a nonlinear optimization problem based on the generalized Kruithof approach which uses the Kullback distance to measure the probabilistic distance between two traffic matrices. In addition, an algorithm using the Affine scaling method is developed to solve the constrained optimization problem. Secondly, for the statistical approach, a series of traffic matrices are obtained by applying a standard deterministic approach. The components of these matrices represent estimates of the volumes of flows being exchanged between all pairs of nodes at the respective measurement points and they form a stochastic counting process. Then, a Markovian Arrival Process of order two (MAP-2) is applied to model the counting processes formed from this series of estimated traffic matrices. Thirdly, for the dynamic approach, the dual problem of the multi-commodity flow problem is formulated to obtain a set of link weights. The new weight set enables flows to be rerouted along new paths, which create new constraints to overcome the under-determined nature of traffic matrix estimation. Since a weight change disturbs a network, the impact of weight changes on the network is investigated by using simulation based on the well-known ns2 simulator package. Finally, we introduce two network applications that make use of the deterministic and the statistical approaches to obtain a traffic matrix respectively and also describe a scenario for the use of the dynamic approach