Anuj, Bajaj (Oakland University)
Title: Solving and Applications of Multi-Facility Location Problems
Abstract:
"We introduce a new approach to solve multi-facility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov’s smoothing.The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets. This is based on joint work with B. Mordukhovich, N. M. Nam, and Tuyen Tran."
Bello-Cruz, Yunier (Northern Illinois University)
Title: On the finite convergence of alternating projections
Abstract:
In this talk, we combine two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names such as Kaczmarz and von Neumann, MAP has the ability to track a pair of points realizing minimum distance between two given closed convex sets. Unfortunately, MAP may suffer from arbitrarily slow convergence, and sublinear rates are essentially only surpassed in the presence of some Lipschitzian error bound, which is our first ingredient. The second one is a seemingly unfavorable and unexpected condition, namely, infeasibility. For two non-intersecting closed convex sets satisfying an error bound, we establish finite convergence of MAP. In particular, MAP converges in finitely many steps when applied to a polyhedron and a hyperplane in the case in which they have empty intersection. Moreover, the farther the target sets lie from each other, the fewer are the iterations needed by MAP for finding a best approximation pair. Insightful examples and further theoretical and algorithmic discussions accompany our results, including the investigation of finite termination of other projection methods.
Anuj Bajaj (Wayne State University)
Title: Solving Multifacility Location Problems by DC Algorithms
Abstract:
This talk presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov’s smoothing. The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets.
Goebel, Rafal (**Plenary Speaker**, Loyola University Chicago)
Title: Convex analysis, optimization, and switching dynamics for the consensus problem
Abstract:
The consensus problem is about controlling the agents in a multiagent system so that, asymptotically, all agents arrive at the same location. The challenges may lie in the agents only communicating their current locations to their neighbors, in the communication structure changing over time, in constraints on the location of each agent, etc. The talk will introduce the consensus problem and show how elements of convex analysis and switching dynamical system theory lead to a framework for establishing convergence of the agents to the same location. A key result is that appropriate switching between the subdifferential flows for several convex functions leads to convergence to a common minimizer of these functions, if any such minimizers exist. The talk is based on joint work with Ricardo Sanfelice.
Nguyen, Oanh (Wayne State University)
Title: Subdifferential Calculus for ordered set-valued mappings and applications
Abstract:
In this talk we will be presenting the calculus rules including sum and chain rules for subdifferential of set-valued mappings. Then we apply that to achieve the particular conditions for the existence and optimality of multiobjective problems with structural costs and/or constraint such as the problems with geometric constraints; problems with equality and inequality constraints; and problems with operator constraints. This talk is based on the joint work with Prof. Boris Mordukhovich.
Nguyen, Trang (Wayne State University)
Title: Optimization Of Controlled Free-time Sweeping Processes and Applications
Abstract:
This talk addresses a free-time optimal control problem for sweeping processes. We develop a constructive finite-difference approximation procedure that allows us to establish necessary optimality conditions for discrete optimal solutions and then show how these optimality conditions are applied to solving several applications in the real life. This is based on joint work with Boris Mordukhovich and Dao Nguyen.
Nguyen-Truc-Dao Nguyen (University of Michigan)
Title: Optimization of a Controlled Sweeping Process and Applications on General Robotics Models
Abstract:
The talk is mostly devoted to the necessary optimality conditions and applications of a novel optimal control theory for perturbed sweeping/Moreau processes to the general mobile robot dynamics with obstacles. Describing these models as controlled sweeping processes with pointwise/hard control and state constraints and applying new necessary optimality conditions for such systems allow us to develop efficient procedures to solve naturally formulated optimal control problems for the general robotics models and completely calculate optimal solutions in general situations. This is based on joint work with Boris Mordukhovich, Trang Nguyen, and Norma Ortiz-Robinson.
Rios, Vinicio (Louisiana State University)
Title: From one isotropic medium to another in minimal time: Elvis's math instinct?
Abstract:
"The goal of this talk is twofold. On the one hand, we recall the observations of Timothy Pennings about the apparent optimal time trajectory covered by his dog Elvis when fetching his favorite tennis ball thrown into Lake Michigan [1]. Using classical calculus we revisit the connection between Elvis's instinctive strategy and the well-known Snell's Law for isotropic mediums, that is, mediums that have uniformity of refraction in all possible directions. On the other hand, the talk serves as an introduction to the striking generalization of the previous ideas to the context of multiple anisotropic mediums using the powerful tools of convex analysis [2]. The outline of the latter project will be presented by Prof. Peter Wolenski in his corresponding talk at this meeting.
Tran, Dat (Wayne State University)
Title: Generalized Damped Newton Algorithms in Nonsmooth Optimization
Abstract:
The talk proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we design such a globally convergent algorithm to solve a class of nonsmooth convex composite problems with extended-real-valued cost functions, which typically arise in machine learning and statistics. Finally, the obtained algorithmic developments and justifications are applied to solving a major class of Lasso problems with detailed numerical implementations. We present the results of numerical experiments and compare the performance of our main algorithm applied to Lasso problems with those achieved by other first-order and second-order methods. The talk is based on the joint work with Pham Duy Khanh (Ho Chi Minh City University of Education, Vietnam), Boris S. Mordukhovich (Wayne State University, Michigan, US), Vo Thanh Phat (Wayne State University, Michigan, US).
Tuyen, Tran (Loyola University Chicago)
Title: Minimizing Differences of Convex Functions with Applications to Multifacility Location and Clustering
Abstract:
Convex optimization techniques have been the topic of extensive research for more than 50 years, but solving large-scale optimization problems without the presence of convexity remains a challenge. This motivates the search for new optimization methods that are capable of handling broader classes of functions and sets where convexity is not assumed. One successful approach to go beyond convexity is to consider the class of functions representable as differences of convex functions. This talk focuses on optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex with applications to facility location and clustering. Our methods involve DC programming, Nesterov’s smoothing technique, and a numerical algorithm for minimizing differences of convex functions to cope with the nonsmoothness and nonconvexity of these problems. We also provide numerical examples to test our methods in solving multifacility location and clustering problems.
Vo, Phat Thanh (Wayne State University)
Title: Coderivative-Based Newton-type Methods in Composite Optimization
Abstract:
This talk discusses two new globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are coderivative-based and employ generalized Hessians (coderivatives of subgradient mappings) associated with objective functions, which are represented in the form of convex composite optimization, where one of the terms may be extended-real-valued. Problems of convex composite optimization are investigated with and without the strong convexity assumption on of smooth parts of objective functions by implementing the machinery of forward-backward envelopes. Numerical experiments are conducted for a basic class of Lasso problems by providing performance comparisons of the new algorithms with some other first-order and second-order methods that are highly recognized in nonsmooth optimization. (Joint work with Pham Duy Khanh, Boris Mordukhovich and Dat Tran)
Wang, Bigwu (Eastern Michigan University)
Title: Some properties of directionally differentiable functions
Abstract:
We present some properties of directionally differentiable functions in this talk; in particular, we explore some results about the images of sets under directionally differentiable functions.
Wang, Yan (**Special Speaker**, Ford Motors)
Title: Evolution of Automotive Controls and Calibrations with Advanced Optimization Methods
Abstract:
The automotive system started with simple functions but its complexity has been growing exponentially in the last few decades because of stringent governmental regulations on fuel economy and safety, as well as increasing demands from customers. Automotive control systems have become more and more complicated and at the same time more and more advanced, due to the fast advancement of computer technologies and applications of new algorithms. In this talk, we will first give an overview of the evolution of automotive controls and calibrations, then cover different optimization based methods that are suitable to the automotive controls and calibrations. We will demonstrate with results in real hardware with advanced controls and optimization methods.
Wolenski, Peter (**Special Speaker, Louisiana State University)
Title: Generalization of the Elvis Problem
Abstract:
The (classical) Elvis problem refers to a particular type of minimal time problem in which the control dynamics are piece-wise constant and isotropic on two mediums separated by an interface. The somewhat impertinent nomenclature refers to an observation by Timothy Pennings whose dog (named Elvis) enjoyed fetching an object thrown from the shore of Lake Michigan into the water. Elvis was observed to retrieve the object by going in a path that resembled how light would refract (according to Snell's Law) in isotropic mediums. The problem is first generalized to allow for anisotropic velocity sets that are closed, convex, bounded and with 0 in its interior. Tools of Convex Analysis are employed to characterize optimal movement. Further generalizations are then considered with potentially having faster movement on the interface and with more than two mediums.
Zheng, Xiaoming (Central Michigan University)
Title: Optimization of a mixing problem with boundary control
Abstract:
The motivation of this study comes from the mixing problem by minimizing the cost through the stirring force exerted on the container wall. First, we introduce the general optimization of mixing of a scalar field such astemperature or salt stirred by Stokes flow. Second, we introduce a boundary control problem and derive the first order necessary condition of optimal solution. Finally, we present a line search method to solve this problem. This is a collaboration with Dr. Weiweihu at University of Georgia.
Zhu, Qiji (**Plenary Speaker**, Western Michigan University)
Title: Optimization Methods in Bank Balance Sheet Management
Abstract:
Bank balance sheet management is one of the most important tasks for bank managers. In recent years I have been involved in the research of such problems as a mathematician. I found that many optimization methods are needed and targeting an important practical problem has stimulated many new directions of investigation in both theoretical and computational aspects of optimization. In this talk I will share my experience in collaborative investigation on the bank balance sheet management problem which is the most interesting endeavor in my research career. Research reported here involve collaboration with my colleagues from both academic and industry: S. Dewansurendra, P. Judice, S. Maier-Paape, A. Platen, R. Vince, and M. Vazifadan.