Titles & Abstracts

Boris Mordukhovich (Wayne State University)

Title: Optimal Control of the Sweeping Process Generated by Moving Convex Polyhedra

Abstract: We study a new class of optimal control problems of the sweeping (Moreau) process governed by differential inclusions, which are  described by the normal cone mapping to moving polyhedral convex sets in finite-dimensional spaces. The main attention is paid to deriving necessary optimality conditions for such unbounded discontinuous differential inclusions with intrinsic stare constraints. This is done by developing the method of discrete approximations and employing appropriate tools of second-order variational analysis and generalized differentiation.

    Based on the joint work in progress with Giovanni Colombo (University of Padua, Italy),  Rene Henrion (Weierstrass Institute at Berlin, Germany) and Nguyen Dinh Hoang (University of Valparaiso, Chile).

Jim Zhu (Western Michigan University)

Title: How to win the game of Blackjack

Abstract: Counting cards, drinking beer and betting big, that is what movie 21 tell us how to win Blackjack in a casino. In reality more important is how do you bet appropriately. We develop the theory of Kelly and Thorp in analyzing the optimal bet sizes for blackjack by incorporating the practical considerations of players wherein only a finite number of plays shall occur as well as pursuing maximizing risk-adjusted returns. We show that the ratio of return to bet size is approximately proportional to the return / drawdown ratio and is a reasonable proxy for risk-adjusted returns. Thus, bet sizes that maximize this measure or its

marginal increase are also reasonable choices. Both theoretical analysis and computer simulation show that these alternative choices of bet sizes are much more conservative compared to what the Kelly-Thorp theory suggests and makes sense in practice. In principle, the analysis and results here also apply to money management problems for investment as well as more generalized cumulative resource allocation considerations involving risk. This is a joint research with Ralph Vince, The president of LSP Partners, LLC.

Rafal Gobel (Loyola University)

Title: Functions which are quasiconvex under small linear perturbations

Abstract: A quasiconvex function is a function whose sublevel sets are convex. A quasiconvex function which remains quasiconvex after the addition of an arbitrary linear perturbation is convex. A quasiconvex function which remains quasiconvex under any sufficiently small linear perturbation is called robustly quasiconvex. Such functions are the topic of the talk.

    Characterization of quasiconvex and robustly quasiconvex nonsmooth functions through first and second order conditions will be presented. The characterization will rely on viscosity concepts in partial differential equations theory and apply to nondifferentiable and discontinuous functions. Some related uniqueness results for boundary problems will be mentioned. Several convex-analytic properties of robustly quasiconvex functions will be shown. Some of these properties are very similar to what is expected of convex functions. Supporting robustly quasiconvex functions by simpler functions will be discussed, with a duality theory as motivation. Nontrivial examples of robustly quasiconvex functions will be given.

    This is joint work with E.N. Barron and R.R. Jensen.

Ilias Kotsireas (Wilfrid Laurier University)

Title: D-optimal matrices

Abstract: D-optimal matrices are square {-1,+1} matrices with maximal determinant. We will survey various algorithmic techniques for finding such matrices. We will also describe some recent new theoretical results whose algorithmic consequences have yet to be determined.

Dmitriy Drusvyastkiy (Cornel University)

Title: Slope and geometry in variational mathematics

Abstract: Various notions of the "slope" of a real-valued function pervade optimization, and variational mathematics more broadly. In the semi-algebraic setting - an appealing model for concrete variational problems - the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. Highlights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the convergence of von Neumann's alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms. This is joint work with A. Daniilidis (Barcelona), A.D. Ioffe (Technion), M. Larsson (Lausanne), and A.S. Lewis (Cornell).

Gabor Pataki (University of North Carolina)

Title: Nice cones in conic optimization, and their characterizations

Abstract: We call a closed convex cone K nice, if the set  K*+F^\perp is closed for all F faces of K. Nice cones find a variety of uses in convex analysis and optimization:

1. 1)The facial reduction algorithm to reduce a conic linear to a strictly feasible system becomes simpler if the underlying cone is nice.
   

2. 2)The classic question: “when is the linear image of a closed, convex cone closed?” has a very simple answer, if the dual of the cone is nice.
   

3. 3)The question, whether a conic linear system is well-behaved from the standpoint of duality also has a simple characterization, when the underlying cone is nice.
   

Most cones over which we can efficiently optimize – polyhedral, semidefinite, and second order cones – are nice.

    I will review some of the above results, then show recent characterizations of nice cones and a close connection to facial exposedness: nice cones are sandwiched between the class of facially exposed cones, and facially exposed cones, in which the duals of all faces are also facially exposed.

    Finally, I will show a generalization of Ramana’s dual for conic systems over nice cones: given a conic linear system, the dual may have a positive gap with the primal, or nonattainment, if the underlying cone is not polyhedral. If the underlying cone is nice, however, we can write down a perfect dual, that avoids all these pathologies, at the cost of having polynomially many extra variables.

Mehdi Karimi (University of Waterloo)

Title: A Quick-and-Dirty Approach to Robustness in Linear Optimization

Abstract: We introduce methods for dealing with linear programming (LP) problems with uncertain data, using the notion of weighted analytic centers and notions from the area of multi-criteria decision making. Our methods are based on high interaction with the decision maker (DM) and try to find solutions which satisfy most of his/her important criteria. Starting with the drawbacks of different methods for dealing with uncertainty in LP, we explain how our methods improve most of them. We prove that, besides many practical advantages, our approach is theoretically as strong as robust optimization. Interactive cutting-plane algorithms are developed for concave and quasi-concave utility functions.

Yuen-Lam Cheung (University of Waterloo)

Title: Facial reduction for semidefinite programming: theory and practice

Abstract: Semidefinite programming is a powerful modeling tool for a wide range of optimization and feasibility problems. Its prevalent use in practice relies on the fact that a (nearly) optimal solution of a semidefinite program can be obtained efficiently, provided that the semidefinite program and its dual satisfy the Slater condition.

    When the Slater condition does not hold for a given semidefinite program, it is possible to regularize the semidefinite program using facial reduction. In this talk, we give an overview of the facial reduction technique for semidefinite programs, as well as its use in theory and practice. We first study a semidefinite programming relaxation of an NP-hard integer quadratic program that models the side chain positioning problem from protein folding, as a motivating example for the need of regularization via facial reduction. Then we outline the facial reduction technique for general semidefinite programs, consider an algorithmic iplementation of the facial reduction, and study some associated numerical issues. In particular, we show that each iteration of the facial reduction algorithm is backward stable. Finally, we illustrate the theoretical importance of the facial reduction procedure in the topic of sensitivity analysis for semidefinite programs.

Shen Shan (University of Western Ontario)

Title: New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures

Abstract: We investigate a class of Bernstein and a class of completely monotone functions with intriguing applications in convex analysis. We derive representation theorems for Bernstein and completely monotone functions with a convexity condition on their measures. These representation theorems are variants of the classical Bernstein and Levy-Khintchine representation theorems.

Wei Ouyang (Wayne State University)

Title: Metric subregularity of order q for nonconvex functions and its applications to Quasi-Newton method

Abstract: We first provide suficient/necessary conditions in terms of coderivative/weak contigent derivative for a nonconvex multifunction to be metrically subregular of order q in two different settings (with q <= 1 and q > 1 respectively). Then we discuss the stability of strong q-subregularity under partial linearizations and perturbations of q-order approximation. Finally, as application, strong metric q-subregularity is applied to characterize the q-order superlinear convergence for quasi-Newton methods of nonsmooth equations and generalized equations.

Boyan Dimitrov (Kettering University)

Title: From Dependence Between Events to Dependence for Functions for Random Variables

Abstract: We discuss the measures of dependence in uncertainty. Dependence exists and can be quantitatively measured. Some classic concepts in measuring dependence are briefly mentioned for comparison. Main attention is focused on the concept of dependence proposed by Bulgarian mathematician N. Obreshkov in 1963. His ideas were revived by the author in 2010  Originally, dependence between random events is introduced, and then it can be used to define functions of dependence between random variables at any point on the plane. Functions of dependence are defined, and their graphs are treated as surfaces of mutual dependence. This makes possible to visualize the local dependence between two random variables on the plane. This opportunity is immediately used in some examples of recently popular copula distributions to show graphically these functions of dependence.