September 18, 2018

October 14-15, 2011
13th Midwest Optimization Meeting
Workshop on Large Scale Optimization and Applications

Hosted by the Fields Institute

Location: Oct. 14 Banting Institute (U of T), Rm. 131
Oct. 15 Fields Institute

Organizers: Ilias S. Kotsireas, Boris Mordukhovich, Hristo Sendov, Henry Wolkowicz

Plenary Speaker Abstracts

Jonathan Borwein, University of Newcastle
The Douglas Rachford Algorithm in the Absence of Convexity

The Douglas-Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets.
Convergence is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical optimization or feasibility problems in which one or more of the constraints involved is non-convex.

As a first step toward addressing this deficiency, we provide convergence results for a proto-typical non-convex (phase-recovery) scenario: Finding a point in the intersection of the Euclidean sphere and an affine subspace.
Much of my lecture will be interactive using the interactive geometry package Cinderella and the html applets

This is joint work with Brailey Sims. A companion article is Jonathan M. Borwein and Brailey Sims, "The Douglas-Rachford algorithm in the absence of convexity," Chapter 6, pp. 93-109 in Fixed-Point Algorithms for Inverse Problems in Science and Engineering in Springer Optimization and Its Applications, 2011.

Available at:

Panos M. Pardalos,University of Florida,
Network Analysis and Data Mining in Computational Neuro-Science

Over the last decades, network models have been widely appreciated as an essential part in studying the dynamics of complex systems. The dynamics of the system is tightly connected to the structure of the underlying network. Human brain is an apt example of such a complex system, and network modeling may prove to be beneficial in studying the brain function. However, the network models of the brain have additional complexity, due to its time varying structure. In this lecture, preliminary results on the structure of brain network models based on the analysis of physiological data (fMRI, EEG) are presented. Moreover, we will discuss particular results on Parkinson’s and epileptic human brain
network models.

This is joint work with Jui-Hong Chien, Dmytro Korenkevych, and Syed N Mujahid.

Javier Pena, Carnegie Mellon University
First-order Algorithms for Large-scale Convex Optimization

A number of interesting problems in game theory, signal processing, and machine learning can be modeled as convex optimization problems. However, for many practical instances of current interest, these optimization problems are immense and not amenable to traditional optimization techniques. This talk discusses some theoretical and computational developments in the modern class of "accelerated first-order algorithms" for convex optimization. We describe various deterministic and stochastic first-order algorithms and their effectiveness in solving computational challenges in algorithmic game theory and in regularized linear regression.

Alexander Shapiro, Georgia Institute of Technology
Risk Averse Stochastic Programming

In recent years a theory of the so-called coherent risk measures had emerged. Its application to static (two-stage) stochastic programming is well established and generally accepted by the scientific community. The multistage case, on the other hand, is much more delicate. In this talk we discuss various approaches to multistage risk averse stochastic programming and point to involved conceptual and numerical difficulties.

Contributed Speaker Abstracts

Preprocessing and Regularization for Degenerate Semidefinite Programs
Yuen-Lam Cheng
Current popular algorithms for semidefinite programs (SDP) rely on primal-dual interior point methods, and require Slater's constraint qualification to hold for both the primal and dual problems. However, in practice there are many instances of SDP where Slater's constraint qualification fails. These instances, particularly those that have nonzero duality gap or have unattainable optimal values, may present numerical difficulty to standard SDP solvers. We propose a backward stable preprocessing technique using rank-revealing rotations of the problem followed by facial reduction. This results in a smaller, well-posed, nearby problem that can be solved by standard SDP solves.

Risk Management of Portfolios by CVaR Optimization
Thomas Coleman
University Waterloo
Coauthors: Yuying Li, Lei Zhu
The optimal portfolio selection problem is the fundamental optimization problem in finance - optimally balancing risk and return, with possible additional constraints. Unfortunately the classical optimization approach is very sensitive to estimation error, especially with respect to the estimated mean return, and the resulting efficient frontier may be of little practical value. Indeed it may be dangerous from a risk management point of view. A popular alternative, usually under the banner of "robust optimization" is ultra-conservative and, we argue, not really robust! In this sense it may also be of questionable practical value. We propose an alternative optimization approach - a CVaR optimization formulation that is relatively insensitive to estimation error, yields diversified optimal portfolios, and can be implemented efficiently. We discuss this promising approach in this talk and present strongly supportive numerical results.

Identifiable sets in optimization
Dmitriy Drusvyatskiy

School of Operations Research and Information Engineering, Cornell University
Coauthors: Adrian S. Lewis
Seeking only the most essential ingredients of sensitivity analysis leads to the notion of a locally minimal identifiable set. I will discuss how for concrete optimization problems, such sets typically exist and are smooth - thereby allowing one to reduce nonsmooth sensitivity analysis to the classical smooth setting.

Complexity of Low-Rank Matrix Approximations with Weights or Missing Data
Nicolas Gillis

University of Waterloo
Coauthors: François Glineur
Weighted low-rank approximation (WLRA), a dimensionality reduction technique for data analysis, has been successfully used in several applications, such as in collaborative filtering to design recommender systems or in computer vision to recover structure from motion. In this talk, after introducing and motivating the problem, we prove that computing an optimal weighted low-rank approximation is NP-hard, already when a rank-one approximation is sought. In fact, we show that it is hard to compute approximate solutions to the WLRA problem with some prescribed accuracy. Our proofs are based on reductions from the maximum-edge biclique problem, and apply to strictly positive weights as well as to binary weights; the latter corresponding to low-rank matrix approximation with missing data, also known as low-rank matrix completion with noise.

Superlinear elliptic equations with subcritical and critical exponential growth without the Ambrosetti-Rabinowitz condition
Nguyen Lam

Wayne State University
Coauthors: Guozhen Lu
In this talk, we discuss about the existence of a nontrivial nonnegative solution to a class of elliptic equations which do not satisfy the Ambrosetti-Rabinowitz (AR) condition where the nonlinear term is superlinear at 0 and of subcritical or critical exponential growth at #8734;. We will avoid the well-known Ambrosetti-Rabinowitz condition using a suitable version of Mountain Pass Theorem introduced by G. Cerami. The Trudinger-Moser inequality plays an important role in establishing our results.

The Legendre-Fenchel conjugate of a product of positive definite quadratic forms
Minghua Lin

The Legendre-Fenchel transform (or conjugate) plays an important role in convex analysis. In particular, it appears in the analysis of dual versions of a given optimization problem. It is useful to find explicit (if possible) Legendre-Fenchel conjugates of functions. In this talk, we show results on theLegendre-Fenchel conjugate of a product of positive definite quadratic forms. This relates to an open question posed by Hiriart-Urruty in the field of nonlinear analysis and optimization. This problem was also recently studied by Zhao, who showed that the Legendre-Fenchel conjugate can be found provided the quadratic form is convex. We present strengthened conditions that ensures the convexity of the product of positive definite quadratic forms. The result also relates to the generalized Wielandt's inequality.

The talk is based on joint work with G. Sinnamon

Facility Location with Economies of Scale and Congestion: Models and Column Generation Heuristics
Da Lu

Most literature on facility location assumes a fixed set-up and a linear variable cost. It is known, however, that as volume increases cost savings are achieved through economies of scale, but when volume exceeds a certain level, congestion occurs and marginal costs start to increase. This is best captured by an inverse S-shaped cost function that is initially concave and then turns convex. We study such class of location problems and propose solution methods based on Lagrangian relaxation, column generation, and branch-and-bound. We introduce a nonlinear mixed integer programming formulation that is decomposable by the status of the facilities, i.e., closed, operating under economies of scale, or operating under congestion. The resulting concave and convex subproblems are then solved efficiently as piecewise concave and convex bounded knapsack problems respectively. A heuristic solution is found based on primal information from the Dantzig-Wolfe master problems and the solution of the subproblems. Armed with the Lagrangian lower bound and the heuristic solution, the procedure is embedded in a branch-and-price type algorithm. Unfortunately, due to the nonlinearity of the problem, global optimality is not guaranteed, but high quality solutions are achieved depending on the amount of branching performed. The methodology is tested on three function types and four cost settings. Solutions with an average gap of 1.1\% are found within an average of 22 minutes.

Characterizing generalized derivatives of set-valued maps: Extending the Aubin and Mordukhovich criterions
C.H. Jeffrey Pang

Massachusetts Institute of Technology
For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in a certain sense. These results generalize the Aubin and Mordukhovich criterions. Finally, the convexified limiting coderivative has a bijective relationship with the set of possible generalized derivatives.

Generalized Damped Newton's Method For Nonsmooth Equations
Hung Phan

Mathematics, University of British Columbia Okanagan
Coauthors: Boris S. Mordukhovich (Department of Mathematics, Wayne State University)
This paper concerns developing a numerical method of the Newton-type with line search to solve systems of nonlinear equations described by nonsmooth continuous functions. The method is also known as the damped Newton's method. We propose and justify a new generalized damped Newton algorithm based on graphical derivatives, which have recently been used to derive new Newton-type methods for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local and global convergence.

Efficient solutions for large-scale trust region subproblem

Ting Kei Pong

University of Waterloo
Coauthors: Henry Wolkowicz; Heng Ye
The Trust Region Subproblem (TRS) is the minimization of a quadratic (possibly non-convex) function over a ball. It is the main step of the trust region method for unconstrained optimization, and is a basic tool for regularization. Many efficient algorithms have been proposed and implemented. In particular, current algorithms can exploit sparsity for large scale problems and handle degeneracy, i.e., the so-called hard case. In this talk, we show that a shift and deflate approach changes the hard case into a trivial case. We use this approach, in addition to several additional techniques, to derive an efficient algorithm for solving large instances of TRS, based on the algorithm by Rendl and Wolkowicz.
This is an ongoing work with Henry Wolkowicz and Heng Ye.

Subdifferentials of Supremum Lipschitzian Functions and its Applications to Nonsmooth Semi-Infinite and Infinite Programs
Nghia Tran

Department of Mathematics, Wayne State University
Coauthors: Boris Mordukhovich
The paper concerns the study of subdifferentials of a function which is the supremum of an arbitrary family of uniformly Lipschitzian functions in Asplund spaces. As a consequence we get involved first-order optimality conditions for nonsmooth optimization problems of the so-called infinite programming that are generally defined on infinite-dimensional spaces of decision variables and contain infinitely many of inequality constraints with arbitrary index sets. These problems reduce to semi-infinite programs when the spaces of decision variables are finite-dimensional. We also extend the classical Mangasarian-Fromovitz constraint qualification and introduce some new types of closedness conditions to such infinite programs.

An Approximation Algorithm For Generating Representative Pareto Sets For Multiobjective Integer Progragramming Problems
Ozgu Turgut

Coauthors: Alper Murat
In real life most of the problems have to be evaluated by considering more than one criteria which generally have conflicting characteristics. The proposed study develops an algorithm for efficiently generating representative Pareto front for multi-objective integer programming problems. The algorithm proposed aims at obtaining a diverse representation, which is well dispersed on the actual Pareto front. Time requirement of the algorithm can be tuned by the decision maker without disturbing the represantativeness quality of the solution set significantly. The idea behind the algorithm is based on finding an evenly spaced reference points constructing grids on the criteria space. Next, each grid is searched for non-dominated solutions by utilizing a scalarization function which combines the reference point and Tchebyshev methods and guarantees local Pareto optimality. The proposed method is compared with other algorithms in literature in terms of the quality of the solution sets generated and computational efficiency.

An Optimal Trend Following Trading Strategy
Jim Zhu

Modeling an equity market with up and down trends using geometric Browning motions whose drifts switch between two regimes following a markovian chain, we show that if only long and flat positions are allowed the strategy that optimizes the cumulative return is a trend following method characterized by a sequence of stopping times. It is shown that these stopping times can be determined by two threshold curves that can be obtained by solving the associated HJB equations. We also discuss how to handle short selling. Simulations and empirical tests with historical market data are conducted to see how these methods work in practice. This is a joint research with Min Dai (National University of Singapore) and Qing Zhang (University of Georgia, Athens).

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