On Solving Large-Scale Finite Minimax Problems using Exponential Smoothing

Journal of Optimization Theory and Applications, Vol. 148, No. 2, pp. 390-421

X-ray CT Image Reconstruction on Highly-Parallel Architectures.

Model-based image reconstruction (MBIR) methods for X-ray CT use accurate models of the CT acquisition process, the statistics of the noisy measurements, and noise-reducing regularization to produce potentially higher quality images than conventional methods even at reduced X-ray doses. They do this by minimizing a statistically motivated high-dimensional cost function; the high computational cost of numerically minimizing this function has prevented MBIR methods from reaching ubiquity in the clinic. Modern highly-parallel hardware like graphics processing units (GPUs) may offer the computational resources to solve these reconstruction problems quickly, but simply "translating" existing algorithms designed for conventional processors to the GPU may not fully exploit the hardware's capabilities. This thesis proposes GPU-specialized image denoising and image reconstruction algorithms. The proposed image denoising algorithm uses group coordinate descent with carefully structured groups. The algorithm converges very rapidly: in one experiment, it denoises a 65 megapixel image in about 1.5 seconds, while the popular Chambolle-Pock primal-dual algorithm running on the same hardware takes over a minute to reach the same level of accuracy. For X-ray CT reconstruction, this thesis uses duality and group coordinate ascent to propose an alternative to the popular ordered subsets (OS) method. Similar to OS, the proposed method can use a subset of the data to update the image. Unlike OS, the proposed method is convergent. In one helical CT reconstruction experiment, an implementation of the proposed algorithm using one GPU converges more quickly than a state-of-the-art algorithm converges using four GPUs. Using four GPUs, the proposed algorithm reaches near convergence of a wide-cone axial reconstruction problem with over 220 million voxels in only 11 minutes.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113551/1/mcgaffin_1.pd

A dual framework for low-rank tensor completion

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.Comment: Aceepted to appear in Advances of Nueral Information Processing Systems (NIPS), 2018. A shorter version appeared in the NIPS workshop on Synergies in Geometric Data Analysis 201

Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography

Convex-Concave Min-Max Stackelberg Games

Min-max optimization problems (i.e., min-max games) have been attracting a great deal of attention because of their applicability to a wide range of machine learning problems. Although significant progress has been made recently, the literature to date has focused on games with independent strategy sets; little is known about solving games with dependent strategy sets, which can be characterized as min-max Stackelberg games. We introduce two first-order methods that solve a large class of convex-concave min-max Stackelberg games, and show that our methods converge in polynomial time. Min-max Stackelberg games were first studied by Wald, under the posthumous name of Wald's maximin model, a variant of which is the main paradigm used in robust optimization, which means that our methods can likewise solve many convex robust optimization problems. We observe that the computation of competitive equilibria in Fisher markets also comprises a min-max Stackelberg game. Further, we demonstrate the efficacy and efficiency of our algorithms in practice by computing competitive equilibria in Fisher markets with varying utility structures. Our experiments suggest potential ways to extend our theoretical results, by demonstrating how different smoothness properties can affect the convergence rate of our algorithms.Comment: 25 pages, 4 tables, 1 figure, Forthcoming in NeurIPS 202

Multiplier methods for engineering optimization

International audienceMultiplier methods used to solve the constrained engineering optimization problem are described. These methods solve the problem by minimizing a sequence of unconstrained problems defined using the cost and constraint functions. The methods, proposed in 1969, have been determined to be quite robust, although not as efficient as other algorithms. They can be more effective for some engineering applications, such as optimum design and control oflarge scale dynamic systems. Since 1969 several modifications and extensions of the methods have been developed. Therefore, it is important to review the theory and computational procedures of these methods so that more efficient and effective ones can be developed for engineering applications. Recent methods that are similar to the multiplier methods are also discussed. These are continuous multiplier update, exact penalty and exponential penalty methods