CAREER: Reduced-order Methods for Big-Data Challenges in Nonlinear and Stochastic Optimization

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The objective of this Faculty Early Career Development (CAREER) Program project is to develop a set of new reduced-order algorithms to tackle the big-data challenges in optimization. The last several years have seen an unprecedented growth in the amount of available data. While nonlinear, especially convex programming (CP) models are important to extract useful knowledge from raw data, high problem dimensionality, large data volumes and inherent uncertainty present significant challenges to the design of optimization algorithms. This research aims to attack these challenges by investigating: (i) novel first-order methods for deterministic CP that converge faster, require little structural information and do not rely on line search, based on level methods; (ii) stochastic first-order methods that handle data uncertainty in an optimal manner, based on stochastic approximation; (iii) novel randomization schemes for solving certain challenging deterministic CP problems beyond the capability of first-order methods; and (iv) stochastic first- and zeroth-order methods for general, not necessarily convex, stochastic programs. The research focuses on two fundamental issues across these topics: (i) the study of complexity which provides guarantees on algorithmic performance; and (ii) the exploitation of structures that leads to the design of algorithms with stronger complexity and superior practical performance.If successful, a set of new algorithmic schemes will advance the state-of-the-art in nonlinear and stochastic optimization, bringing many practically relevant data analysis problems within the range of tractability. Example applications include algorithms for faster and more accurate medical image reconstruction and classification, which will be beneficial to healthcare. In addition, in seismology, effective stochastic programming methods will help to build predictive models by measuring thousands of earthquakes detected at seismic stations. The project will also support the PI's educational goals to improve students' learning in operations research, broaden the representation of underrepresented groups in the PhD program, and contribute to open research infrastructure through the development of optimization solvers.