First-Order Methods for Large-Scale Optimization and Applications

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Nowadays large-scale optimization problems frequently arise in engineering and sciences, especially in bioinformatics, data mining, machine learning, medical, image processing and statistics. Due to high dimensionality and dense data, they are often very challenging for second-order methods, namely, Newton-type methods as each iteration of the latter methods requires solving a typically dense linear system. It is thus of paramount importance to develop efficient first-order methods for finding low accuracy solutions. This proposal aims at developing efficient first-order methods for large-scale constrained optimization and monotone inclusion problems, and also exploring their applications. The objectives of this proposal consist of: 1) developing first-order methods for solving general convex programming and establishing their iteration complexity; 2) developing first-order methods for solving monotone inclusion problems where inexact or non-Lipschitz continuous operator is involved and/or subproblems are solved inexactly, and establishing their iteration complexity; 3) developing decomposition methods for solving monotone inclusion problem with sum of multiple maximal monotone operators and establishing their iteration complexity; 4) studying first-order line-search methods for solving general nonconvex optimization problems and establishing their iteration complexity; and 5) exploring applications of first-order methods in the aforementioned areas. The anticipated significance of the proposed work includes: 1) providing new insights into first-order methods and enhancing their efficiency and capability for solving much broader convex and nonconvex optimization problems; 2) developing efficient first-order methods for solving the monotone inclusion problems, which will provide powerful tools for solving general saddle point, equilibrium, variational inequality and optimization problems; and 3) developing efficient software of first-order methods for various applications.