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# On the Extensions of the Predictor-Corrector Proximal Multiplier (PCPM) Algorithm and Their Applications

Many real-world application problems can be modeled mathematically as constrained convex optimization problems. The scale of such problems can be extremely large, posing significant challenges to traditional centralized algorithms and calling for efficient and scalable distributed algorithms. However, most of the existing works on distributed optimization have been focusing on block-separable problems with simple, linear constraints, such as the consensus-type constraints. The focus of this dissertation is to propose distributed algorithms to solve (possibly non-separable) large-scale optimization problems with more complicated constraints with parallel updating (aka in Jacobi fashion), instead of sequential updating in the form of Gauss-Seidel iterations. In achieving so, this dissertation extends the predictor corrector proximal multiplier method (PCPM) to address three issues when solving a large-scale constrained convex optimization problem: (i) non-linear coupling constraints; (ii) asynchronous iterative scheme; (iii) non-separable objective function and coupling constraints.

The idea of the PCPM algorithm is to introduce a predictor variable for the Lagrangian multiplier to avoid the augmented term, hence removing the coupling of block variables while still achieving convergence without restrictive assumptions. Building upon this algorithmic idea, we propose three distributed algorithms: (i) N-block PCPM algorithm for solving N-block convex optimization problems with both linear and nonlinear coupling constraints; (ii) asynchronous N-block PCPM algorithm for solving linearly constrained N-block convex optimization problems; (iii) a distributed algorithm, named PC

^{2}PM, for solving large-scale convex quadratically constrained quadratic programs (QCQPs). The global convergence is established for each of the three algorithms. Extensive numerical experiments, using various data sets, are conducted on either a single-node computer or a multi-node computer cluster through message passing interface (MPI). Numerical results demonstrate the efficiency and scalability of the proposed algorithms.A real application of the N-block PCPM algorithm to solve electricity capacity expansion models is also studied in this dissertation. A hybrid scenario-node-realization decomposition method, with extended nonanticipativity constraints, is proposed for solving the resulting large-scale optimization problem from a multi-scale, multi-stage stochastic program under various uncertainties with different temporal scales. A technique of orthogonal projection is exploited to simplify the iteration steps, which leads to a simplified N-block PCPM algorithm amenable to massive parallelization during iterations. Such an algorithm exhibits much more scalable numerical performance when compared with the widely used progressive hedging approach (PHA) for solving stochastic programming problems.