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| Author(s): |
Felipe Eduardo Atenas Maldonado
Total Authors: 1
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| Document type: | Doctoral Thesis |
| Press: | Campinas, SP. |
| Institution: | Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica |
| Defense date: | 2023-04-09 |
| Examining board members: |
Paulo José da Silva e Silva;
Andrzej Piotr Ruszczynski;
Radu Ioan Bot;
José Mario Martínez Pérez;
Welington Luis de Oliveira
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| Advisor: | Claudia Alejandra Sagastizabal; Paulo José da Silva e Silva |
| Abstract | |
Complex large-scale optimization problems usually display a block-separable structure that allows the use of decomposition techniques. Decomposition methods appropriately handle couplings in the constraints or in the variables in order to exploit the separable structure. This thesis explores decomposition methods for convex and nonconvex optimization problems, that first solve decentralized subproblems, and then coordinate the distributed information. Our goal is two-fold. On one hand, we propose a general unifying framework for convergence analysis of descent methods in nonconvex optimization, including rates of convergence under mild regularity assumptions. Decomposition algorithms frequently provide descent for some improvement measure throughout iterations, such as reduction of some merit function or the distance to the set of solutions. Approaches included in this framework are proximal bundle methods, and the Douglas-Rachford splitting method for weakly convex optimization. On the other hand, we extend the unifying analysis to constrained optimization problems over a linear subspace. This allows us to develop decomposition techniques that capitalize on the scenario separability of multistage stochastic optimization problems, since the linking constraints for these problems are represented by certain linear subspace, related to nonanticipativity in the decision process. For convex problems, we propose two novel methods that replace certain Augmented Lagrangian by separable approximations inducing separability in the dual problem. The coupling constraints are modeled with a linear function constructed using forward-backward steps. These methods are variants of the Progressive Hedging algorithm by Rockafellar and Wets, with the key difference of being convergent also with varying stepsizes along iterations. The two proposed methods differ in the way that improvement along iterations is evaluated. The first method corresponds to a proximal bundle-like adaptation of the Progressive Hedging algorithm, that measures sufficient descent of the cost function. The second one is a splitting technique that employs a relative-error acceptance test, assessing infeasibility and model accuracy. Both methods are shown to generate primal-dual sequences that convergence to solutions to the primal and dual problems, respectively, with linear speed of convergence (AU) | |
| FAPESP's process: | 19/20023-1 - Sthochastic variational analysis applied to the energy industry |
| Grantee: | Felipe Eduardo Atenas Maldonado |
| Support Opportunities: | Scholarships in Brazil - Doctorate |