The alternating direction method of multipliers for mixed-integer optimization applications
- A very large number of practical optimization problems have been expressed as minimization of a convex objective function over a nonconvex set, specifically discrete sets such as the set of integer points. These problems arise in a variety of fields such as statistics and modeling, data analysis, and control. Two general approaches have been used for solving mixed integer optimization problems. One approach is to solve the problem globally using methods such as branch-and-bound, which suffer from exponential worst case time complexity. Another approach is to use heuristics such as semidefinite relaxation hierarchies which terminate in polynomial time, but do not guarantee finding the global solution. In this dissertation, we discuss a generic system of heuristic solutions for such problems. We describe an implementation of these methods in a package called NCVX, as an extension of CVXPY, a Python package for formulating and solving convex optimization problems. Our heuristics, which employ convex relaxations, convex restrictions, local neighbor search methods, and the alternating direction method of multipliers (ADMM), require the solution of a modest number of convex problems, and are meant to apply to general problems, without much tuning. Several numerical examples such as regressor selection, circle packing, the traveling salesman problem, and factor analysis modeling are studied. We also describe a fast optimization algorithm for approximately minimizing convex quadratic functions over the intersection of affine and separable constraints. We discuss the favorable computational aspects of our algorithm, which allow it to run quickly even on very modest computational platforms such as embedded processors. We study numerical examples including hybrid vehicle control and power converter control, and we compare our methods with existing open source and commercial alternatives. Finally we discuss how randomized linear programming heuristics can be used to solve graph isomorphism problems. We motivate our heuristics by showing guarantees under some conditions, and present numerical experiments that show effectiveness of these heuristics in the general case.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Electrical Engineering.
|Boyd, Stephen P
|Boyd, Stephen P
|Statement of responsibility
|Submitted to the Department of Electrical Engineering.
|Thesis (Ph.D.)--Stanford University, 2017.
- © 2017 by Mohammad Reza Takapoui
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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