Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding

We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the inte...

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Veröffentlicht in:Journal of optimization theory and applications Jg. 169; H. 3; S. 1042 - 1068
Hauptverfasser: O’Donoghue, Brendan, Chu, Eric, Parikh, Neal, Boyd, Stephen
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.06.2016
Springer Nature B.V
Schlagworte:
Accuracy
Algorithms
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Engineering
Freeware
Intersections
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Operators
Optimization
Source code
Splitting
Studies
Subspaces
Theory of Computation
90C25
Cone programming
Operator splitting
90C06
49M29
49M05
First-order methods
Optimization
ISSN:0022-3239, 1573-2878
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Zusammenfassung:We introduce a first-order method for solving very large convex cone programs. The method uses an operator splitting method, the alternating directions method of multipliers, to solve the homogeneous self-dual embedding, an equivalent feasibility problem involving finding a nonzero point in the intersection of a subspace and a cone. This approach has several favorable properties. Compared to interior-point methods, first-order methods scale to very large problems, at the cost of requiring more time to reach very high accuracy. Compared to other first-order methods for cone programs, our approach finds both primal and dual solutions when available or a certificate of infeasibility or unboundedness otherwise, is parameter free, and the per-iteration cost of the method is the same as applying a splitting method to the primal or dual alone. We discuss efficient implementation of the method in detail, including direct and indirect methods for computing projection onto the subspace, scaling the original problem data, and stopping criteria. We describe an open-source implementation, which handles the usual (symmetric) nonnegative, second-order, and semidefinite cones as well as the (non-self-dual) exponential and power cones and their duals. We report numerical results that show speedups over interior-point cone solvers for large problems, and scaling to very large general cone programs.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-016-0892-3