Abstract
In rigorous constrained global optimization, upper bounds on the objective function help to reduce the search space. Obtaining a rigorous upper bound on the objective requires finding a narrow box around an approximately feasible solution, which then must be verified to contain a feasible point. Approximations are easily found by local optimization, but the verification often fails. In this paper we show that even when the verification of an approximate feasible point fails, the information extracted from the results of the local optimization can still be used in many cases to reduce the search space. This is done by a rigorous filtering technique called constraint aggregation. It forms an aggregated redundant constraint, based on approximate Lagrange multipliers or on a vector valued measure of constraint violation. Using the optimality conditions, two-sided linear relaxations, the Gauss–Jordan algorithm and a directed modified Cholesky factorization, the information in the redundant constraint is turned into powerful bounds on the feasible set. Constraint aggregation is especially useful since it also works in a tiny neighborhood of the global optimizer, thereby reducing the cluster effect. A simple introductory example demonstrates how our new method works. Extensive tests show the performance on a large benchmark.
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Notes
This is true for simple consistency algorithms such as 2B, HC4, Box consistency and the like. However, for this simple example, constraint propagation with 3B-consistency would (as our new technique) directly reduce the domains to the optimal point. This is a coincidence that is unlikely to happen with more complex quadratic constraints.
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Acknowledgments
This research was supported by the Austrian Science Fund (FWF) under the contract numbers P23554-N13 and P22239-N13. Numerous suggestions by the referees, which markedly improved the presentation of the paper, are gratefully acknowledged.
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Domes, F., Neumaier, A. Constraint aggregation for rigorous global optimization. Math. Program. 155, 375–401 (2016). https://doi.org/10.1007/s10107-014-0851-4
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DOI: https://doi.org/10.1007/s10107-014-0851-4
Keywords
- Constraint aggregation
- Global optimization
- Constraint satisfaction
- Filtering method
- Verified computing
- Interval analysis