Convergence-order analysis of branch-and-bound algorithms for constrained problems

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Abstract

The performance of branch-and-bound algorithms for deterministic global optimization is strongly dependent on the ability to construct tight and rapidly convergent schemes of lower bounds. One metric of the efficiency of a branch-and-bound algorithm is the convergence order of its bounding scheme. This article develops a notion of convergence order for lower bounding schemes for constrained problems, and defines the convergence order of convex relaxation-based and Lagrangian dual-based lower bounding schemes. It is shown that full-space convex relaxation-based lower bounding schemes can achieve first-order convergence under mild assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at Slater points, and at infeasible points when second-order pointwise convergent schemes of relaxations are used. Lagrangian dual-based full-space lower bounding schemes are shown to have at least as high a convergence order as convex relaxation-based full-space lower bounding schemes. Additionally, it is shown that Lagrangian dual-based full-space lower bounding schemes achieve first-order convergence even when the dual problem is not solved to optimality. The convergence order of some widely-applicable reduced-space lower bounding schemes is also analyzed, and it is shown that such schemes can achieve first-order convergence under suitable assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at unconstrained points in the reduced-space, and at infeasible points under suitable assumptions when the problem exhibits a specific separable structure. The importance of constraint propagation techniques in boosting the convergence order of reduced-space lower bounding schemes (and helping mitigate clustering in the process) for problems which do not possess such a structure is demonstrated.

Keywords

Global optimization Constrained optimization Convergence order Convex relaxation Lagrangian dual Branch-and-bound Lower bounding scheme Reduced-space 

Mathematics Subject Classification

49M20 49M29 49M37 49N15 65K05 68Q25 90C26 

Notes

Acknowledgements

The authors would like to thank Garrett Dowdy and Peter Stechlinski for helpful discussions.

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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Process Systems Engineering Laboratory, Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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