A Global Optimization Algorithm using Lagrangian Underestimates and the Interval Newton Method
 Tim Van Voorhis
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Convex relaxations can be used to obtain lower bounds on the optimal objective function value of nonconvex quadratically constrained quadratic programs. However, for some problems, significantly better bounds can be obtained by minimizing the restricted Lagrangian function for a given estimate of the Lagrange multipliers. The difficulty in utilizing Lagrangian duality within a global optimization context is that the restricted Lagrangian is often nonconvex. Minimizing a convex underestimate of the restricted Lagrangian overcomes this difficulty and facilitates the use of Lagrangian duality within a global optimization framework. A branchandbound algorithm is presented that relies on these Lagrangian underestimates to provide lower bounds and on the interval Newton method to facilitate convergence in the neighborhood of the global solution. Computational results show that the algorithm compares favorably to the Reformulation–Linearization Technique for problems with a favorable structure.
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 Title
 A Global Optimization Algorithm using Lagrangian Underestimates and the Interval Newton Method
 Journal

Journal of Global Optimization
Volume 24, Issue 3 , pp 349370
 Cover Date
 20021101
 DOI
 10.1023/A:1020383700229
 Print ISSN
 09255001
 Online ISSN
 15732916
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Lagrangian dual
 Interval Newton method
 Convex underestimate
 Quadratically constrained quadratic program
 Industry Sectors
 Authors

 Tim Van Voorhis ^{(1)}
 Author Affiliations

 1. Department of Industrial and Manufacturing Systems Engineering, Iowa State University, 2019 Black Engineering, Ames, IA, 50011, USA