Extended duality for nonlinear programming
Duality is an important notion for nonlinear programming (NLP). It provides a theoretical foundation for many optimization algorithms. Duality can be used to directly solve NLPs as well as to derive lower bounds of the solution quality which have wide use in other high-level search techniques such as branch and bound. However, the conventional duality theory has the fundamental limit that it leads to duality gaps for nonconvex problems, including discrete and mixed-integer problems where the feasible sets are generally nonconvex.
In this paper, we propose an extended duality theory for nonlinear optimization in order to overcome some limitations of previous dual methods. Based on a new dual function, the extended duality theory leads to zero duality gap for general nonconvex problems defined in discrete, continuous, and mixed spaces under mild conditions. Comparing to recent developments in nonlinear Lagrangian functions and exact penalty functions, the proposed theory always requires lesser penalty to achieve zero duality. This is very desirable as the lower function value leads to smoother search terrains and alleviates the ill conditioning of dual optimization.
Based on the extended duality theory, we develop a general search framework for global optimization. Experimental results on engineering benchmarks and a sensor-network optimization application show that our algorithm achieves better performance than searches based on conventional duality and Lagrangian theory.
KeywordsNonlinear programming Global optimization Duality gap Extended duality
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- 11.Gould, N.I.M., Orban, D., Toint, P.L.: An interior-point ℓ 1-penalty method for nonlinear optimization. Technical Report RAL-TR-2003-022, Rutherford Appleton Laboratory Chilton, Oxfordshire, UK, November (2003) Google Scholar
- 14.Luo, Z.Q., Pang, J.S.: Error bounds in mathematical programming. Math. Program. Ser. B, 88(2) (2000) Google Scholar
- 16.Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997) Google Scholar
- 23.Wang, X., Xing, G., Zhang, Y., Lu, C., Pless, R., Gill, C.: Integrated coverage and connectivity configura-tion in wireless sensor networks. In: Proc. First ACM Conference on Embedded Networked Sensor Systems (2003) Google Scholar
- 24.Xing, G., Lu, C., Pless, R., Huang, Q.: On greedy geographic routing algorithms in sensing-covered networks. In: Proc. ACM International Symposium on Mobile Ad Hoc Networking and Computing (2004) Google Scholar
- 25.Xing, G., Lu, C., Pless, R., O’Sullivan, J.A.: Co-Grid: An efficient coverage maintenance protocol for distributed sensor networks. In: Proc. International Symposium on Information Processing in Sensor Networks (2004) Google Scholar