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.