Duality theory interpretation

This chapter introduces the fundamentals of duality theory. Section 4.1 presents the formulation of the primal problem, defines the perturbation function associated with the primal problem and discuss its properties, and establishes the relationship between the existence of optimal multipliers and the stability of the primal problem. Section 4.2 presents the formulation of the dual problem and introduces the dual function and its associated properties along with its geometrical interpretation. Section 4.3 presents the weak and strong duality theorems, while section 4.4 defines the duality gap and establishes the connection between the continuity of the perturbation function and the existence of the duality gap. Further reading in these subjects can be found in Avriel (1976), Bazaraa et al. (1993), Geoffrion (1971), Geoffrion (1972b), Minoux (1986), and Walk (1989). 

Experiments are not only difficult to perform but also lead to alternative interpretations and consequently controversies (Criterion 3). Davisson-Germer had considerable difficulty in understanding their experimental data before it could be accepted as evidence for wave-particle duality. Even leading scientists (Planck, Bohr) were reluctant to accept the wave-particle duality as it conflicted with some of the predictions based on the previous dominant paradigm, namely the wave theory of light. 

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