Some results on Lipschitz properties of the optimal values in semi-infinite programming

Abstract

This paper studies certain Lipschitz properties of the optimal value function of a linear semi-infinite programming problem and its dual problem in the sense of Haar. In this setting, it is already known that the optimal value function of the so-called primal problem was Lipschitz continuous around a given stable solvable problem, when perturbations of all the coefficients are allowed. Recently, a Lipschitz constant, depending only on the nominal problem data, has been computed and, the no duality gap under certain stability conditions ensures moreover that the obtained constant still holds for the dual optimal value function. Our approach here is focused on obtaining Lipschitz constants for both primal and dual optimal value functions under weaker hypothesis of stability, which do not preclude, in all the cases, the existence of duality gap. Now, the allowed perturbations are restricted to the coefficients of the objective function of the corresponding problems.

Keywords:

• semi-infinite programming
• Lipschitz properties
• optimal value

AMS Subject Classification :

• 90C31
• 90C34
• 49K40
• 90C46
• 15A39

Acknowledgements

This research has been partially supported by grants MTM2005-08572-C03-02 from MEC (Spain) and FEDER (EU), and ACOMP/2007/247 from Generalitat Valenciana (Spain). The author wishes to thank the anonymous referees and the associate editor for their valuable comments and suggestions.

Notes

^†In this case, π ^d is called dual of π in the sense of Haar.

Additional information

Notes on contributors

F.J. Toledo

Email: [email protected]