Duality of optimization problems with gauge functions

ABSTRACT

Recently, Yamanaka and Yamashita proposed the so-called positively homogeneous optimization problem, which includes many important problems, such as the absolute-value and the gauge optimization problems. They presented a closed form of the dual formulation for the problem, and showed weak duality and the equivalence to the Lagrangian dual under some conditions. In this work, we focus on a special positively homogeneous optimization problem, whose objective function and constraints consist of some gauge and linear functions. We prove not only weak duality but also strong duality. We also study necessary and sufficient optimality conditions associated to the problem. Moreover, we give sufficient conditions under which we can recover a primal solution from a Karush-Kuhn-Tucker point of the dual formulation. Finally, we discuss how to extend the above results to general optimization problems by considering the so-called perspective functions.

KEYWORDS:

• Gauge optimization
• duality theory
• convex optimization
• positively homogeneous functions

Acknowledgments

The authors are grateful to anonymous referees for their remarkably careful reading and helpful suggestions especially in the duality theory in Section 3. The authors are also grateful to Prof. Ellen. H. Fukuda for helpful and detailed comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by a Grant-in-Aid for Scientific Research (C) [grant number 17K00032] from Japan Society for the Promotion of Science.