No Gap Second-Order Optimality Conditions for Circular Conic Programs

Abstract

In this article, we study the second-order optimality conditions for a class of circular conic optimization problem. First, the explicit expressions of the tangent cone and the second-order tangent set for a given circular cone are derived. Then, we establish the closed-form formulation of critical cone and calculate the “sigma” term of the aforementioned optimization problem. At last, in light of tools of variational analysis, we present the associated no gap second-order optimality conditions. Compared to analogous results in the literature, our approach is intuitive and straightforward, which can be manipulated and verified. An example is illustrated to this end.

Keywords:

• Circular cone
• no gap second-order optimality conditions
• second-order tangent set
• “sigma” term
• tangent cone

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 While finalizing a first version of this work, the authors became aware of an important observation made in Bonnans et al. [5], mainly focus on perturbation analysis on second-order cone programming. One possible way to obtain the results discussed in this article is to transform the circular conic constraints to the second-order cone constraints via the relation (1.3) and then adapt the conclusions based on the framework of second-order cone programming [5]. However, in this article we adopt a constructive way to deal with our mentioned issues. We have the following two reasons: (a) Through these qualitative analysis, we can learn more details on the structure of circular cone, which plays a crucial role on developing optimization theory for nonsymmetric cones. (b) The parameters in our discussion have an important effect on establishing the associated error bound analysis as Drusvyatskiy and Lewis [8] and consequently analyzing convergence rate of numerical algorithms such as proximal point method and its variants.

Additional information

Funding

The first author is supported by the Doctoral Foundation of Tianjin Normal University [grant number 52XB1513], the National Natural Science Foundation of China [grant number 11601389, 11871372]. The second author is supported by Ministry of Science and Technology, Taiwan.