TY - JOUR

T1 - Exact formula for the second-order tangent set of the second-order cone complementarity sET∗

AU - Chen, Jein Shan

AU - Ye, Jane J.

AU - Zhang, Jin

AU - Zhou, Jinchuan

N1 - Funding Information:
∗Received by the editors July 25, 2017; accepted for publication (in revised form) July 30, 2019; published electronically November 21, 2019. https://doi.org/10.1137/17M1140479 Funding: The first author’s work is supported by the Ministry of Science and Technology, Taiwan. The research of the second author is supported by the NSERC. The third author’s work is supported by the NSFC (grants 11971220, 11601458). The fourth author’s work is supported by the National Natural Science Foundation of China (grants 11771255, 11801325) and by the Young Innovation Teams of Shandong Province (2019KJI013). †Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan (jschen@ math.ntnu.edu.tw). ‡Corresponding author. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 2Y2 (janeye@uvic.ca). §Department of Mathematics, Southern University of Science and Technology, Shenzhen, People’s Republic of China (zhangj9@sustech.edu.cn). ¶Department of Statistics, School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, People’s Republic of China (jinchuanzhou@163.com).

PY - 2019

Y1 - 2019

N2 - The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the SOC, and calculating the second-order directional derivative of the projection operator over the SOC. As an application, we derive second-order necessary optimality conditions for the mathematical program with SOC complementarity constraints.

AB - The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the SOC, and calculating the second-order directional derivative of the projection operator over the SOC. As an application, we derive second-order necessary optimality conditions for the mathematical program with SOC complementarity constraints.

KW - Mathematical program with second-order cone complementarity constraints

KW - Projection operator

KW - Second-order cone complementarity sets

KW - Second-order directional derivatives

KW - Second-order necessary optimality conditions

KW - Second-order tangent sets

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U2 - 10.1137/17M1140479

DO - 10.1137/17M1140479

M3 - Article

AN - SCOPUS:85076174096

VL - 29

SP - 2986

EP - 3011

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 4

ER -