Second order sufficient conditions for aclass of bilevel programs with lower level second-order coneprogramming problem

Xiaoni Chi; Zhongping Wan; Zijun Hao

doi:10.3934/jimo.2015.11.1111

Article Contents

2015, Volume 11, Issue 4: 1111-1125. Doi: 10.3934/jimo.2015.11.1111

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Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem

1.
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072

Received: September 2012

Revised: July 2014

Published: October 2015

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Abstract

When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel program with lower level second-order cone programming problem (SOCBLP). In this paper, we show that the Lagrange multiplier set mapping of the lower level problem of a class of the SOCBLPs is upper semicontinuous under suitable assumptions. Based on this fact, we detect the similarities and relationships between the SOCBLP and its KKT reformulation. Then we derive the specific expression of the critical cone at a feasible point, and show that the second order sufficient conditions are sufficient for the second order growth at an M-stationary point of the SOCBLP under suitable conditions.

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Mathematics Subject Classification: Primary: 90C30; Secondary: 90C46.

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