A COMPARISON OF THREE NONDEGENERACY CONDITIONS FOR SEMIDEFINITE PROGRAMS

Nondegeneracy assumptions are often needed in order to prove local fast convergence of suitable algorithms as well as in the sensitivity analysis for semidefinite programs. Here we investigate the precise relation between three nondegeneracy concepts introduced in the literature. The nondegeneracy conditions considered here are called KSS-, AHO-, and KN-nondegeneracy since they were first introduced by Kojima, Shida, and Shindoh [Mathematical Programming, 80 (1998), pp. 129–160], Alizadeh, Haeberly, and Overton [Mathematical Programming, 77 (1997), pp. 111–128], and Kanzow and Nagel [SIAM Journal on Optimization, to appear], respectively. While all three conditions are equivalent if strict complementarity holds at a solution of the semidefinite program, we show that KSSnondegeneracy cannot hold without this assumption, whereas the other two nondegeneracy conditions are still equivalent even without strict complementarity. This result provides considerable new insight into both the AHOand the KN-nondegeneracy conditions since the two corresponding definitions are completely different in nature.

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