For composite optimization problems, strong variational sufficiency is equivalent to a generalized strong second-order sufficient condition and to positive definiteness of a generalized augmented Lagrangian Hessian.
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Characterizations of Strong Variational Sufficiency in General Models of Composite Optimization
For composite optimization problems, strong variational sufficiency is equivalent to a generalized strong second-order sufficient condition and to positive definiteness of a generalized augmented Lagrangian Hessian.