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To Appear in SIAM Journal of Optimization Constructing Generalized Logarithmic-exponential Functions Using Convex Functions with Regularity Conditions

The logarithmic-exponential (log-exp) function has been widely used in convex analysis and mathematical programming. This paper studies a natural generalization of the log-exp function. Certain necessary and sufficient conditions are obtained for establishing such a generalization. The derived sufficient conditions are explicitly expressed in terms of the first and second derivatives of the functions involved, and hence can be easily checked. We show that some classes of convex functions with certain regularity (such as S∗-regularity and self-regularity) can be used to construct such generalized log-exp functions.