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Exponents and logarithms are fundamental components in many importantapplications such as logistic regression, maximum likelihood, relative entropy,and so on. Since the exponential cone can be viewed as the epigraph ofperspective of the natural exponential function or the hypograph of perspectiveof the natural logarithm function, many mixed-integer convex programs involvingexponential or logarithm functions can be recast as mixed-integer exponentialconic programs (MIECPs). However, unlike mixed-integer linear programs (MILPs)and mixed-integer second-order conic programs (MISOCPs), MIECPs are still underdevelopment. To harvest the past efforts on MILPs and MISOCPs, this paperpresents second-order conic (SOC) and polyhedral approximation schemes for theexponential cone with application to MIECPs. To do so, we first extend andgeneralize existing SOC approximation approaches in the extended space, proposenew scaling and shifting methods, prove approximation accuracies, and derivelower bounds of approximations. We then study the polyhedral outerapproximation of the exponential cones in the original space using gradientinequalities, show its approximation accuracy, and derive a lower bound of theapproximation. When implementing SOC approximations, we suggest learning theapproximation pattern by testing smaller cases and then applying it to thelarge-scale ones; and for the polyhedral approximation, we suggest using thebranch and cut method for MIECPs. Our numerical study shows that the proposedmethods show speed-ups over solver MOSEK for MIECPs, and the scaling, shifting,and polyhedral outer approximation methods work very well.

Second-Order Conic and Polyhedral Approximations of the Exponential Cone: Application to Mixed-Integer Exponential Conic Programs