Exponents and logarithms exist in many important applications such as logistic regression, maximum likelihood, relative entropy and so on. Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer nonlinear convex programs involving exponential or logarithm functions can be recast as mixed-integer exponential conic programs (MIECPs). Recently, solver MOSEK is able to solve large-scale continuous exponential conic programs (ECPs). However, unlike mixed-integer linear programs (MILPs) and mixed-integer second-order conic programs (MISOCPs), MIECPs are far beyond development. To harvest the past efforts on MILPs and MISOCPs, this paper presents second-order conic (SOC) and polyhedral approximation schemes for the exponential cone with application to MIECPs. To do so, we first extend and generalize existing SOC approximation approaches in the extended space, propose new scaling and shifting methods, prove approximation accuracies, and derive lower bounds of approximations. We then study the polyhedral outer approximation of the exponential cones in the original space using gradient inequalities, show its approximation accuracy, and derive a lower bound of the approximation. When implementing SOC approximations, we suggest learning the approximation pattern by testing small cases and then applying to the large-scale cases; and for the polyhedral approximation, we suggest using the cutting plane method when solving the continuous ECP and branch and cut method for MIECPs. Our numerical study shows that the proposed scaling, shifting, and polyhedral outer approximation methods outperform solver MOSEK for both continuous ECPs and MIECPs and can achieve up to 20 times speed-ups compared to solver MOSEK when solving MIECPs.

## Citation

Ye, Q., Xie, W. (2021). Second-Order Conic and Polyhedral Approximations of the Exponential Cone: Application to Mixed-Integer Exponential Conic Programs. Available at Optimization Online.