A Smooth Equality Reformulation of Nonlinear Exponential Cone Optimization

Abstract

Exponential cone optimization has received considerable attention in the modern literature, but studies involving nonlinear objective and constraint functions are still limited. This work can be seen as one of the first steps toward understanding nonlinear exponential cone optimization (NECO) problems. Here, we ask the question "Can smooth equality constraints replace nonlinear exponential cone constraints?". From a practical point of view, even squared slack reformulations in nonlinear programming introduce difficulties. However, such reformulations can be useful for studying optimality conditions. Motivated by this observation, in this work, we use a smooth surjective parameterization of the exponential cone to reformulate NECO as an equality-constrained nonlinear program (RNLP). We then analyze Karush-Kuhn-Tucker (KKT) and second-order optimality conditions for NECO through RNLP. Although feasibility is represented exactly, the KKT conditions for RNLP do not in general guarantee those for NECO. We characterize this correspondence through multiplier restrictions that are equivalent to a second-order condition involving only slack variables. We also establish an exact curvature identity showing that RNLP reproduces the sigma term of NECO without additional assumptions. Finally, we prove the relationships between second-order necessary and sufficient conditions for the two problems. We identify when RNLP can be used to rule out local minimizers of NECO or establish strict local optimality for NECO without explicitly evaluating the sigma term.