  Parameterized max-affine (PMA) and parameterized log-sum-exp (PLSE) networks
are proposed for general decision-making problems. The proposed approximators
generalize existing convex approximators, namely, max-affine (MA) and
log-sum-exp (LSE) networks, by considering function arguments of condition and
decision variables and replacing the network parameters of MA and LSE networks
with continuous functions with respect to the condition variable. The universal
approximation theorem of PMA and PLSE is proven, which implies that PMA and
PLSE are shape-preserving universal approximators for parameterized convex
continuous functions. Practical guidelines for incorporating deep neural
networks within PMA and PLSE networks are provided. A numerical simulation is
performed to demonstrate the performance of the proposed approximators. The
simulation results support that PLSE outperforms other existing approximators
in terms of minimizer and optimal value errors with scalable and efficient
computation for high-dimensional cases.
