Parameterized Convex Universal Approximators for Decision-Making Problems
This work addresses decision-making problems in optimization and control by providing scalable convex approximators, though it appears incremental as it builds on existing max-affine and log-sum-exp networks.
The authors tackled the problem of approximating parameterized convex functions for decision-making by proposing parameterized max-affine (PMA) and parameterized log-sum-exp (PLSE) networks, which generalize existing convex approximators and are proven to be universal approximators, with simulation results showing PLSE outperforms existing methods in minimizer and optimal value errors for high-dimensional cases.
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.