Parameterized convex minorant (PCM) method is proposed for the approximation of the objective function in amortized optimization. In the proposed method, the objective function approximator is expressed by the sum of a PCM and a nonnegative gap function, where the objective function approximator is bounded from below by the PCM convex in the optimization variable. The proposed objective function approximator is a universal approximator for continuous functions, and the global minimizer of the PCM attains the global minimum of the objective function approximator. Therefore, the global minimizer of the objective function approximator can be obtained by a single convex optimization. As a realization of the proposed method, extended parameterized log-sum-exp network is proposed by utilizing a parameterized log-sum-exp network as the PCM. Numerical simulation is performed for parameterized non-convex objective function approximation and for learning-based nonlinear model predictive control to demonstrate the performance and characteristics of the proposed method. The simulation results support that the proposed method can be used to learn objective functions and to find a global minimizer reliably and quickly by using convex optimization algorithms.

翻译：针对摊销优化中的目标函数逼近问题，本文提出参数化凸下界（PCM）方法。该方法将目标函数逼近器表示为PCM与非负间隙函数之和，其中目标函数逼近器受限于优化变量凸函数PCM的下界。该目标函数逼近器是连续函数的通用逼近器，且PCM的全局最小值点能够达到目标函数逼近器的全局最小值。因此，通过单一凸优化即可获得目标函数逼近器的全局最小值点。作为该方法的实现，本文利用参数化对数-求和-指数网络作为PCM，提出扩展型参数化对数-求和-指数网络。通过参数化非凸目标函数逼近及基于学习的非线性模型预测控制数值仿真，验证了所提方法的性能与特性。仿真结果表明，该方法可有效学习目标函数，并借助凸优化算法可靠快速地定位全局最小值点。