Parameterized Convex Minorant for Objective Function Approximation in Amortized Optimization

TL;DR

This paper introduces a Parameterized Convex Minorant (PCM) method for approximating objective functions in amortized optimization, enabling reliable and fast global minimization through convex optimization techniques.

Contribution

The paper proposes a novel PCM-based objective function approximator that is a universal approximator and allows global minimizers to be found via convex optimization.

Findings

PCM approximator effectively learns objective functions.

Global minimizers can be obtained by convex optimization.

Numerical simulations demonstrate the method's efficiency and reliability.

Abstract

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.