# Parameterized Convex Minorant for Objective Function Approximation in Amortized Optimization

arXiv: https://arxiv.org/abs/2310.02519 · HTML: https://mlsanity.com/paper/2310.02519/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 4 Oct 2023.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Tutorial on amortized optimization](https://mlsanity.com/paper/2202.00665/) (2022)
- [A Universal Approximation Result for Difference of log-sum-exp Neural Networks](https://mlsanity.com/paper/1905.08503/) (2019)
- [Parameterized Convex Universal Approximators for Decision-Making Problems](https://mlsanity.com/paper/2201.06298/) (2022)
- [Monte Carlo Information Geometry: The dually flat case](https://mlsanity.com/paper/1803.07225/) (2018)

## Abstract (arXiv)

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.