NLP Solutions as Asymptotic Values of ODE Trajectories

TL;DR

This paper demonstrates that solutions to constrained optimization problems can be obtained as asymptotic solutions of ODEs, enabling efficient, parallel, and scalable computation methods including analog circuits.

Contribution

It introduces a novel ODE-based framework for solving optimization problems using an exact penalty approach, avoiding multiple problem solves.

Findings

ODE solutions approximate constrained optimization solutions

Efficient single-ODE solution replaces iterative optimization

Applications include combinatoric problems and analog circuit implementations

Abstract

In this paper, it is shown that the solutions of general differentiable constrained optimization problems can be viewed as asymptotic solutions to sets of Ordinary Differential Equations (ODEs). The construction of the ODE associated to the optimization problem is based on an exact penalty formulation in which the weighting parameter dynamics is coordinated with that of the decision variable so that there is no need to solve a sequence of optimization problems, instead, a single ODE has to be solved using available efficient methods. Examples are given in order to illustrate the results. This includes a novel systematic approach to solve combinatoric optimization problems as well as fast computation of a class of optimization problems using analogic circuits leading to fast, parallel and highly scalable solutions.