An Efficient Local Optimizer-Tracking Solver for Differential-Algebriac Equations with Optimization Criteria

TL;DR

This paper introduces a fast, local optimizer-tracking solver for differential-algebraic equations with embedded optimization, improving efficiency and convergence without relying on costly global optimization at each step.

Contribution

The paper presents a novel local optimizer tracking method for DAEOs that maintains convergence order and reduces computational cost compared to global optimization approaches.

Findings

Solver accurately solves DAEOs with jump events.

Runs significantly faster than global optimization-based solvers.

Preserves the convergence order of the integration method.

Abstract

A sequential solver for differential-algebraic equations with embedded optimization criteria (DAEOs) was developed to take advantage of the theoretical work done by Deussen et al. Solvers of this type separate the optimization problem from the differential equation and solve each individually. The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events. These jump events occur when the global solution to the optimization problem jumps to a new value. Without explicit treatment, these events will reduce the order of convergence of the integration step to one. The solver implements a "local optimizer tracking" procedure to detect and correct these jump events. Local optimizer tracking is much less expensive than running a deterministic global optimizer at every time step. This preserves the order of convergence of the integrator component without sacrificing performance to perform deterministic global optimization at every time step. The newly developed solver produces correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.