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

Alexander Fleming, Jens Deussen, Uwe Naumann

TL;DR

The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events to produce correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.

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.

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

TL;DR

The new solver relies on the reduction of a DAEO to a sequence of differential inclusions separated by jump events to produce correct solutions to DAEOs and runs much faster than sequential DAEO solvers that rely only on global optimization.

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.

Paper Structure

This paper contains 12 sections, 1 theorem, 18 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description.
  3. Deterministic Global Optimization with Interval Arithmetic.
  4. Time Stepping
  5. Local Optimizer Tracking
  6. Jump Event Detection
  7. Numerical Experiments
  8. Performance Testing
  9. Robustness Testing
  10. Conclusions
  11. Further Work
  12. Code Reference

Key Result

Theorem 3.1

(Sylvester's Criterion) A matrix $A\in\mathbb{R}^{n\times n}$ is positive-definite if and only if each of the leading principal minors (the upper left blocks of increasing size) of $A$ have positive determinants.

Figures (5)

  • Figure 1: The solution to \ref{['eq:the-easy-one']} as computed by the solver both with and without event correction enabled. $x(t)$ plots \ref{['eq:easy-daeo-solution']}, and the vertical line is located at $\tau_1$, the location of the first event. The data points $x^+(t)$ and $x^-(t)$ show the computed trajectory for $x(t)$ with and without events, respectively. The event detection procedure generates an extra data point at $t={\tau_1}$ that lies on the boundary between two of the component IVPs in \ref{['eq:the-easy-one']}. $\Delta t$ was set to $0.25$ for this computation.
  • Figure 2: Event locator error $\left|\tau - \tau_{exact}\right|$. The event locator error converges with order $\Delta t^2$ to a user-set tolerance, which is used for every root-finding routine in the solver.
  • Figure 3: Convergence rates of the solver with (marked in blue crosses, converges with order 2) and without (marked in orange exes, converges with order 1).
  • Figure 4: Run time of the solver (ms) compared against the time step size. The cost of global optimization at every time step, even in one dimension, quickly becomes prohibitive as $\Delta t$ decreases. There is little difference between the cost of the solver with event correction enabled or disabled, since the solution to \ref{['eq:the-easy-one']} only involves one event correction.
  • Figure 5: Results of running the solver on \ref{['eq:complicated-example']}. The solver correctly detects the emergence and disappearance of local optimizers.

Theorems & Definitions (3)

  • Definition 3.1
  • Theorem 3.1
  • Definition 4.1