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Prediction-Correction Splittings for Nonsmooth Time-Varying Optimization

Nicola Bastianello, Andrea Simonetto, Ruggero Carli

Abstract

We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function. We design an online algorithmic framework based on prediction-correction, which employs splitting methods to solve the sampled instances of the time-varying problem. We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory. Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm.

Prediction-Correction Splittings for Nonsmooth Time-Varying Optimization

Abstract

We address the solution of time-varying optimization problems characterized by the sum of a time-varying strongly convex function and a time-invariant nonsmooth convex function. We design an online algorithmic framework based on prediction-correction, which employs splitting methods to solve the sampled instances of the time-varying problem. We describe the prediction-correction scheme and two splitting methods, the forward-backward and the Douglas-Rachford. Then by using a result for generalized equations, we prove convergence of the generated sequence of approximate optimizers to a neighborhood of the optimal solution trajectory. Simulation results for a leader following formation in robotics assess the performance of the proposed algorithm.

Paper Structure

This paper contains 19 sections, 4 theorems, 59 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prediction-Correction and Splittings
  3. Prediction-correction
  4. Splitting methods
  5. Solving generalized equations
  6. Convergence Analysis
  7. Proof of Theorems \ref{['th:linear-convergence']} and \ref{['th:quadratic-convergence']}
  8. Single-valuedness
  9. Lipschitz continuity
  10. Simulations
  11. Problem formulation
  12. Numerical results
  13. Conclusions and Future Directions
  14. Taylor approximation error
  15. Early termination error
  16. ...and 4 more sections

Key Result

Lemma 1

The sequence $\{\mathbold{x}(k)\}_{k \in \mathbb{N}}$ generated by the DRS according to eq:drs satisfies the following inequality $\square$

Figures (3)

  • Figure 1: The prediction-correction scheme.
  • Figure 1: Tracking error in different settings.
  • Figure 2: Asymptotical error of the FBS as a function of $T_\mathrm{s}$.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Remark 3
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 4 more