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Prediction-Correction for Nonsmooth Time-Varying Optimization via Forward-Backward Envelopes

Nicola Bastianello, Andrea Simonetto, Ruggero Carli

Abstract

We present an algorithm for minimizing the sum of a strongly convex time-varying function with a time-invariant, convex, and nonsmooth function. The proposed algorithm employs the prediction-correction scheme alongside the forward-backward envelope, and we are able to prove the convergence of the solutions to a neighborhood of the optimizer that depends on the sampling time. Numerical simulations for a time-varying regression problem with elastic net regularization highlight the effectiveness of the algorithm.

Prediction-Correction for Nonsmooth Time-Varying Optimization via Forward-Backward Envelopes

Abstract

We present an algorithm for minimizing the sum of a strongly convex time-varying function with a time-invariant, convex, and nonsmooth function. The proposed algorithm employs the prediction-correction scheme alongside the forward-backward envelope, and we are able to prove the convergence of the solutions to a neighborhood of the optimizer that depends on the sampling time. Numerical simulations for a time-varying regression problem with elastic net regularization highlight the effectiveness of the algorithm.

Paper Structure

This paper contains 16 sections, 3 theorems, 77 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prediction-Correction with Envelopes
  3. Prediction-correction
  4. Forward-backward envelope
  5. Proposed algorithm
  6. Convergence Analysis
  7. Simulations
  8. Conclusion
  9. Proofs
  10. Approximation error
  11. Computation of $J$
  12. Computation of $K$
  13. Approximation error bound
  14. Early termination errors
  15. Overall error bound
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

Let Assumptions as:first-as:first-bis hold, and choose the parameters $P$ and $C$ in such a way that Then the trajectory $\{\mathbold{x}_k\}_{k \in \mathbb{N}}$ generated by Algorithm alg:envelope-prediction-correction converges to a neighborhood of the optimal trajectory $\{\mathbold{x}_k^*\}_{k \in \mathbb{N}}$ as $\square$

Figures (1)

  • Figure 1: Experimental results.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Dini dontchev2014implicit
  • Remark 7