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Online Interior-point Methods for Time-varying Equality-constrained Optimization

Jean-Luc Lupien, Iman Shames, Antoine Lesage-Landry

TL;DR

This work proposes the first projection-free OCO algorithm admitting time-varying linear constraints and convex generalized inequalities: the online interior-point method for time-varying equality constraints (OIPM-TEC).

Abstract

An important challenge in the online convex optimization (OCO) setting is to incorporate generalized inequalities and time-varying constraints. The inclusion of constraints in OCO widens the applicability of such algorithms to dynamic and safety-critical settings such as the online optimal power flow (OPF) problem. In this work, we propose the first projection-free OCO algorithm admitting time-varying linear constraints and convex generalized inequalities: the online interior-point method for time-varying equality constraints (OIPM-TEC). We derive simultaneous sublinear dynamic regret and constraint violation bounds for OIPM-TEC under standard assumptions. For applications where a given tolerance around optima is accepted, we employ an alternative OCO performance metric -- the epsilon-regret -- and a more computationally efficient algorithm, the epsilon-OIPM-TEC, that possesses sublinear bounds under this metric. Finally, we showcase the performance of these two algorithms on an online OPF problem and compare them to another OCO algorithm from the literature.

Online Interior-point Methods for Time-varying Equality-constrained Optimization

TL;DR

This work proposes the first projection-free OCO algorithm admitting time-varying linear constraints and convex generalized inequalities: the online interior-point method for time-varying equality constraints (OIPM-TEC).

Abstract

An important challenge in the online convex optimization (OCO) setting is to incorporate generalized inequalities and time-varying constraints. The inclusion of constraints in OCO widens the applicability of such algorithms to dynamic and safety-critical settings such as the online optimal power flow (OPF) problem. In this work, we propose the first projection-free OCO algorithm admitting time-varying linear constraints and convex generalized inequalities: the online interior-point method for time-varying equality constraints (OIPM-TEC). We derive simultaneous sublinear dynamic regret and constraint violation bounds for OIPM-TEC under standard assumptions. For applications where a given tolerance around optima is accepted, we employ an alternative OCO performance metric -- the epsilon-regret -- and a more computationally efficient algorithm, the epsilon-OIPM-TEC, that possesses sublinear bounds under this metric. Finally, we showcase the performance of these two algorithms on an online OPF problem and compare them to another OCO algorithm from the literature.
Paper Structure (10 sections, 10 theorems, 57 equations, 2 figures, 2 algorithms)

This paper contains 10 sections, 10 theorems, 57 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions
  4. Assumptions
  5. Newton Step
  6. Algorithms
  7. Numerical Experiment
  8. Conclusion
  9. Proof of Lemma \ref{['lem:nred']}
  10. Proof of Lemma \ref{['lem:eta']}

Key Result

Lemma 1

Let $\Gamma_\eta^t(\mathbf{y}) = d_\eta(\mathbf{x})+\bm{\nu}^\top(\mathbf{A}\mathbf{x} -\mathbf{b}_t)$ be a self-concordant function. The following holds:

Figures (2)

  • Figure 1: Regret comparison of OIPM-TEC, $\epsilon$OIPM-TEC, and MOSP
  • Figure 2: Constraint violation comparison of OIPM-TEC, $\epsilon$OIPM-TEC, and MOSP

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 8 more