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Chasing Convex Functions with Long-term Constraints

Adam Lechowicz, Nicolas Christianson, Bo Sun, Noman Bashir, Mohammad Hajiesmaili, Adam Wierman, Prashant Shenoy

TL;DR

This work tackles online convex optimization with long-term constraints in multidimensional decision spaces, introducing two problem families: CFL and MAL. It develops a threshold-based pseudo-cost minimization algorithm (ALG1) that is provably $\alpha$-competitive, with $\alpha$ determined by problem parameters via $\frac{U-L-2\beta}{U-U/\alpha-2\beta}=\exp(1/\alpha)$, and extends the approach to a learning-augmented setting with the CLIP algorithm achieving the optimal consistency–robustness trade-off. A formal connection between CFL and MAL is established, allowing the competitive guarantees to transfer between settings through a simplex transformation. Numerical experiments on synthetic CFL instances validate the theoretical results, showing strong performance and robustness to advice quality, and demonstrating dimension-free behavior in practice. The work advances online optimization by integrating long-term constraints, multidimensional decision spaces, and switching costs, with potential impact on carbon-aware resource management and sustainable computing systems.

Abstract

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions $\mathbf{x}_t$ in a metric space $(X,d)$ to simultaneously minimize their hitting cost $f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the time horizon $T$, the player must satisfy a long-term demand constraint $\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ denotes the fraction of demand satisfied at time $t$. Such problems can find a wide array of applications to online resource allocation in sustainable energy/computing systems. We devise optimal competitive and learning-augmented algorithms for the case of bounded hitting cost gradients and weighted $\ell_1$ metrics, and further show that our proposed algorithms perform well in numerical experiments.

Chasing Convex Functions with Long-term Constraints

TL;DR

This work tackles online convex optimization with long-term constraints in multidimensional decision spaces, introducing two problem families: CFL and MAL. It develops a threshold-based pseudo-cost minimization algorithm (ALG1) that is provably -competitive, with determined by problem parameters via , and extends the approach to a learning-augmented setting with the CLIP algorithm achieving the optimal consistency–robustness trade-off. A formal connection between CFL and MAL is established, allowing the competitive guarantees to transfer between settings through a simplex transformation. Numerical experiments on synthetic CFL instances validate the theoretical results, showing strong performance and robustness to advice quality, and demonstrating dimension-free behavior in practice. The work advances online optimization by integrating long-term constraints, multidimensional decision spaces, and switching costs, with potential impact on carbon-aware resource management and sustainable computing systems.

Abstract

We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions in a metric space to simultaneously minimize their hitting cost and switching cost as determined by the metric. Over the time horizon , the player must satisfy a long-term demand constraint , where denotes the fraction of demand satisfied at time . Such problems can find a wide array of applications to online resource allocation in sustainable energy/computing systems. We devise optimal competitive and learning-augmented algorithms for the case of bounded hitting cost gradients and weighted metrics, and further show that our proposed algorithms perform well in numerical experiments.
Paper Structure (45 sections, 17 theorems, 70 equations, 12 figures, 1 algorithm)

This paper contains 45 sections, 17 theorems, 70 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Problem Formulation and Preliminaries
  4. Convex function chasing with a long-term constraint.
  5. Assumptions.
  6. An example motivating application.
  7. Online metric allocation with a long-term constraint.
  8. Assumptions.
  9. Competitive analysis.
  10. Learning-augmented consistency and robustness.
  11. A connection between $\mathsf{CFL}$ and $\mathsf{MAL}$.
  12. Designing Competitive Algorithms
  13. Challenges.
  14. Algorithm description.
  15. On the tightness of competitive ratios.
  16. ...and 30 more sections

Key Result

Lemma 2.2

For any $\mathsf{MAL}$ instance on a weighted star metric $(X, d)$, there is a corresponding $\mathsf{CFL}$ instance on $(\mathbb{R}^{n-1}, \lVert \cdot \rVert_{\ell_1 ({\mathbf{w'}})})$ that preserves $f_t^a(\cdot) \ \forall t, c(\cdot) \ \forall a \in X$. Furthermore, for any points $(a,b) \in X$,

Figures (12)

  • Figure 1: CDFs of empirical competitive ratios for various algorithms.
  • Figure 2: Varying adversarial factor $\xi$, with $U/L$ = $250, \beta$ = $50, d$ = $5$, and $\sigma$ = $50$.
  • Figure 3: Varying $U/L$, with $\beta = U/5, d = 5, \xi = 0$, and $\sigma = U/5$.
  • Figure 4: Varying $\beta$, with $U/L = 250, d = 5, \xi = 0$, and $\sigma = 50$.
  • Figure 5: Varying $d$ with $\beta$ = $50, U/L$ = $250, \sigma$ = $50,$ and $\xi$=$0$.
  • ...and 7 more figures

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Definition 3.1: Pseudo-cost threshold function $\phi$ for $\mathsf{CFL}$
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Lemma 4.1
  • Definition 4.2: Pseudo-cost threshold function $\phi^\epsilon$
  • ...and 27 more