{
  "abstract": "Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \\cite{CONESVaze} where the objective function \\(f\\) remains fixed but the feasible region evolves over time as a nested sequence \\(S_1 \\supseteq S_2 \\supseteq \\cdots \\supseteq S_T\\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known \\emph{nested convex body chasing} (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost, respectively, for any $β\\in [0,1)$, over a time horizon of $T$. We also show that any {\\it weakly adaptive} online algorithm with $O(T^β)$ regret has a movement cost of $Ω\\left(T^{\\frac{1-β}{2}}\\right)$ for any $β\\in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\\log T)$. To complement this, we show that any online algorithm with sublinear {\\it anytime} regret has a movement cost of $Ω\\left(\\log T\\right)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11207",
  "author_total": 1,
  "authors": [
    {
      "name": "Rahul Vaze"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29p47kke3k1nw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11207",
  "fulltext_url": null,
  "id": "29p47kke3k1nw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11207",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1803|Advanced Bandit Algorithms Research"
    ]
  ],
  "openalex": "W7212251348",
  "pdf_url": "https://arxiv.org/pdf/2609.11207",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29p47kke3k1nw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11207",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11207"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Convex Optimization with Nested Evolving Feasible Sets (CONES) under Time-Varying Loss Functions",
  "title_text": "Convex Optimization with Nested Evolving Feasible Sets (CONES) under Time-Varying Loss Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/29p47kke3k1nw",
  "venue": "arXiv",
  "versions": []
}