Multi-Objective and Multi-Level Optimization: Algorithms and Applications (3rd Edition)

A special issue of Algorithms (ISSN 1999-4893). This special issue belongs to the section "Combinatorial Optimization, Graph, and Network Algorithms".

Deadline for manuscript submissions: 30 November 2026 | Viewed by 735

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Department of Enterprise Engineering, University of Rome "Tor Vergata", 00133 Rome, Italy
Interests: scheduling; graph theory; optimization; mathematical modeling; supply chain optimization; logistics; transportation; production systems
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Special Issue Information

Dear Colleagues,

Decision-making in real-world applications often requires considering more than one objective to find an effective solution. When (conflicting) objectives are associated with either a single decision-maker or cooperative decision-makers, this typically leads to multi-objective optimization. Here, optional solutions do not have the same image value, as happens in single-objective optimization, but are non-dominated and equivalent and allow the definition of the Pareto front. When objectives are associated with different non-cooperative decision-makers, we fall into the game theory arena; furthermore, when objectives and/or decision-makers have a hierarchy among them, we must cope with nested optimization problems and, therefore, multi-level optimization.

All these problems are computationally difficult to solve, and their resolution typically involves reformulating the latter into several single-objective problems or one single-objective problem by introducing additional (non-linear) constraints. Moreover, to limit the computational burden, before their resolution, it is worthwhile to reduce the number of objectives to a very limited (significative) number by applying proper methodologies.

This Special Issue will collect original manuscripts dealing with multi-objective and multi-level optimization; we invite original papers presenting innovative applications and/or contributing to the wider theory.

Prof. Dr. Massimiliano Caramia
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Algorithms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • multi-objective optimization
  • multi-level optimization
  • decision-making

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Research

36 pages, 2369 KB  
Article
Certified Adaptive Triangulation Sampling for Deterministic Pareto-Surface Reconstruction
by Massimiliano Caramia
Algorithms 2026, 19(6), 476; https://doi.org/10.3390/a19060476 - 11 Jun 2026
Viewed by 439
Abstract
Many deterministic multi-objective optimization methods generate Pareto outcomes by repeatedly solving scalarized subproblems for different preference or reference vectors. When the number of objectives is m3, the resulting samples lie on an (m1)-dimensional Pareto surface [...] Read more.
Many deterministic multi-objective optimization methods generate Pareto outcomes by repeatedly solving scalarized subproblems for different preference or reference vectors. When the number of objectives is m3, the resulting samples lie on an (m1)-dimensional Pareto surface in objective space. For tasks such as visualization, trade-off exploration, interactive decision making, and sensitivity analysis, a finite cloud of non-dominated points may be insufficient; one often needs a continuous surrogate of the Pareto surface together with a quantitative control of its reconstruction error. This paper studies the corresponding outer-loop reconstruction problem: how should new reference vectors be selected so as to reconstruct the Pareto surface to a prescribed uniform accuracy while using as few scalarized solves as possible? We propose Certified Adaptive Triangulation Sampling (CATS), a curvature-aware adaptive triangulation method for reconstructing a Pareto surface from an oracle uz(u), uΔd, where d=m1. CATS builds a simplicial mesh over the reference simplex and refines the cell with the largest local interpolation quantity η(τ)=12maxkMτ,kdiam(τ)2, where Mτ,k is an upper bound on the Hessian norm of the kth component of the oracle-induced map over τ. This quantity matches the natural error scale of affine interpolation for C2 maps. The rigorous certified interpretation of CATS applies when the preference-to-Pareto map is single-valued, C2, and equipped with reliable local Hessian-norm upper bounds. If such bounds are replaced by numerical curvature estimates, the same rule can still be used as an adaptive refinement indicator, but the resulting stopping test is not a formal certificate unless those estimates are themselves validated. Under the certified assumptions, we prove that the stopping condition maxτη(τ)ε guarantees supuΔdz(u)z^(u)ε, and that the oracle complexity of certified simplicial piecewise-affine reconstruction is Θ(εd/2). On the rigorously certified core tests, CATS uses 2.7×3.8× fewer oracle calls than uniform reference-direction sampling and 1.2×1.6× fewer than an AWS-inspired patch-area refinement rule. Additional benchmark studies, evaluated with the same interpolation quantity as a practical stopping indicator, show the same qualitative advantage, especially on anisotropic and localized surface geometries. Full article
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