[Paper Review] Computing a Finite Size Representation of the Set of Approximate Solutions of an MOP

Oliver Schuetze, Carlos A. Coello Coello+2 more|ArXiv.org|Apr 3, 2008

Advanced Multi-Objective Optimization Algorithms8 references3 citations

TL;DR

This paper proposes a novel archiving strategy for stochastic search algorithms to compute a finite, tight approximation of the set of $ε$-efficient solutions in multi-objective optimization (MOP). By leveraging probabilistic convergence guarantees and Hausdorff distance-based quality bounds, the method ensures finite-size archives with controlled approximation error, validated empirically on continuous and discrete MOPs including truss design and knapsack problems.

ABSTRACT

Recently, a framework for the approximation of the entire set of $ε$-efficient solutions (denote by $E_ε$) of a multi-objective optimization problem with stochastic search algorithms has been proposed. It was proven that such an algorithm produces -- under mild assumptions on the process to generate new candidate solutions --a sequence of archives which converges to $E_ε$ in the limit and in the probabilistic sense. The result, though satisfactory for most discrete MOPs, is at least from the practical viewpoint not sufficient for continuous models: in this case, the set of approximate solutions typically forms an $n$-dimensional object, where $n$ denotes the dimension of the parameter space, and thus, it may come to perfomance problems since in practise one has to cope with a finite archive. Here we focus on obtaining finite and tight approximations of $E_ε$, the latter measured by the Hausdorff distance. We propose and investigate a novel archiving strategy theoretically and empirically. For this, we analyze the convergence behavior of the algorithm, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting approximation, and present some numerical results.

Motivation & Objective

• To address the practical limitation of infinite or large archives in continuous multi-objective optimization (MOP), where the set of $ε$-efficient solutions forms an $n$-dimensional object.
• To develop a finite-size representation of the set of approximate solutions ($E_{ε}$) that is both tight and computationally manageable.
• To provide theoretical bounds on the approximation quality (measured by Hausdorff distance) and the cardinality of the resulting archive.
• To empirically validate the method on benchmark MOPs, including continuous truss design and discrete knapsack problems, demonstrating enhanced solution diversity and decision support.

Proposed method

• Introduces a new archiving strategy that dynamically maintains a finite set of candidate solutions while ensuring convergence to the set of $ε$-efficient solutions in the probabilistic sense.
• Employs a distance-based selection mechanism using the Hausdorff distance to measure approximation quality and control archive size.
• Applies a probabilistic convergence analysis to derive theoretical bounds on both the Hausdorff error and the maximum cardinality of the archive.
• Uses a modified evolutionary strategy with the new archiver to explore the parameter space and collect solutions within a compact domain $Q$.
• Incorporates a tolerance parameter $\Delta$ to define a region of interest around a reference solution, enabling the identification of approximate solutions near a preferred point.
• Employs scalarization and dominance-based filtering to maintain non-dominated and $ε$-efficient solutions within the archive.

Experimental results

Research questions

• RQ1 Can a finite-size archive be constructed that tightly approximates the set of $ε$-efficient solutions in continuous MOPs with theoretical guarantees?
• RQ2 What are the theoretical bounds on the Hausdorff distance between the finite archive and the true set of $ε$-efficient solutions?
• RQ3 How does the cardinality of the archive scale with respect to the problem dimension and desired approximation accuracy?
• RQ4 Can the proposed archiving strategy enhance decision-making by offering a diverse set of approximate solutions near a preferred design?

Key findings

• For a continuous truss design problem, the algorithm achieved a Hausdorff distance of approximately 0.0005 with a final archive of 8377 elements, converging in 4 minutes and 21 seconds when $\Delta = 0$.
• With $\Delta = 0.0001$, the same problem required 11 minutes and 17 seconds, yielding a tighter approximation with a smaller Hausdorff error.
• In the {0,1}-knapsack problem with $n=30$, a final archive of 182 elements was obtained for $\epsilon = (2,2)$ and $\Delta = 0.1$, including seven approximate solutions within a 0.6% tolerance of the reference point $x_0$.
• The seven approximate solutions near $x_0$ differed significantly in parameter space—up to 12 items different—demonstrating that approximate solutions offer meaningful diversity for decision-making.
• The method produced a finite, high-quality approximation of $E_{\epsilon}$ with theoretical bounds on both approximation error and archive size, ensuring practical usability.
• The numerical results confirm that considering approximate solutions increases the number of viable options for the decision maker without sacrificing solution quality.