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For solids whose cross-sectional areas vary in a predictable way, volume can be determined by integrating these areas along an axis perpendicular to the slices. This approach is particularly useful for polyhedral solids, where classical geometric formulas may not be immediately applicable. A tetrahedron provides a clear example of how cross-sectional integration can be applied to a three-dimensional object with continuously changing geometry.Consider a tetrahedron with height h and a base that...
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Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms.

Tobias Kuhn1, Carlos M Fonseca2, Luís Paquete2

  • 1Mathematical Institute, University of Kaiserslautern, Germany.

Evolutionary Computation
|July 3, 2015
PubMed
Summary
This summary is machine-generated.

This study introduces new methods for the hypervolume subset selection problem in multiobjective optimization. Efficient algorithms were developed to find optimal subsets, improving computational time for evolutionary algorithms.

Keywords:
Multiobjective optimizationhypervolumek-link shortest pathsubset selection

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Area of Science:

  • Multiobjective Optimization
  • Computational Geometry
  • Algorithm Design

Background:

  • The hypervolume subset selection problem is crucial for evolutionary algorithms in multiobjective optimization.
  • Efficient algorithms are needed for selecting k-cardinality subsets that maximize the hypervolume indicator.
  • Existing methods require significant computational resources.

Purpose of the Study:

  • To develop novel, efficient formulations for the two-dimensional hypervolume subset selection problem.
  • To improve the computational complexity of finding optimal subsets for evolutionary algorithms.
  • To provide practical algorithmic solutions for a key problem in multiobjective optimization.

Main Methods:

  • Introduced a linear integer programming formulation solvable via its linear programming relaxation.
  • Developed a k-link shortest path formulation on a Monge property digraph.
  • Applied dynamic programming to solve the shortest path formulation.

Main Results:

  • Achieved a dynamic programming solution with O(nk) time complexity for the two-dimensional problem.
  • This improves upon existing O(nk log n) and O(nk log k) algorithms.
  • Numerical results demonstrate efficiency for various n and k values.

Conclusions:

  • The proposed formulations offer significant improvements in computational efficiency for the hypervolume subset selection problem.
  • These advancements are vital for enhancing the performance of evolutionary algorithms in multiobjective optimization.
  • The new algorithms provide practical and faster solutions for selecting nondominated points.