Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

The Midpoint Rule for Double Integrals01:30

The Midpoint Rule for Double Integrals

The midpoint rule for a double integral provides a practical method for estimating volume over a rectangular region when the surface height varies continuously. In civil engineering, this method is useful for approximating the amount of soil to be moved when planning a road across uneven terrain. The road footprint is represented as a rectangle in the xy-plane. At the same time, the terrain elevation above a flat reference level is described by a continuous height function f(x,y). The objective...
Hyperbolic and Inverse Hyperbolic Functions: Problem Solving01:30

Hyperbolic and Inverse Hyperbolic Functions: Problem Solving

An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
Calculation of First-Law Quantities II01:24

Calculation of First-Law Quantities II

The first law of thermodynamics establishes that the change in internal energy of a system is given by ΔU = q + w, where q is the heat exchanged, and w is the work performed. For a perfect gas, both internal energy (U) and enthalpy (H) depend solely on temperature. Consequently, for any change of state, whether reversible or irreversible, the internal energy change is determined by integrating the heat capacity at constant volume, and the enthalpy change by integrating the heat capacity at...
Local Maximum and Minimum Values01:31

Local Maximum and Minimum Values

In multivariable calculus, a function of two variables can exhibit local maximum or minimum values at certain points on its surface. A local maximum occurs when the function's value at a point is greater than at all nearby points, while a local minimum occurs when the function’s value is less than at all nearby locations. These points are referred to as local extrema and are of central importance in optimization problems.Local extrema are found at critical points, where the surface becomes...
Fundamental Theorem of Calculus I: Problem Solving01:22

Fundamental Theorem of Calculus I: Problem Solving

In many engineering and environmental applications, accumulated quantities are determined from rates that vary over time. A common example arises in water management, where a supply system pumps water into a storage tank at a rate that changes with time. Accurately determining how much water has entered the tank over a given period is essential for maintaining proper pressure, scheduling operations, and ensuring system safety.The flow rate of water into the tank is described by a time-dependent...
Problem Solving: Volume01:13

Problem Solving: Volume

The volume of a fuel tank mounted on the wing of a jet aircraft can be modeled using the concept of solids of revolution. In this case, the tank is formed by rotating a two-dimensional region, defined by a mathematical function, about the x-axis. The region extends along the axis from zero to two meters, and the resulting three-dimensional shape is symmetric about the axis of rotation. Because the boundary curve lies directly against the axis, the disk method is an appropriate technique for...

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same journal

Computing Optimal Populations for Binary Problems using Logic Minimization.

Evolutionary computation·2026
Same journal

Enhancing Generalization and Scalability for Multi-Objective Optimization with Population Pre-Training.

Evolutionary computation·2026
Same journal

XCS for Sequential Perceptual Aliasing in Multi-Step Decision Making.

Evolutionary computation·2026
Same journal

A dynamic multi-objective evolutionary algorithm using dual-space prediction and surrogate-based sampling.

Evolutionary computation·2026
Same journal

Adapting MOEA/D to CMA-ES for Dealing with Ill-conditioned Multiobjective Problems.

Evolutionary computation·2026
Same journal

Editorial of the Special Issue: Parallel Problem Solving from Nature PPSN 2024 Extended Versions of Best Paper Candidates.

Evolutionary computation·2026

Related Experiment Video

Updated: Jun 18, 2026

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data
09:37

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data

Published on: April 26, 2016

S-Metric calculation by considering dominated hypervolume as Klee's measure problem.

Nicola Beume1

  • 1Faculty of Computer Science, Technische Universität Dortmund, 44221 Dortmund, Germany. nicola.beume@tu-dortmund.de

Evolutionary Computation
|November 18, 2009
PubMed
Summary
This summary is machine-generated.

This study simplifies calculating the S-metric, a key measure for multi-objective optimization, by adapting Klee's measure problem algorithm. The new method offers improved efficiency for evolutionary algorithms using the S-metric.

More Related Videos

Quantitative Optical Microscopy: Measurement of Cellular Biophysical Features with a Standard Optical Microscope
14:09

Quantitative Optical Microscopy: Measurement of Cellular Biophysical Features with a Standard Optical Microscope

Published on: April 7, 2014

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography
08:02

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography

Published on: February 25, 2015

Related Experiment Videos

Last Updated: Jun 18, 2026

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data
09:37

Extracting Metrics for Three-dimensional Root Systems: Volume and Surface Analysis from In-soil X-ray Computed Tomography Data

Published on: April 26, 2016

Quantitative Optical Microscopy: Measurement of Cellular Biophysical Features with a Standard Optical Microscope
14:09

Quantitative Optical Microscopy: Measurement of Cellular Biophysical Features with a Standard Optical Microscope

Published on: April 7, 2014

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography
08:02

Reservoir Condition Pore-scale Imaging of Multiple Fluid Phases Using X-ray Microtomography

Published on: February 25, 2015

Area of Science:

  • Multi-objective optimization
  • Computational geometry
  • Algorithm analysis

Background:

  • The S-metric (dominated hypervolume) is crucial for evaluating Pareto front approximations in multi-objective optimization.
  • Fast S-metric computation is essential for evolutionary algorithms that use it iteratively.

Purpose of the Study:

  • To present a novel, efficient algorithm for calculating the S-metric.
  • To adapt a known algorithm for Klee's measure problem to the specific case of the S-metric.

Main Methods:

  • The S-metric calculation is framed as a specialized instance of Klee's measure problem (KMP).
  • An existing O(n log n + n(d/2) log n) algorithm for KMP is adapted and simplified for the S-metric.
  • The adapted algorithm avoids complex data structures, achieving an O(n(d/2) log n) upper bound for d >= 3.

Main Results:

  • A new algorithm for S-metric calculation is developed with a reduced complexity of O(n(d/2) log n).
  • Conceptual simplifications enable implementation without intricate data structures.
  • Empirical performance is evaluated against a state-of-the-art algorithm using test functions.

Conclusions:

  • The proposed method provides a more efficient approach to S-metric computation.
  • This advancement can benefit evolutionary algorithms and other multi-objective optimization techniques.
  • The adaptation of Klee's measure problem offers a valuable geometric perspective for optimization quality assessment.