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Related Concept Videos

Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
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A silo with a cylindrical base, flat bottom, and hemispherical roof is a common design in agricultural and industrial storage due to its structural efficiency and ease of construction. Optimizing its dimensions to maximize storage capacity for a given amount of material—i.e., a fixed surface area—is a classic problem in applied calculus and engineering design. The key parameters are the radius r of the base and the height h of the cylindrical section.The total volume of the silo is obtained by...
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Optimization Problems

Optimization problems often involve identifying maximum or minimum values under specific constraints. A well-known example is determining the longest horizontal pipe that can be moved around a right-angled corner, where a 3-meter-wide hallway meets a 2-meter-wide hallway. This scenario, common in architectural design and industrial transport, can be understood conceptually through geometric and trigonometric reasoning.To visualize the problem, consider the pipe as a straight line that touches...
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Lagrange Multipliers: One Constraint

In constrained optimization, the objective is to maximize or minimize a quantity while satisfying a fixed condition. A standard example is a rectangular pen built against a barn wall using 100 meters of fencing. Because the wall provides one side of the enclosure, only the other three sides require fencing. The problem is to find the dimensions that produce the greatest possible area.Let L represent the length parallel to the wall and W the width perpendicular to it. The area of the pen is A =...
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Related Experiment Video

Updated: Jun 16, 2026

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
11:53

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm

Published on: December 9, 2012

Analyzing Quadratic Unconstrained Binary Optimization Problems Via Multicommodity Flows.

Di Wang1, Robert D Kleinberg

  • 1Department of Computer Science Cornell University Ithaca, NY 14853 Telephone: 607-379-1538

Discrete Applied Mathematics (Amsterdam, Netherlands : 1988)
|February 18, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a new method for Quadratic Unconstrained Binary Optimization (QUBO) using multicommodity flow, offering better lower bounds and insights into variable relationships for NP-complete problems.

Related Experiment Videos

Last Updated: Jun 16, 2026

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
11:53

Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm

Published on: December 9, 2012

Area of Science:

  • Optimization
  • Computer Science
  • Operations Research

Background:

  • Quadratic Unconstrained Binary Optimization (QUBO) problems are NP-complete.
  • Existing methods for computing lower bounds C(k) for k > 2 rely on linear programming.

Purpose of the Study:

  • To develop a polynomial-time computable lower bound C(3) for QUBO problems.
  • To explore the use of network flow algorithms for QUBO analysis.
  • To identify relational persistencies among variables in optimal solutions.

Main Methods:

  • Formulating C(3) computation as a maximum multicommodity flow problem.
  • Analyzing saturated edges in the multicommodity flow network.
  • Utilizing single-commodity flow problems to detect variable persistencies.

Main Results:

  • C(3) can be computed efficiently using a maximum multicommodity flow algorithm.
  • The multicommodity flow approach provides lower bounds and information on optimal variable assignments.
  • Relational persistencies (variable relationships) can be identified through flow analysis.

Conclusions:

  • Maximum multicommodity flow offers a novel and efficient method for computing C(3) bounds in QUBO.
  • Network flow analysis provides deeper insights into QUBO problem structures and solutions.
  • This approach enhances the understanding and solvability of NP-complete optimization problems.