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Introduction to Scalars

Introduction to Scalars

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period lasts 50 min," or "the gas tank in my car holds 65 L," or "the distance between the two posts is 100 m." A physical quantity that can be specified completely in this manner is called a scalar quantity. The word "scalar" is a synonym for "number." Time, mass, distance, length, volume,...

Pareto Chart

Pareto Chart

A Pareto chart is a bar graph or a combination of both line and bar graphs. The bar lengths represent the individual values or the frequency, while the lines represent the cumulative total values. In this chart, the longest bars are arranged on the left and the shortest bars on the right, which makes it easier to read and interpret the data. It can also be called a Pareto diagram or Pareto analysis.
The Pareto chart is named after the Italian economist Vilfredo Pareto, who described the Pareto...

Couples: Scalar and Vector Formulation

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One might wonder how the captain of a large ship can navigate through the ocean with just a turn of the steering wheel. The answer lies in the concept of two parallel forces that are equal in magnitude and opposite sense, creating a couple moment.
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Cartesian Form for Vector Formulation

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The Cartesian form for vector formulation is a process to calculate the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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In mechanics, commonly used terms like force, speed, velocity, and work can be classified as either scalar or vector quantities. A scalar is a physical quantity that can be described by its magnitude alone and does not require any directional components. Examples of scalar quantities are mass, area, and length.
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A framework for controllable Pareto front learning with completed scalarization functions and its applications.

Tran Anh Tuan¹, Long P Hoang², Dung D Le²

¹School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Ha Noi, Viet Nam.

Neural Networks : the Official Journal of the International Neural Network Society

|November 1, 2023

Summary

This study introduces Controllable Pareto Front Learning (CPFL) for Multi-Objective Optimization (MOO). CPFL accurately approximates Pareto fronts with improved computational efficiency compared to existing methods.

Keywords:

Hypernetwork Multi-objective optimization Multi-task learning Pareto front learning Scalarization problem

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Area of Science:

Multi-Objective Optimization (MOO)
Computational Intelligence
Machine Learning

Background:

Pareto Front Learning (PFL) approximates Pareto fronts in MOO.
Previous PFL methods lack clear mapping between preference vectors and Pareto optimal solutions.

Purpose of the Study:

To develop a comprehensive framework for PFL called Controllable Pareto Front Learning (CPFL).
To address the ambiguity in previous PFL methods by combining pseudoconvex scalarization with Hypernetworks.

Main Methods:

Demonstrated convergence and completion for MOO using pseudoconvex scalarization functions.
Integrated these aspects into a Hypernetwork framework for CPFL.

Main Results:

The proposed CPFL framework achieves high accuracy in approximating Pareto fronts.
CPFL significantly reduces computational cost during inference compared to prior methods.

Conclusions:

CPFL offers a robust and efficient approach to Multi-Objective Optimization.
The framework provides a comprehensive solution for learning Pareto fronts with controllable aspects.