Related Experiment Video
Updated: Jul 29, 2025
19:53
Single-stage Dynamic Reanimation of the Smile in Irreversible Facial Paralysis by Free Functional Muscle Transfer
Published on: March 1, 2015
105.9K
Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs
Hao Hu1, Renata Sotirov2, Henry Wolkowicz3
1Clemson, South Carolina 29634 USA School of Mathematical and Statistical Sciences, Clemson University.
Mathematical Programming
|May 22, 2023
Summary
Facial reduction (FR) and symmetry reduction (SR) techniques enhance semidefinite programming (SDP) via alternating direction method of multipliers (ADMM). This approach improves numerical stability and running time for solving complex combinatorial problems.
Area of Science:
- Optimization
- Numerical Analysis
- Combinatorial Optimization
Background:
- Semidefinite programming (SDP) is a powerful tool for solving complex problems.
- Existing methods for SDP can be computationally intensive and suffer from numerical instability.
- Doubly nonnegative (DNN) relaxations are used for hard combinatorial problems.
Purpose of the Study:
- To integrate facial reduction (FR) and symmetry reduction (SR) techniques into an alternating direction method of multipliers (ADMM) framework.
- To solve doubly nonnegative (DNN) relaxations of hard combinatorial problems more efficiently.
- To improve the numerical stability and reduce the running time of SDP solvers.
Main Methods:
- Combining FR and SR techniques within an ADMM framework.
- Incorporating nonnegativity constraints to solve DNN relaxations.
- Analyzing the singularity degree of DNN relaxations before and after reduction.
Main Results:
- The integrated FR and SR approach fits well within the ADMM framework.
- The method allows for solving DNN relaxations of hard combinatorial problems, including large-scale quadratic assignment problems.
- Significant improvements in numerical stability and running time were observed for both ADMM and interior point methods.
Conclusions:
- The combination of FR and SR offers a significant advantage for solving DNN relaxations of combinatorial problems.
- The proposed method enhances both the efficiency and stability of SDP solvers.
- This approach is effective for large-scale problems with high-dimensional semidefinite constraints.
Related Concept Videos
01:28Symmetry in Maxwell's Equations
3.5K
Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...
3.5K
01:21Plastic Deformations of Members with a Single Plane of Symmetry
113
When a structural member undergoes plastic deformation due to bending, it is crucial to understand the position of the neutral axis and the stress distribution. This member, characterized by a single plane of symmetry, exhibits a uniform stress distribution, with negative stress above the neutral axis and positive stress below. Notably, the neutral axis does not align with the centroid of the cross-section. This misalignment is typical in cases where the cross-section is not rectangular or...
113
01:21Properties of Fourier series II
202
Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
A function f(t) is...
A function f(t) is...
202
01:12Reduced Mass Coordinates: Isolated Two-body Problem
1.4K
In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an...
1.4K
01:27Gauss's Law: Planar Symmetry
8.0K
A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
8.0K
01:19Routh-Hurwitz Criterion II
306
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
306