Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs
Abstract
:1. Introduction
2. Formulation Symmetry for QuadraticallyConstrained Quadratic Optimization Problems
2.1. Symmetry Detection with Expression Graphs
2.2. Symmetry Detection with Tensors
2.2.1. Sparse Tensor Representation
2.2.2. Converting Matrices to EdgeLabeled, VertexColored Graphs
 If ${I}_{i}={J}_{i}$, i.e., a quadratic term, then $E=E\cup \{\left\{{({v}_{{I}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{\left({v}_{{I}_{i}}\right)}^{r}\right\}\}$.
 else for bilinear term, ${I}_{i}\ne {J}_{i}$, then $E=E\cup \{\left\{{({v}_{{I}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{({v}_{{J}_{i}},{v}_{{K}_{i}})}^{r}\right\}\cap \left\{{({v}_{{I}_{i}},{v}_{{J}_{i}})}^{r}\right\}\}.$
3. Formulation Symmetry Detection via Binary Layered Graphs
Algorithm 1 Algorithm constructing the vertex set 

Algorithm 2 Algorithm constructing the edge set 

4. Numerical Discussion and Comparison to the StateoftheArt
5. Exploiting Symmetry in the Point Packing Problem
Exhaustive Search and 2D Symmetry Removal
Algorithm 3 Algorithm 1—Exhaustive Search 
Input: number of points n, number of grids k, number of occupied grids m. Output: The optimal solution ${d}^{*}$

Algorithm 4 Algorithm 2—2D Symmetry Removal 
Input: number of points n, number of grids k, number of occupied grids m. Output: The optimal solution ${d}^{*}$

 One point on ${V}_{1}$, 2 points on ${V}_{4}$ and 2 points on ${V}_{5}$. This is from ${O}_{3}$.
 Two points on ${H}_{1}$, 1 point on ${H}_{2}$ and 2 points on ${H}_{5}$. This is from ${O}_{4}$.
6. Results and Comparisons
7. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
MINLP  Mixedinteger nonlinear optimization 
QCQP  Quadraticallyconstrained quadratic optimization 
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Kouyialis, G.; Wang, X.; Misener, R. Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs. Processes 2019, 7, 838. https://doi.org/10.3390/pr7110838
Kouyialis G, Wang X, Misener R. Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs. Processes. 2019; 7(11):838. https://doi.org/10.3390/pr7110838
Chicago/Turabian StyleKouyialis, Georgia, Xiaoyu Wang, and Ruth Misener. 2019. "Symmetry Detection for Quadratic Optimization Using Binary Layered Graphs" Processes 7, no. 11: 838. https://doi.org/10.3390/pr7110838