Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization
<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow>...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-05-01
|
| Series: | Mathematical and Computational Applications |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2297-8747/27/3/42 |
| _version_ | 1797484633909100544 |
|---|---|
| author | János D. Pintér Frank J. Kampas Ignacio Castillo |
| author_facet | János D. Pintér Frank J. Kampas Ignacio Castillo |
| author_sort | János D. Pintér |
| collection | DOAJ |
| description | <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the largest small polygon with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> vertices, is a polygon with a unit diameter that has a maximal of area <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is known that for all odd values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is a regular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-polygon; however, this statement is not valid even for values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>. Finding the polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>6</mn></mrow></semantics></math></inline-formula> has been a long-standing challenge. In this work, we developed high-precision numerical solution estimates of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists in the efficient numerical solution of the model-class considered. This is followed by results for an illustrative sequence of even values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>, up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>1000</mn></mrow></semantics></math></inline-formula>. Most of the earlier research addressed special cases up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>20</mn></mrow></semantics></math></inline-formula>, while others obtained numerical optimization results for a range of values from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>100</mn></mrow></semantics></math></inline-formula>. The results obtained were used to provide regression model-based estimates of the optimal area sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> of interest, thereby essentially solving the LSP model-class numerically, with demonstrably high precision. |
| first_indexed | 2024-03-09T23:08:14Z |
| format | Article |
| id | doaj.art-c9b89dc33ee6411ab2d727d4d2ffd979 |
| institution | Directory Open Access Journal |
| issn | 1300-686X 2297-8747 |
| language | English |
| last_indexed | 2024-03-09T23:08:14Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematical and Computational Applications |
| spelling | doaj.art-c9b89dc33ee6411ab2d727d4d2ffd9792023-11-23T17:50:41ZengMDPI AGMathematical and Computational Applications1300-686X2297-87472022-05-012734210.3390/mca27030042Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical OptimizationJános D. Pintér0Frank J. Kampas1Ignacio Castillo2Department of Management Science and Information Systems, Rutgers University, 57 US Highway 1, New Brunswick, NJ 08901, USAPhysicist at Large Consulting LLC, Bryn Mawr, PA 19010, USALazaridis School of Business and Economics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, ON N2L 3C5, Canada<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the largest small polygon with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> vertices, is a polygon with a unit diameter that has a maximal of area <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. It is known that for all odd values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is a regular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>-polygon; however, this statement is not valid even for values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>. Finding the polygon <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>S</mi><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>6</mn></mrow></semantics></math></inline-formula> has been a long-standing challenge. In this work, we developed high-precision numerical solution estimates of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></semantics></math></inline-formula>, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists in the efficient numerical solution of the model-class considered. This is followed by results for an illustrative sequence of even values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula>, up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>1000</mn></mrow></semantics></math></inline-formula>. Most of the earlier research addressed special cases up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>20</mn></mrow></semantics></math></inline-formula>, while others obtained numerical optimization results for a range of values from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>6</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>100</mn></mrow></semantics></math></inline-formula>. The results obtained were used to provide regression model-based estimates of the optimal area sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mrow><mi>A</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, for even values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>n</mi></semantics></math></inline-formula> of interest, thereby essentially solving the LSP model-class numerically, with demonstrably high precision.https://www.mdpi.com/2297-8747/27/3/42nonlinear programminglargest small polygons (LSP){<i>LSP</i>(<i>n</i>)} model-classoptimal area sequence {<i>A</i>(<i>n</i>)}revised LSP modelmathematica model development environment |
| spellingShingle | János D. Pintér Frank J. Kampas Ignacio Castillo Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization Mathematical and Computational Applications nonlinear programming largest small polygons (LSP) {<i>LSP</i>(<i>n</i>)} model-class optimal area sequence {<i>A</i>(<i>n</i>)} revised LSP model mathematica model development environment |
| title | Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization |
| title_full | Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization |
| title_fullStr | Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization |
| title_full_unstemmed | Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization |
| title_short | Finding the Conjectured Sequence of Largest Small <i>n</i>-Polygons by Numerical Optimization |
| title_sort | finding the conjectured sequence of largest small i n i polygons by numerical optimization |
| topic | nonlinear programming largest small polygons (LSP) {<i>LSP</i>(<i>n</i>)} model-class optimal area sequence {<i>A</i>(<i>n</i>)} revised LSP model mathematica model development environment |
| url | https://www.mdpi.com/2297-8747/27/3/42 |
| work_keys_str_mv | AT janosdpinter findingtheconjecturedsequenceoflargestsmallinipolygonsbynumericaloptimization AT frankjkampas findingtheconjecturedsequenceoflargestsmallinipolygonsbynumericaloptimization AT ignaciocastillo findingtheconjecturedsequenceoflargestsmallinipolygonsbynumericaloptimization |