A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm

<table class="fixed-width table-group" id="tab21"><tr><td><table class="table"><colgroup><col style="width:7.14em"/><col style="width:5.91em"/><col style="width:3.80em"/><col style="width:3.83em"/><col style="width:3.78em"/><col style="width:3.99em"/></colgroup><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="3"><span style="width: 19.477ptpx;"><svg height="11.5564pt" id="M770" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 19.477 11.5564" width="19.477pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path></g><g transform="matrix(.013,0,0,-0.013,7.347,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,11.845,0)"><path d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z" id="g113-46"></path></g><g transform="matrix(.013,0,0,-0.013,14.809,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg></span></td><td class="align_center" colspan="5">QPR-T1-1</td></tr><tr class="thead"><td class="align_center">IED + fmin</td><td class="align_center" colspan="4">IED + GA</td></tr><tr class="thead"><td class="align_center">result</td><td class="align_center">best</td><td class="align_center">worst</td><td class="align_center">mean</td><td class="align_center">sd</td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td class="align_left"><span style="width: 18.7472ptpx;"><svg height="12.2284pt" id="M771" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -9.04692 18.7472 12.2284" width="18.7472pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,5.363,0)"><path d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z" id="g113-46"></path></g><g transform="matrix(.013,0,0,-0.013,8.327,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.701,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g></svg></span></td><td class="align_center">—</td><td class="align_center">1.1349</td><td class="align_center">1.4578</td><td class="align_center">1.2706</td><td class="align_center">0.0846</td></tr><tr><td class="align_left"><span style="width: 18.7472ptpx;"><svg height="12.2284pt" id="M772" style="vertical-align:-3.18148pt" version="1.1" viewbox="-0.0498162 -9.04692 18.7472 12.2284" width="18.7472pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,5.363,0)"><path d="M170 255C170 288 146 313 114 313S58 288 58 255C58 221 82 198 114 198S170 221 170 255Z" id="g113-46"></path></g><g transform="matrix(.013,0,0,-0.013,8.327,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g 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class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Comparison of the results obtained by both versions of the IED for problem QPR-T1-1.</div>

Mathematical Problems in Engineering

tab21

Table 21

Table 21: A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm 

Table 21 | A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm 