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="tab20"><tr><td><table class="table"><colgroup><col style="width:7.14em"/><col style="width:5.91em"/><col style="width:4.53em"/><col style="width:4.53em"/><col style="width:4.47em"/><col style="width:3.83em"/></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="M762" 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">QQR-T1-2</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="M763" 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">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−8.3391</td><td class="align_center">−8.4729</td><td class="align_center">0.0481</td></tr><tr><td class="align_left"><span style="width: 18.7472ptpx;"><svg height="12.2284pt" id="M764" 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="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g></svg></span></td><td class="align_center">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−8.3391</td><td class="align_center">−8.4729</td><td class="align_center">0.0481</td></tr><tr><td class="align_left"><span style="width: 46.224ptpx;"><svg height="13.8027pt" id="M765" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -10.5116 46.224 13.8027" width="46.224pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-.271)"><path d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z" id="g119-37"></path></g><rect height="0.65243" width="10.2955" x="9.96925" y="-9.80939"></rect><g transform="matrix(.013,0,0,-0.013,9.969,0)"><path d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z" id="g113-110"></path></g><g transform="matrix(.013,0,0,-0.013,22.444,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,27.807,0)"><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,-0.013,30.771,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,36.145,3.132)"><path d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z" id="g50-30"></path></g></svg></span></td><td class="align_center">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−8.3391</td><td class="align_center">−8.4729</td><td class="align_center">0.0481</td></tr><tr><td class="align_left"><span style="width: 39.2925ptpx;"><svg height="12.6081pt" id="M766" style="vertical-align:-0.3499002pt" version="1.1" viewbox="-0.0498162 -12.2582 39.2925 12.6081" width="39.2925pt" 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="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.382,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.204,-5.741)"><path d="M178 262C178 297 153 323 118 323C84 323 59 297 59 262C59 225 84 201 118 201C153 201 178 225 178 262Z" id="g50-46"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.361,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0065,0,0,-0.0065,15.183,-3.641)"><path d="M452 139L416 148C386 81 373 77 319 77H141L282 220C372 312 414 373 414 464C414 569 332 635 240 635C181 635 132 609 101 574L42 493L70 467C98 506 143 562 209 562C280 562 324 515 324 435C324 347 261 272 197 197C148 139 89 84 34 29V0H417L452 139Z" id="g176-51"></path></g><g transform="matrix(.013,0,0,-0.013,22.344,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,32.881,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg></span></td><td class="align_center">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−7.5694</td><td class="align_center">−8.2675</td><td class="align_center">0.3201</td></tr><tr><td class="align_left"><span style="width: 39.2925ptpx;"><svg height="12.6081pt" id="M767" style="vertical-align:-0.3499002pt" version="1.1" viewbox="-0.0498162 -12.2582 39.2925 12.6081" width="39.2925pt" 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="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.382,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.204,-5.741)"><path d="M178 262C178 297 153 323 118 323C84 323 59 297 59 262C59 225 84 201 118 201C153 201 178 225 178 262Z" id="g50-46"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.361,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0065,0,0,-0.0065,15.183,-3.641)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z" id="g176-50"></path></g><g transform="matrix(.013,0,0,-0.013,22.344,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,32.881,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g></svg></span></td><td class="align_center">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−7.5694</td><td class="align_center">−8.2675</td><td class="align_center">0.3201</td></tr><tr><td class="align_left"> <svg height="14.8949pt" id="M768" style="vertical-align:-2.636701pt" version="1.1" viewbox="-0.0498162 -12.2582 72.2536 14.8949" width="72.2536pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-.271)"><path d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z" id="g119-37"></path></g><rect height="0.65243" width="10.2955" x="9.96925" y="-9.80939"></rect><g transform="matrix(.013,0,0,-0.013,9.969,0)"><path d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z" id="g113-110"></path></g><g transform="matrix(.013,0,0,-0.013,20.265,0)"><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,-0.013,24.763,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.145,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.967,-5.741)"><path d="M178 262C178 297 153 323 118 323C84 323 59 297 59 262C59 225 84 201 118 201C153 201 178 225 178 262Z" id="g50-46"></path></g><g transform="matrix(.0091,0,0,-0.0091,36.124,-5.741)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path></g><g transform="matrix(.0065,0,0,-0.0065,39.946,-3.641)"><path d="M1017 229C1017 354 919 454 794 454C671 454 601 383 539 282C472 388 394 454 266 454C171 454 39 379 39 212C39 98 123 -12 266 -12C388 -12 456 59 518 159C585 52 665 -12 794 -12C893 -12 1017 64 1017 229ZM493 201C440 112 375 29 276 29C193 29 137 120 137 242C137 357 195 413 254 413C361 413 432 304 493 201ZM917 204C917 74 866 29 804 29C695 29 624 138 564 240C616 329 682 413 782 413C868 413 917 322 917 204Z" id="g176-30"></path></g><g transform="matrix(.013,0,0,-0.013,50.774,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,61.31,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,67.55,0)"><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></td><td class="align_center">−8.4985</td><td class="align_center">−8.4958</td><td class="align_center">−7.5694</td><td class="align_center">−8.2675</td><td class="align_center">0.3201</td></tr><tr><td class="align_left"><span style="width: 76.3192ptpx;"><svg height="15.4384pt" id="M769" style="vertical-align:-3.8339pt" version="1.1" viewbox="-0.0498162 -11.6045 76.3192 15.4384" width="76.3192pt" 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="M797 0V26C739 32 732 36 732 103V296C732 394 682 449 605 449C576 449 550 437 529 423C504 407 475 389 446 366C425 418 382 449 334 449C303 449 279 437 253 421C222 403 201 385 180 371V452C135 432 85 419 41 411V388C99 379 102 374 102 310V103C102 38 93 32 27 26V0H238V26C189 32 180 38 180 103V338C210 363 250 390 289 390C351 390 377 348 377 275V103C377 37 368 32 306 26V0H520V26C465 32 456 38 456 101V296C456 314 455 326 453 338C491 369 529 390 565 390C628 390 653 345 653 274V107C653 36 642 32 583 26V0H797Z" id="g121-107"></path></g><g transform="matrix(.013,0,0,-0.013,10.608,0)"><path d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z" id="g121-95"></path></g><g transform="matrix(.013,0,0,-0.013,16.315,0)"><path d="M474 0V26C414 34 401 43 364 100L267 248C300 297 324 332 345 358C381 400 394 405 455 411V437H272V411C316 406 323 401 305 370C287 337 267 306 247 276L188 369C169 397 173 405 215 411V437H16V411C71 404 83 396 114 348L201 212C171 167 144 127 116 92C77 42 66 34 4 26V0H190V26C139 34 136 43 156 77C175 113 198 150 220 183L294 66C311 39 302 31 260 26V0H474Z" id="g121-118"></path></g><g transform="matrix(.013,0,0,-0.013,22.477,0)"><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,-0.013,26.975,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,32.338,0)"><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,-0.013,35.302,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,40.676,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.013,0,0,-0.013,47.802,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path></g><g transform="matrix(.013,0,0,-0.013,52.945,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.013,0,0,-0.013,58.308,0)"><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,-0.013,61.272,.241)"><path d="M162 -163V703H102V-163H162ZM309 -163V703H249V-163H309Z" id="g113-10"></path></g><g transform="matrix(.0091,0,0,-0.0091,66.646,-5.755)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.0091,0,0,-0.0091,66.646,3.784)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.013,0,0,-0.013,71.592,0)"><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">−8.4985</td><td class="align_center">−8.4980</td><td class="align_center">−8.0434</td><td class="align_center">−8.3737</td><td class="align_center">0.1564</td></tr><tr 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 QQR-T1-2.</div>

Mathematical Problems in Engineering

tab20

Table 20

Table 20: A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm 

Table 20 | A Hybrid Epigraph Directions Method for Nonsmooth and Nonconvex Constrained Optimization via Generalized Augmented Lagrangian Duality and a Genetic Algorithm 