A Framework for Constrained Optimization Problems Based on a Modified Particle Swarm Optimization

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/mpe/volume-2016/8627083/figures/8627083.fig.007a.svgz" name="8627083.fig.007a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Best fitness curves for <svg height="11.927pt" id="M447" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 18.3545 11.927" width="18.3545pt" 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="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.259,3.132)"><path d="M453 623H65C60 580 56 530 46 472H80C103 538 110 548 175 548H389C310 380 195 168 90 0L98 -12L175 -2C273 206 366 408 462 610L453 623Z" id="g50-56"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg></td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/mpe/volume-2016/8627083/figures/8627083.fig.007b.svgz" name="8627083.fig.007b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Best fitness curves for <svg height="11.927pt" id="M448" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 18.3544 11.927" width="18.3544pt" 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="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.259,3.132)"><path d="M247 635C116 635 39 518 39 421C39 315 113 237 221 237C239 237 258 241 281 253L349 289C315 140 204 42 60 19L66 -15C90 -15 153 -5 207 16C344 70 447 199 447 385C447 521 373 635 247 635ZM231 598C329 598 356 478 356 390C356 371 355 347 352 325C330 309 297 297 262 297C179 297 127 369 127 456C127 515 156 598 231 598Z" id="g50-58"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg></td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/mpe/volume-2016/8627083/figures/8627083.fig.007c.svgz" name="8627083.fig.007c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>Best fitness curves for <svg height="11.927pt" id="M449" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 18.3545 11.927" width="18.3545pt" 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="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z" id="g113-72"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="684" vert-adv-y="684"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><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><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.259,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z" id="g50-49"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg></td></tr><tr class="fig-image" id="d"><td><object data="https://static.hindawi.com/articles/mpe/volume-2016/8627083/figures/8627083.fig.007d.svgz" name="8627083.fig.007d" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(d) </b>Best fitness curves for <svg height="11.8174pt" id="M450" style="vertical-align:-3.1815pt" version="1.1" viewbox="-0.0498162 -8.6359 13.0189 11.8174" width="13.0189pt" 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="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 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<div>The best fitness curves of different PSO algorithms for the four benchmark test functions.</div>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: A Framework for Constrained Optimization Problems Based on a Modified Particle Swarm Optimization 