FAIPA_SAND: An Interior Point Algorithm for Simultaneous ANalysis and Design Optimization

Transcription

1 FAIPA_AN: An Ineror Pon Algorhm for mlaneos ANalyss an esgn Opmzaon osé Hersos*, Palo Mappa* an onel llen** *COPPE / Feeral Unersy of Ro e anero, Mechancal Engneerng Program, Caa Posal 6853, Ro e anero, razl. e-mal: com.frj.br, Web page: hp:// **RENAUT, Research erce A. Golf, 7888 Gyancor cee, France e-mal: renal.fr. Absrac In he classcal approach for Engneerng esgn Opmzaon, a Mahemacal Program ha epens on he esgn arables s sole. Tha s, he objece fncon an he consrans epen eclsely of he esgn arables. Ths, he sae eqaon ha represens he sysem o be esgne ms be sole a each of he eraons of he opmzaon process. The smlaneos analyss an opmal esgn echnqe (AN) consss on ang he sae arables o he esgn arables an nclng he sae eqaon as aonal eqaly consrans [6,7,8,9]. In hs way, he sae eqaon s sole a he same me as he opmzaon problem. We presen a new Algorhm for AN opmzaon ha soles he enlarge problem n a ery effcen way an aes aanage of nmercal ools normally ncle n Engneerng Analyss sofware. Ths one s an eenson of he Feasble Arc Ineror Pon Algorhm, FAIPA.. Keywors: esgn Opmzaon, Nonlnear Programmng, Nmercal Opmzaon, Engneerng esgn. 3. Inrocon We conser he Opmal esgn of Engneerng ysems represene by a ae Eqaon e (, ), n r e R. r The eqaon epens on he parameers R, ha we call esgn arables, beng R he sae arables. The classcal moel for hs problem can be represene by he Nonlnear Program Mnmze f (, ()) sbjec o g(, ()) an h(, ( )), () () soles he sae eqaon for, f s he Objece Fncon, g R m an h R p are he neqaly an he eqaly consrans respecely. We assme ha f, g an h are connos, as well as her frs eraes. The problem () s sole eraely an, a each eraon, he sae eqaon ms be sole an he sensy of he sae arables ms be compe. If he solon of he sae s erae, he whole process can be ery panfl. The smlaneos analyss an opmal esgn echnqe (AN) consss on ang he sae arables o he esgn arables an nclng he sae eqaon as aonal eqaly consrans. Then, he sae eqaon s sole a he same me as he opmzaon problem. Ths s ery aanageos n he case of nonlnear sysems b, on he oher han, he sze of he Mahemacal Program s grealy ncrease. The Nonlnear Program for AN Opmzaon s sae as follows: Mnmze f (, ), sbjec o g(, ) () h(, ) an e(,). We presen a new Nonlnear Programmng Algorhm for AN opmzaon ha soles he enlarge problem n a ery effcen way an aes aanage of nmercal ools normally ncle n Engneerng Analyss sofware. Ths one s an eenson of he Feasble Arc Ineror Pon Algorhm, FAIPA [,,3].

2 FAIPA maes eraons n he prmal an al arables of he opmzaon problem o sole Karsh - Khn - Tcer opmaly conons. Gen an nal neror pon, efnes a seqence of neror pons wh he objece monooncally rece. A each pon, a feasble escen arc s obane an an neac lne search s one along hs one. A each of hese eraons, o compe a feasble arc, FAIPA soles hree lnear sysems wh he same mar. There s classcal a qas-newon erson of FAIPA an also a me Memory qas-newon algorhm. In he presen conrbon we presen a echnqe o rece he sze of he lnear sysems an of he qas-newon mar o he same magnes as n classcal esgn opmzaon. The presen meho can be consere as Rece Newon le Algorhm. In general rece algorhms reqre feasbly of he eqaly consrans a each erae. Ths means ha, a each eraon, feasbly ms be resore wh an erae procere. Ths procere s aoe n he presen meho. 4. FAIPA The Feasble Arc Ineror Pon Algorhm Conser now he sanar Nonlnear Program: R n Mnmze f () sbjec o g() an h(),, g R m an h R p. The Feasble arc neror Pon Algorhm reqres an nal esmae of a he neror of he feasble regon efne by he neqaly consrans, an generaes a seqence of pons also a he neror of hs se. A each eraon, FAIPA efnes an arc ha s feasble wh respec o he neqaly consrans an escen regarng he objece or anoher approprae fncon. Tha s, we can wal along he feasble arc recng he objece an remanng feasble. When only neqaly consrans are consere, FAIPA reces he objece A each eraon. In he complee problem, an ncrease of he objece may be necessary n orer o hae he eqales sasfe. In hs conrbon we conser a qas Newon erson of FAIPA. The Algorhm: Parameers: ϕ >, (, ) α an r >, n n p r R. aa. Inalze, λ > an R symmerc an pose efne. s a feasble pon. ep. Compaon of he recon Compe (,λ, ) an (,λ, ) by solng he lnear sysems: (3) + g() λ Λg () h () If, sop. + h() + G() λ h() f ( ),. an + g() λ + h(), Λg () + G() λ λ. h () If r, ae r >,,,,p. Tae (,r) f ( ) + r sgn[h( ) ]h ( ) φ. [ φ ] If φ(,r) > ae nf ( ϕ) ;( α ) φ(,r) / (,r) ρ else ρ ϕ. + ρ

3 ep. Compaon of he escen feasble arc Tae w I w E g ( ) g () g an h ( ) h () h + +.,...,m. Compe an λ by solng he followng lnear sysem: + g() λ + h() I Λg () + G() λ λw E h () w ep 3. Crlnear search Fn a sep sasfyng a gen consrane lne search creron n he alary fncon φ(, r) an sch ha g ( + + ) < f λ or g ( + + ) g (), oherwse. < ep 4. Upaes e + + an efne new ales for λ >, > Go bac o sep. an symmerc an efne pose. 5. FAIPA_AN Algorhm To smplfy hs presenaon we conser he AN Opmzaon problem wh only neqaly consrans: Mnmze, sbjec o an f (, ) g(, ) e(,). (4) When apple o hs problem, FAIPA soles he lnear sysems: Λ g (, ) e (, ) Λ g e (, ) (, ) g(, ) g(, ) G e(, ) e(, ) λ λ f f λ (, ) (, ) e(, ) λ λ ω ω (5) (6) In general he nmber of egrees of freeom of he moel s mch larger han he nmber of esgn arables. In conseqence, he sze of he sysems aboe an of he qas - Newon mar are grealy ncrease n AN approach. In hs conrbon e presen a new echnqe ha reces he sae arables an he sae eqaons from he lnear sysem also reces he qas Newon mar of he sze of he esgn arables. Now we call From he frs sysem of eqaon (5), we ge δ [ e (, ) ] e(,) an [ e (,)] e (,) δ [ ] e (,) [ f(,) + + (8) [ ] ] (7) (9)

4 Then, we can elmnae he sae eqaon an he corresponng agrange mlplers from he frs sysem n Eq.(5). If we efne M [I ], () MM, () I I n n + r [ I ] R R, () r n+ r [ I] R R, (3) II an we can wre he frs sysems of he Eq. (5 ) as I I, (4) Λ Λ b g g, (5) G λ Λ gδ [ ] f (, ) {I + [ ] I }I δ b f (, ) +. (6) The presen approach wll be effece f b an are compe who nee of sorng. Esng Rece Algorhms resore he sae eqaon a each eraon. Then he hr erm of he rgh se of he Eq. (6) s nll, snce δ. We presen a formlaon ha aos hs procere, ha s eqalen o sole he sae eqaon a each eraon. Or meho also aos he sorage of he qas-newon mar o ealae n he Eq. (5). Wh hs objec, we se lme memory represenaon of qas-newon marces. 5. Usng he lme memory echnqe [4]. e be s + an y f ( + ) f( ), (7) he rec FG pae formla s gen by ss y y +. (8) s s ys + Conserng he q pars { s, y}, q,...,, we efne [y,...,y ] an [s,...,s ]. Y q Ths, we can wre he rec FG pae formla as s he lower ranglar mar ha sasfes q [ ] Y I Y (9) q q ag[s y,..., s y ], s q + y q f > j oherwse q I an + ( ) j, +. We can proe ha ess an s nonsnglar.

5 Then, gen a ecor we can ealae he proc who sorng from Eq.(9) by means of he followng proceres:. Upae, Y an compe,,.. Compe he Cholesy facorzaon of + o oban.. Compe Y p.. Perform a forwar an hen a bacwar sole o oban p Y q. Compe q ] [Y. We se hs procere for ealae b, who nee of resorng he eqlbrm a each eraon. Gen he ecors an, we can se he Eq. (9) o ealae he proc : W W, () Y W. Now we represen each elemen of he mar for j an each row of he mar M for M. Then we wre he mar from he Eq. () as j j M M. () Ths, we ban ealae who sorng by means of (). The secon an hr of he sysems (5) proce he same rece sysems wh he same mar an wh rgh ha we compe n a smlar way as b. 6. Conclsons. We obane a new algorhm ha aos resorng he eqlbrm a each eraon an ha reces he sze of he lnear sysems an of he qas-newon mar o he same magnes as n classcal esgn opmzaon. Ths formlaon maes AN opmzaon of nonlnear engneerng sysems ery effcen, when compare wh he classcal approach. Moreoer, solers of commercal smlaon coes can be employe. These solers ae aanage of he srcre of he lnear sysems an, n general, hey are ery effcen. 7. References.. Hersos an anos G. Feasble Arc Ineror Pon Algorhms for Nonlnear Opmzaon, Forh Worl Congress on Compaonal Mechancs, (n C-ROM), enos Ares, Argenna, ne-ly, Hersos. A Vew on Nonlnear Opmzaon, pg. 7-6, Chaper of he boo " Aances n rcral Opmzaon ",. Hersos E., KUWER Acaemc Pblshers, Hollan, ne, Hersos.A Feasble recons Ineror Pon Technqe for Nonlnear Opmzaon, OTA - ornal of Opmzaon Theory an Applcaons, Vol. 99, Nº, pg. -46, Ocober, 998.

6 4. yr R H, Noceal an chanabel R H. Represenaon of Qas-Newon Marces an her Use n me Memory Mehos, Techncal Repor CU-C-6-9, Unersy of Colorao (oler, CO, 99). 5. enberg G. near an Nonlnear programmng, n prne, lsson-wesley, eone A an Hersos. Ineror Pon Technqes for Opmal Conrol of Varaonal Ineqaly, rcral Opmzaon Research ornal, Vol. 4, Nº /3, pg. -7, Ocober, Hersos, as G P, anos G an Moa oares CM. hape rcral Opmzaon wh an Ineror Pon Mahemacal Programmng Algorhm, rcral Opmzaon an mlscplnary ornal, pg7-5, Ocober. 8. as G, Hersos, Rochnha F. mlaneos hape Opmzaon an Nonlnear Analyss of Elasc ols, Forh Worl Congress on Compaonal Mechancs (n C-ROM), enos Ares, Argenna. ne-ly, Hersos, eone A, as G, anos G. Conac hape Opmzaon: A Mahemacal Programmng Approach, Apple Mechancs n he Amercas. Vol. 6, pg , prne for AAM an ACM - Ro e anero, razl, 4-8 anary, 999. h Pan-Amercan Congress of Apple Mechancs an Eghh Inernaonal Conference on ynamc Problems n Mechancs.