INTERIOR POINT ALGORITHMS FOR NONLINEAR CONSTRAINED LEAST SQUARES PROBLEMS

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1 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, INTERIOR POINT ALGORITHMS FOR NONLINEAR CONSTRAINED LEAST SQUARES PROBLEMS José Hersovs*, Verase Dubeu* *Mechacal Egeerg Progra, COPPE Federal Uversy of Ro de Jaero, UFRJ Ro de Jaero, RJ, Brazl. jose@ oze.ufrj.br verase@ oze.ufrj.br Crsovão M. Moa Soares**, Aurélo L. Araújo*** ** IDMEC/IST, Isue of Mechacal Egeerg, Pole IST, Lsbo, Porugal. *** ESTIG, Polyechc Isue of Bragaça, Porugal. csoares@alfa.s.ul. aaraujo@b. ABSTRACT We cosder Nolear Leas Squares robles wh equaly ad equaly cosras ad roose a uercal echque ha egraes ehods for ucosraed robles, based o Gauss-Newo algorh, wh FAIPA, he Feasble Arc Ieror Po Algorh for cosraed ozao. We also rese soe uercal resuls o es robles avalable he leraure ad coare he wh he quas- Newo verso of FAIPA. We also descrbe a alcao o he defcao of echacal araeers of coose aerals. The rese algorhs are globally coverge, very robus ad effce. INTRODUCTION I hs aer we cosder Nolear Leas Squares Probles wh equaly ad equaly cosras, whe olear sooh fucos are volved. Callg [,,, ] he desg varables, f( he objecve fuco, g( [g (, g (,, g (] he equaly cosras ad h( [h (, h (,, h (] he equaly cosras, he roble ca be deoed as: ze ad subjec o f, R g ;,..., h ;,..., ( The fuco f( s a su of squares of he olear fucos r ( ;,, s. s f [ r ] r( ( Probles of hs ye occur whe fg odel fucos o eereal daa [, ]. I hs case r ( s called a resdual fuco. I rereses he dscreacy bewee he rue value ad he aroae value, redced by a olear odel. If he odel s o have ay valdy, we ca eec ha f ( * wll be sall, ad ha s, he uber of daa os, wll be uch greaer ha. We assue ha s >. Noe ha, f he se of equaly cosras verfes regulary codos [3], o have a soluo us be. A large uber of secal urose algorhs s avalable he ucosraed case, bu oly very few ehods were develoed for he olearly cosraed case [4, 5, 6]. A uercal echque ha egraes wellow ehods for ucosraed robles a geeral ehod for Nolear Cosraed Ozao s reseed hs aer. Ths ehod s he Feasble Arc Ieror Po Algorh, FAIPA, ha aes eraos he ral ad dual varables of he ozao roble o solve Karush-Kuh-Tucer oaly codos. Gve a al eror o, FAIPA defes a sequece of eror os wh he objecve reduced a each of he eraos. A each o, a feasble desce arc s obaed ad a eac le search s doe alog hs arc. To coue he feasble arc, FAIPA solves hree lear syses wh he sae ar. These syses clude he secod dervave of he Lagraga fuco. There s also a quas- Newo verso of FAIPA. I hs oe, he Hessa of he Lagraga s relaced by a quas- Newo aroao. I he rese algorh, sead of he Hessa, we eloy a aroao based o

2 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, Gauss-Newo ehod ad soe of her odfcaos. I he followg secos we descrbe FAIPA, soe esg ehods for Leas Square ad we rese he algorh roosed here. Fally we descrbe he uercal resuls o soe es robles ad a raccal alcao sold echacs. B Λ g h( g G( h d f λ - µ h( where d R, λ R, µ R. (9 FAIPA, THE FEASIBLE ARC INTERIOR POINT ALGORITHM FAIPA, roosed by Hersovs [3, 7, 8], s a eror o ehod ha solves geeral robles of olear ozao. FAIPA aes eracos he ral ad dual varables of he ozao roble o solve Karush - Kuh - Tucer (KKT oaly codos. KKT codos corresodg o Proble ( ca be wre as follows: f g( λ h( µ (3 G( λ (4 (5 λ (6 g( (7 h(, (8 where λ R ad µ R are he Lagrage ullers corresodg o he equaly ad he equaly cosras resecvely, G( R deoes a dagoal ar such ha G g. I wha follows we call Λ R a dagoal ar wh Λ λ. FAIPA requres a feasble al o ad defes a sequece of feasble os, wh a oooe reduco of he objecve fuco. The Feasble Arc Ieror Po Algorh o solve Proble ( s descrbed ow: FAIPA ALGORITHM Paraeer. α (,. Daa. Ω a, λ >, λ R, µ >, µ R, B R syerc ad osve defe ad c, c R. Se. Couao of a feasble desce dreco. ( Solve he lear syse (d, λ, µ : f d, so. ( Solve he lear syse (d, λ, µ : Λ B g h d g G λ - λ ( h µ µ where d R, λ R, µ R. ( ( If c µ, ae c >. µ, for,...,. (v Le be φ (, c f c h( ( f d φ (, c, se: else > ( α d φ(, c ρ f d ; ( d φ (, c (v Coue d ρ d. d d ρd Se. Couao of a feasble desce arc. ( Le be h ( d h h d where :,...,. (3 (4 w g ( d g g d (5 where :,..., ; I w E (6

3 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, ( Solve he lear syse ( d, µ : B g h d I Λ g( G( λ - λ w (7 h( E µ µ w ( Fd a se legh sasfyg a gve le search crero o he aulary fuco φ (, c such ha: g ( d d < f λ, (8 s he secod dervave of he Lagraga, a Newo algorh s obaed. A very effce algorh, whou eed of secod dervaves couao, s obaed wh B equal o a quas- Newo aroao of H (, µ. g ( Feasble arc d d d d ρ d g( f( or oherwse. Se 3. Udaes. ( Se d < g ( d g d d (9 ( ad defe ew values for: w >, λ >, µ > ad B syerc ad osve defe. ( Go o bac o se. The sze of lear syses ( ad ( s equal o he su of he uber of varables lus he uber of equaly ad equaly cosras. I [9] s rovde ha ( ad ( had a uque soluo. I Fgure he Feasble Arc s rereseed he case whe here s a acve equaly cosra, ha s g (. I s roved ha s ossble o wal fro alog he arc o ge a ew feasble o wh a lower objecve value. The algorh has global covergece for ay B syerc ad osve defe. However, ag B H (, µ, where Fgure : Feasble Arc. ABOUT THE UNCONSTRAINED LEAST SQUARES PROBLEM To udersad he basc feaures of he algorh rese here, we cosder he ucosraed olear leas square roble: s f r ( R where r( rerese he resdual vecor. The Jacoba Mar of he resdual s J ( r... rs ad he Hessa ar of f( Where r..., (3 rs f J J Q( (4 l Q( : r r. (5 H (, λ, µ f λ g µ h ( I he Gauss-Newo ehod, Q( s gored ad he Hessa s sly aroae by

4 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, f J J. (6 The eraos for Gauss-Newo ehod are he J ( J ( ( J ( r( (7 Gauss-Newo ehod s based o Newo s ehod ad ca fal for he sae reasos as Newo s ehod does. I arcular, whe J ( J ( s o osve defe or whe s badly codoed. Gauss-Newo ehod assues ha, ear of he soluo, J J s a good aroao o f,. e. Q( ca be egleced. Ths assuo s o jusfed for robles wh a large resdual. A ossble sraegy hs case, s o clude a quas-newo aroao M of he uow secod dervave er Q( [4]. The search dreco wh a quas-newo aroao o Q(, called M, s gve by: [ J ( J ( M ] d J ( r(. (8 Le be s ( (9 y J ( r( J ( r( (3 The followg forula for M s based o he BFGS udae [9]: M y y (3 y s K M W s sw sws aroaely equal o J J. However, J ( J ( M ca be sgular or badly codoed, resulg a o-desce search dreco, ad he erao fals. Leveberg Marquard ehod cosss o addg a osve dagoal ar ε I where ε > s ae bg eough o have J ( J ( M εi osve defe. The a dffculy o aly hs echque s o ge a way of choosg ε o very large order o aa as well as ossble he seed of covergece of Gauss Newo algorh. LEVENBERG MARQUARDT METHOD WITH CHOLESKY DECOMPOSITION Le be ar B J ( J ( M. If B s syerc ad osve-defe, ca be obaed a Cholesy facorzao B LL (33 where L s lower-ragular ar. The odfed Cholesy facorzao s a uercally sable ehod o coue ε ha roduces a osve-defe ar [4]. The elees of L ca be eressed by a sle recurrece relao: ad j b ljlj l l for,,, - (34 l b l (35 j j where W M (3 J ( J ( I he case whe B s o osve defe, s roved ha oe or ore dagoal elees are such ha I s roved ha f s esured ha y s >, he he udag forula has he roery ha f J M s a osve-defe ar, ( J ( he so s J ( J ( M, see [4]. Ths roery s used asyocally whe J J s b l j. (36 j I cosequece, l obaed (35 s o a real uber

5 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, Addg o b a bg eough osve uber, a osve defe ar B s he obaed. Ths rocedure s equvale o Leveberg Marquard ad allows o defe very recsely he erurbao requred o ge a osve defe ar. ABOUT CONSTRAINED LEAST SQUARE PROBLEMS The algorh ha we roose here s based o FAIPA. Isead of ag B equal o a quas Newo aroao of H (, µ, we cosruc a ar ha cludes a Gauss-Newo aroao of he objecve fuco. λ g H(, λ, µ f µ h (37. We eloy he sae udae forula (3, bu ag where y l, λ, µ l(, λ, µ, (38 ( l(, µ f g( λ h( µ. (39 I ucosraed ozao s roved ha f s osve defe a a local u. Whe here are cosras, we have ha geeral H (, µ s o osve-defe. I effec, s oly esured ha H (, µ a a local soluo s osve defe he sace age o he acve cosras. However, FAIPA requres a osve defe ar B. We eloy Leveberg-Marquard ehod wh Cholesy decooso o oba B osve defe. NUMERICAL TESTS We rese soe uercal resuls obaed wh he algorh for Cosraed Leas Squares robles, FAIPA_LS reseed hs corbuo. These resuls are coared wh a quas Newo verso of FAIPA. We also descrbe a alcao o a verse roble solds echacs. Proble 5: Source: Holza [], Helblau []. Objecve Fuco: 99 f,...,99. ( r r (. e ( u u 5 Cosras: Sar (feasble: Proble 57: ( 5l( (,.5,3 f ( Source: Bes [], Gould [3]. Objecve Fuco: f,...,44. a, b 44 ( r r b Cosras: :aed A of [4] (.49 e( ( a 8 4 Sar (feasble: Proble 7: (.4,5.9 f ( Source: Helblau [, 4]. Objecve Fuco:

6 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, f y, cal ( y 6.83 c bc b e [( c bc e b 3 ( 3 4 c, y :aed A of [4], obs 9 Cosras:, cal y, obs 3b.4 4 Sar (feasble:.49 (.4,5.5.9 f ( Table. Nuercal Resuls o Probles Hoc/Schows ad Lear Equaly Cosraed Leas Square Proble. Proble 5 ( 3, 6, Udae of B cfv ofv er FAIPA_qN FAIPA_LS Proble 57 (, 3, Udae of B cfv ofv er FAIPA_qN FAIPA_LS Proble 7 ( 4, 9, Udae of B cfv ofv er FAIPA_qN FAIPA_LS Lear Proble ( 4,, 3 Udae of B cfv ofv er FAIPA_qN FAIPA_LS.. 4 We reor here our eerece wh 4 es robles. Three robles coled by Hoc e. al. [5] ad he las oe roble s a lear equaly cosraed leas square (LSE roble descrbed (h:// The LSE roble s where A b A B 3 subjec o B d, b 6, 3 3 ad d. The resuls are suarzed Table, where s he uber of varables, he uber of equaly cosras, he uber of equaly cosras, er s he uber of erao; cfv s he coued objecve fuco value, ofv s he ou fuco value, FAIPA_qN s he quas- Newo verso of FAIPA ad FAIPA_LS, he rese algorh. All es robles were solved wh he sae value for he araeer α. Ths was ae: α.7, as he geeral verso of FAIPA. The so crero adoed was a olerace o he oal objecve fuco ε -5. The al o, ofv ad oher characersc of he es robles are descrbed Hoc e. al. [5]. Idefcao of aeral araeers: Ths eale s eded o llusrae he alcao of he descrbed ozao echques o a class of verse robles, aely he defcao or esao of aeral araeers coose laaed laes ade of wo dffere aerals. The roble cosss of esag he elasc roeres of he wo aerals ha ae u he lae by fg a se of eereally easured udaed egevalues ( λ o hose obaed hrough a hgher order fe elee odel ( λ.

7 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, The objecve fuco s a weghed leas squares esaor: f ( where [,] I λ λ ( w λ (4 w eresses he cofdece he eereal daa ad, hs eale s ae as uy. The roble s he forulaed as a o lear cosraed zao roble, where he desg varables are o desoal fucos of he elasc roeres of each aeral ad he cosras are osed order o ee he cosuve ar osve defe: f s.. l g u (4 Full deals regardg hs defcao echque ca be foud Araújo e al. [,6]. The lae hs eale s ade of udrecoal layers of E glass ad T3 carbo fbres eoy ar. The re-regs used o buld he lae were Srucl g/ VEE R368, for he glass layers ad Srucl 35g/ CTE35 R367, for he carbo layers. The sacg sequece s [ 94 C, V 3 ] S ad he recagular lae desos ad ass are a9, b54, h3.89 ad 89.85g. The al esaes for he elasc roeres of he glass ad carbo layers corresod o he roeres of ycal udrecoal layers of hese aerals for 5% V f : E glass: E 45GPa ; E 4. 5GPa ; G G G 3. 7GPa ; ν.8. 3 T3 carbo: 3 E 7. GPa ; E 8. 8GPa ; G G G 3. GPa ; ν For he fe elee dscresao a regular 6 esh was used ad he roble was solved 7 eraos usg he FAIPA (Wolfe crero for le search ad he sog crero was he reduco of he ealy fuco (less ha -6. Resuls are reseed Tables hrough 4. Resduals r ω o he aural frequeces were obaed fro easured ( ω λ π ad defed ( ω λ ( π aural frequeces, usg he followg eresso: r ω ω ω ω (4 A good agreee s sough bewee he defed global roeres ad her avalable sra gauge couerars ad oe ca coclude ha he defed roeres for each aeral are reasoably wh wha oe could eec for hese aerals, ece for he rasverse shear odulus G 3 ad G 3, because he lae s o hc eough for hese shear effecs o be oceable, hece ay resuls for her defcao are o ruly relable [, 6]. Also, he dffere aeral deses were o ae o accou hs eale, whch could ar ela he ably o f he ffh aural frequecy wh suffce accuracy. Table. Idefed global roeres ad sra gauge easurees Idefed Sra gauge E [GPa] 7. E [GPa] y G y [GPa] 4. G [GPa] z. G [GPa] 3.8 yz ν y Table 3. Idefed roeres er aeral E glass T3 carbo E [GPa] E [GPa] G [GPa] G [GPa] G [GPa] ν.7.45

8 4 h Ieraoal Coferece o Iverse Probles Egeerg Ro de Jaero, Brazl, Table 4. Eereal frequeces ad resduals obaed afer defcao ω [Hz] r ω [%] CONCLUSIONS The rese s a srog a effce echque ha eeds o cosraed robles he advaages of Gauss-Newo ehods. The uercal resuls suded here show a rovee of he couer effor whe coared wh he classcal quas Newo verso of FAIPA. We oe ha FAIPA s very robus ad effce ad was esed wh ore ha es robles he leraure ad was aled several raccal alcaos [, 6]. Uforuaely, we dd fd ore Cosraed Leas Squares es robles. ACKNOWLEDGMENTS The auhors wsh o acowledges CNPq (Brazl, FAPERJ (Brazl ad FCT/ICCTI (Porugal for he facal suor rovded o hs research. REFERENCES. A. L. Araújo, C. M. Moa Soares & M. J. Morera de Freas, Characerzao of Maeral Paraeers of Coose Seces Usg Ozao ad Eereal Daa. Cooses: ar B, Vol. 7B (,. 85-9, A. L. Araújo, C. M. Moa Soares & M. J. Morera de Freas, P. Pederse, J. Hersovs, Cobed uercal-eereal odel for he defcao of echacal roeres of laaed srucures. Coose Srucures, Vol. 5, ,. 3. J. Hersovs, A Feasble Arc Ieror Po Techque for Nolear Ozao, JOTA Joural of Ozao Theory ad Alcaos, Vol. 99, N,. -46, Ocober, P. E. Gll ad W. Murray, Algorhs for he Soluo of he Nolear Leas-Squares Proble, SIAM Joural o Nuercal Aalyss, Vol. 5, No 5, , Ocober, P. E. Gll, W. Murray ad M. H. Wrgh. Praccal Ozao, A.P., K. Schows, Solvg Cosraed Nolear leas Square Probles by a Geeral Purose SQP-Mehod, Ieraoal Seres of Nuercal Maheacs, Brhauseer Verlag Basel, Vol. 84 (c, , J. Hersovs, A Vew o Nolear Ozao, Ca. Advaces Srucural Ozao, J. Hersovs Ed., KLUWER Acadec Publshers, Hollad,. 7-6, Jue, J. Hersovs ad G. Saos, Feasble Arc Ieror Po Algorhs for Nolear Ozao, Fourh World Cogress o Couaoal Mechacs, ( CD-ROM, Bueos Ares, Argea, Jue, D. G. Lueberger, Lear ad Nolear Prograg, a ed., Sddlsso-Wesley, G. Holza, Coarave aalyss of olear rograg codes wh he Wesa algorh, SRCC Reor No. 3, Uversy of Psburgh, Psburgh, D. M. Helblau, Aled olear rograg, Mc-Graw Hll boo-coay, New Yor, 97.. J. T. Bes, A acceleraed uller ehod for olear rograg, Joural of Ozao Theory ad Alcaos, Vol., No., 47-74, F. J. Gould, Nolear olerace rograg, : Nuercal Mehods for Nolear Ozao Theory ad Alcaos, Vol. 6, No. /, 49, 66, D. M. Helblau ad Yaes R. V., A ew ehod of flow roug, Waer Resources Reseach, Vol. 4,.93, New Yor, W. Hoc, Schows K. Tes Eales for Nolear Prograg Codes. Lecure Noes Ecoocs ad aheacal Syses. No. 87. Srger-Verlag Berl Hedelberg New Yor, A. L. Araújo, C. M. Moa Soares, J. Hersovs, P. Pederse, Develoe of a fe elee odel for he defcao of echacal ad ezoelecrc roeres hrough grade ozao ad eereal vbrao daa. Coose Srucures, o be ublshed.