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Gb"/c_%+@*$q'pl+,tH2-<3c1'AEhti?N2Uo:Tr;CMR_O06DRepCnM].tbss=`.H:4=4]jB(h^u=^c1"]Q)Ro@=]Z9)iLLO$Y2at8!_%je!AZ-8VY'5"f75MAfHC,M_"[Vi'\p*k48V28ODD.R]D%"BH@"]Gke6_=q`8!3+BrF"@ksD%(RjSWP_7JrRes#".3TSS4oGCq86;e.l1RAQiPrYT1;MRk^F!V%cfr7WYVGq4s"\kLu'+L=+mXf^^W! /FormXob.64d9f2eb61bb80d35e448f9c6682b2b4 63 0 R stream 78 0 obj endobj 62 0 R Gb"/#bAP3O%*S[#FKe,`U2(IrPt4A&`+)m==mnHlNCK\jB^Cet_UGCAfIW&>9IR65e[sO^nRO98I#l9*M!tW0hq.@@p"`^Ds+WOXEqJF:b$r-5SPR+%Z1!prdcPtrU.r@RV]H46[++;A1RjAQr!.l[BJ3kjcKHPQ!3kQI5!!EG065JU]=LmY! /Rect [ 202.4988 65 0 obj /Type /XObject 293.9127 The web of transition dynamics backup diagram state … endobj /Type /XObject % 'FormXob.1be64776d9130247f6ffe27a4595d846': class PDFImageXObject /Length 491 /Type /Annot >> << /A << /S /URI /URI (http://en.wikipedia.org/w/index.php?title=Bellman_equation) >> 49 0 R /Border [ 0 Bellman’s dynamic programming was a successful attempt of such a paradigm shift. % 'Page3': class PDFPage % 'Annot.NUMBER54': class PDFDictionary 0 /Subtype /Image % 'FormXob.1fd5a13b437ebf8e4289315e2c04096a': class PDFImageXObject /Filter [ /ASCII85Decode By applying the principle of dynamic programming the first order nec-essary conditions for this problem are given by the Hamilton-Jacobi-Bellman (HJB) equation, V(xt) = max ut {f(ut,xt)+βV(g(ut,xt))} which is usually written as V(x) = max u {f(u,x)+βV(g(u,x))} (1.1) If an optimal control u∗ exists, it has the form u∗ = h(x), where h(x) is << /A << /S /URI /FormXob.64d9f2eb61bb80d35e448f9c6682b2b4 63 0 R /Border [ 0 /Length 520 337.3969 /Type /Annot >> /Subtype /Link /URI (http://en.wikipedia.org/w/index.php?title=Backward_induction) >> endobj /Type /XObject /Subtype /Link 0 /ColorSpace /DeviceRGB 512.6969 ] 0 ] %PDF-1.5 7 0 obj /Subtype /Link endobj 0 ] /Border [ 0 Gb"0M_%$GV$j0?+s3]gWAu=r*9@bV? Little has been done in the study of these intriguing questions, and I do not wish to give the impression that any extensive set of ideas exists that could be called a "theory." 329.1828 /Rect [ 498.6927 50 0 obj /Type /Annot >> 4 0 obj 0 ] 446.9469 /Subtype /Image 0 stream 26 0 R << /A << /S /URI Gb"/g_%+@:#Xe4`+,tH41bEE6N.Eh)\#%&']Je4c;C[r(V,D#C[BDL_I'*=$=dUW.*i%,t9nh,,8J22FQJ5lJorAqGS+Z!>cOets9CNl$m+4ELN7"C$aa/EZEKSEMDON*!1PtMQL^E;EBL!2WcuI6o%W=Gj/6+5+N"OEM'?)&8E*k06dfISTN1PL@8_uaGpK:sY#2DQ#Q`,&$GLse.]*C%I8jl.`MoCK#$DXY_\W"f6,`H)"]u(>=NWCEf^ORk&2O7j3)Ccl'EXU%E0Lkd6/AuDtbm9P=/^bLg3SPEKU)M3ZGLFc0oLY@g\=,F:OTeZ;UK]ub=-ZA,_:P]QVY%>64"34c\_A\\I0k(d`>iAE=^2@W7J6Rd`)QnF/T9NQ-N9.f$:0%.8PJ(K]Oq*\p!R;hb0tLOaB4Q+71W]9KqFg\#6q.1e%Fs6q(fJJ3!aRISM.VpgRjj^K)>3f^I9jh7J=5agX(&G8'gJiA! /Rect [ 343.2578 62 0 obj 527.6969 ] Let us recall Bellman’s statement, noting that this statement was made in the context of certain decision processes where the notion of optimality regarding policies was associated with a preassigned criterion function defined over the final state variables. % 'FormXob.e1f9c30ef779cd2dc4b6dca75cf910de': class PDFImageXObject /Height 40 endobj /Type /Action 59 0 obj % 'FormXob.0cc17d20790591600c58c36146e9e427': class PDFImageXObject /ImageB /Type /XObject Gb"/cd;f=4$j5tu5qq)g6@k2i1ri_b_@!UA:E/eV6+Q5A%+up@$NTn's'R*#F7.+`GdJOfAAhOGE*gID.DrD25bqdV=*dU8>sYZcmM9(0,',V5DPr$IK+ZKG+]c;>`0@nOpq55,#O9=7hBE*C=>b9d["3j)VamL^d-7gN91m(D8=OM?=&7BNc9`N%rendstream % 'FormXob.1fd5a13b437ebf8e4289315e2c04096a': class PDFImageXObject /URI (http://en.wikipedia.org/w/index.php?title=Scheme_%28programming_language%29) >> 700.9469 % 'FormXob.624787f328f7c1907bc84fae77480f51': class PDFImageXObject 535.9469 /Subtype /Link /Type /Annot >> /Border [ 0 /Height 43 589.126 stream /Length 631 Dynamic programming is ... Let’s take a look at what kind of problems dynamic programming can help us solve. /Type /Action endobj /Width 27 >> << /BitsPerComponent 8 /Length 2579 /Type /Annot >> /Subtype /Image /Trans << >> Gb"/hgMYq'#Xn:a+!"?rV(2%i^jp92c["904*\?;]l-4C+bUE4"4o3,X!#H&[9-_al'N7FggJU%2t.R!GK42aLhRZs%2SZM!BnU2eQc/5#+%8>rd_u@W!77N>/&]1gif&-J,?fY5W)Aoj)l,k3Uo&Q/3*GdCZ7/+Os"WoU[t\Lr/5AX[2[8TT%6*cJC]#V=t=RT5A[s,>9$lK[Vh-%k=%kGbh!g&89`I0@/FFjK-Lb4E0CO5-_qD]QTiY+Ak4h/-plC"5*LlRh*&b.F,:916D*LHauHulJhj^XlnlhBF;KPSbM!TFX,0@`aWZilaN2m0%K6CAI6SWO62r[:cr-eYKcO'(XcD!-iE;CRUZQ9JA','QB'A6?T]o"R#lb)n9hmtk-O"X?^Rc$-#;CHA;3Lgq#?MNCH3AXfDC2e]mF\l.@npWku*"\rXjT.n>s?NYr/Cb!X586AD.KR5[V0C[*JB?ia>?K2GJPnS?*BS-hcAS/I:-or);\]JWddgSNEr,'aG4l1Gq]in+Vj^$Vgpd&qMj2#'RRGVOjgMZmZ3ZTHHA3/1bK;MQ&'q8jPuO"l\TX"nA+_riWKa?Or6%8W. /Subtype /Link % Page dictionary /ColorSpace /DeviceRGB endobj stream /Subtype /Image 730.9469 "i56ti;~>endstream % 'Annot.NUMBER18': class PDFDictionary %���� The mathematical state-ment of principle of optimality is remembered in his name as the Bellman Equation. /FlateDecode ] 42 0 obj /Height 47 << /A << /S /URI 520.9469 /FlateDecode ] 9 0 R /Width 263 >> stream /Border [ 0 85 0 obj /Type /Action /Length 147 4 0 R /Subtype /Link /Subtype /Image /URI (http://en.wikipedia.org/w/index.php?title=Programming_language) >> endobj 75 0 obj /Length 144 /FormXob.259737109518d6104fe35d5cd9a74d71 58 0 R 638.3219 Bellman (or DP) backup operator Denote p ij (u) = p (j | i, u), and take any vector J = (J (1), . << /Annots [ 2 0 R 44 0 R 47 0 obj endobj /Width 173 >> 412.9469 ] /Border [ 0 "I1Qendstream 47 0 R /Subtype /Link 99.59687 endobj 459.5427 endobj /Type /Action << /A << /S /URI endobj /Type /Annot >> stream 0 ] % 'FormXob.febd74aab40296455b234ff3672335b5': class PDFImageXObject /Type /Annot >> 53 0 obj /Length 594 366.2916 /Type /Annot >> 16 0 obj There are good many books in algorithms which deal dynamic programming quite well. % 'Annot.NUMBER41': class PDFDictionary 50 0 obj 2 0 obj << /BitsPerComponent 8 442.9469 /Border [ 0 << /BitsPerComponent 8 endobj endobj << /BitsPerComponent 8 /Rect [ 449.6234 /Resources << /Font 1 0 R In the 1950’s, he refined it to describe nesting small decision problems into larger ones. endobj /Type /XObject stream 382.9469 /Width 267 >> 745.9469 ] /Border [ 0 Gb"/db8,^Q$jHV0O7m[((Hhs(jE8l,fbHgd``UQN&44)N/^R4#BWIpuojT9V4]gl:V:]U\]lf/#Rd-/1jc(BmSD?GbdUott'qeIX;J[:08r&.PKF_q";SuWOJRYsNR`?B84$.BZnE_Jo=!qFD`"OAJD'^O\_[2GS;gDjE8H$&2f2&&5REuVB7k0G1?2\RFYpPs.blP`'Xd%[t1@nqB",;qYrV'rmdK5iBd-C%0,[>&O9Nc@LbE/NFWDo_e17P+e/!Cj*q2a+!C:I079VmN5:6\qR2Lm,u@&OdfeYAhdE2YSqBobIV5X(3]&40^C#=NWpci~>endstream stream /FormXob.51835b45461ebaacf37b68d61d5bc546 71 0 R << /A << /S /URI /Subtype /Link 0 /ImageC /Type /Action 0 /ColorSpace /DeviceRGB endobj R. Bellman, Some applications of the theory of dynamic programming to logistics, Navy Quarterly of Logistics, September 1954. /Rect [ 74.69291 /Border [ 0 /URI (http://en.wikipedia.org/w/index.php?title=Divide_and_conquer_algorithm) >> << /BitsPerComponent 8 /Type /Action /FlateDecode ] /Type /Annot >> /Border [ 0 endobj 51 0 obj /Filter [ /ASCII85Decode /Width 167 >> endobj /Type /Action /Type /Action /Subtype /Image /Type /Action /Length 812 /URI (http://en.wikipedia.org/w/index.php?title=IEEE) >> /Subtype /Image 638.3219 << /A << /S /URI 748.9469 << /BitsPerComponent 8 531.9982 0 << /A << /S /URI The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and also because it sounded impressive. << /A << /S /URI /Type /XObject /Height 50 Bellman: \Try thinking of some combination that will possibly give it a pejorative meaning. /Length 162 82 0 obj /Type /XObject << /A << /S /URI 50 0 R /FlateDecode ] Gb"/hbHA`:#XnXk+)P$CU5Ir`^];Alk3Qu#(`nlcEI0'd&4-XGCXKkL-]H;A;G#RYSZi^*bCYp:&M+]7ir@Dba>m/P?u\;-%9LrU@tC$8Ha!te*is!! Dynamic Programming - Dover Publications In addition to his fundamental and far-ranging work on dynamic programming, Bellman … 66 0 R endobj /Border [ 0 /Rotate 0 % 'Annot.NUMBER23': class PDFDictionary /Type /XObject 0 ] 153, Issue 3731, pp. /Subtype /Image /Length 197 /Width 40 >> endobj /Filter [ /ASCII85Decode /Length 491 87 0 obj /Length 1970 /Subtype /Image /Rect [ 255.6488 RICHARD BELLMAN ON THE BIRTH OF DYNAMIC PROGRAMMING STUART DREYFUS University of California, Berkeley, IEOR, Berkeley, California 94720, dreyfus@ieor.berkeley.edu W hat follows concerns events from the summer of 1949, when Richard Bellman first became inter-ested in multistage decision problems, until 1955. 43 0 R 21 0 obj 350.4329 45 0 obj /Height 47 Gb"/c_%+@*$q'pl+,tH22GLQ3,^,?:iS/&5QoA\o>?b<9>6R&O\pi^8*K-%6ZX'"R%c-n4;n`PG6#h`T[ATV16Fq1#gG;*Y2QV[RS^ZJ^Xt86;^9f/Z!]rZpXSJ2;'hB@Hh%3M]:Bkno]q@(.*p(ehCd4KQ:b"`1"A_f@IWNht0d*bGVYh,d=^(;_KD)B0ODQ.T:Q]3CcWf%(r. endobj % 'FormXob.64d9f2eb61bb80d35e448f9c6682b2b4': class PDFImageXObject /Trans << >> /ColorSpace /DeviceRGB /Rect [ 186.6846 /Type /Action *4Rs4Hj$O)*NsP#!H:["6i!DU6, OgJQ\!HrEA[NuOCVQd_]EsL!2BZ^X9+mHZJt>&kO/pT)1\F[fc6goAX^EEUCJ#cZWPe(sk`!g=uZB\@p+oo^RITn>5:*fYHbmYL.\:W&-Zd*Hi-a59Wt!1pBLCH.U"Tolg"*S:@F^q5G>Rq?icg@neELeLA\h6*9Cq_@Y!ouCLn3um-:@7AQ#"WWtWDY%_[]oVm3*urMn9$-3#*fE/u@0-ebbe!A%[W&ise;/>)UtPG9`poBP)q[!MAURFGj/\1ITY5FkPZ9kjISc6ZgR$7jnl"TLVA*@-p0HaPWKB5XD@U=$riXmet2ZmkuSuc;tS4]]iUu;+"b(_N8/;Jo"70?6ZR2dYNbhJt?o_6NNm"ig/ZLt7FCT*I99+%h/]ue[)P]?b*)Fj(0^`^V8%/Euti&`q9h2qY0(b%G7ofYdCU>Ym5mEK/U3riLA(8Jge$ot_AE5n/Tk*sJ/"<9Ug7V;ck_=ak(ZsSXmfY[?B*/]2&ZVK''/Jtt]9X7jo#Jf<5VU:es;/#oC?#/h/+p#.E&Ym:\$otPtDX(fQ1W(5M^3rpQ>#7=3C7\MVpT[D$l*3:JKu#NaZaMBp^puBmL^n2fcel,ikg%DUQPl>D:2g^/gClsC=23+2,oC#bh"tXgAQYEIn$Xm0\buS6USai2W\D7\Hb%^q9MSF_lKc%8Y64sOq2#Lt[O0+k2&6bi_VMBo?/JDl.X^5ZBeh\L#IQ6f(2e=S'bq*AGR7sB&MNf@D]@*\lP7nNn$EI%$lTRg9)\'-fELdTfD)5VnEX*0En5_=cW7e1I-al_4]:"O.%"dV6-?&iIs?Gp\&ZFQ21iE,o:`uB\LVMK_"4^OX-=YSmGV^3Y%.WMJc?01;B0ArjdMt,1!G:%M@,2(4_`=C;1Ac0Sj^iF*k!fq%Qr;n_A/6?r?d(7c7!PVG!`o#jDk=Z)I@H@6fVC?63U9mqV:FZ7Y7"$q5&,?OQAI5Ish:J<9c7O1$S'^fn74>>f(IkZ/puT$3,HX#M4h/V$d/@XkMH?">dJC$BOb$5L@N[_0ohBBR0m253*`m3Q>5nmO,>rMoA"4/EqE)?%/dHO9sQ#p5q0*!impW()UD\o]7-V"@>u;%-t'0=fUI3Ri5eg6V'\CJKu%AQ5oX9F*'IJRK4@MFU$i)Wce:=b,Td%nFe;9FNE(3+&s>Db`NoP#;SUm-85@ki^:-2l&5H.H/Lk9'.^.1ZJ`6q(>I=+1-t3egj:3`br&L+lH>'8+h7G0!nri;@nBSOhj.5`9khgQaDlU[$.`QB[-3^FD]l!OnFjGTqJd^_$d]:s+HA*jMX:.pmg%S]<<2%=aJnY1E(-U0k<1mQrB6r&s$sD-,QoX_P`M,+2EC;#Gb)Ci1hoo"R(t!KQ[dqb"EWD\@h>^[_WneM(l4M>=ea[J1u"rP!Q9Rsrnmd_8[-fUKg$\(D5fDI"_.lqQJ]D-&n%h0OTU;F2G'd&l.&>!K^1qD"LD'=kN8Ycn_`DOPS'.#"R9CZU/[9:VdEat&QWnE\!3q:3(&iKdB![9ZD;^SsW=ga;1IT3%TbrRtE,Qe&FcY($+=lQt=i8snM_?,,ssV@j>ZS*7'Cato,_G`Z-.!`#arI1s)S;Cm:HnGK-r")jg!"^aU"rCT&rZ-N@H>X#dr>n3t>4lj6VRiaMLS'")V[1\%D&W:]rVeWj7%V20dc)o*TZ]^V>iE[pIX_FUUWE5,A9XV3s=!7dZej8#V5uAo0!4%&2h==&F,a+k0[$$R$/&uZ?VSAp9!Q;FI\MY7fZi\>F's\9h[T\4!2-FTa.iCplEi\=Ongc25JA,mo3O^=kP#_Q_^O"a_/Kb/hZ_NeGUg?%rZZ9:8.k6iMN'!G3P8QqBJ>cgTlsRRDi./5WId=@/;QlUJ)+o[+6>&SIt1*UE9&Gh49dP/IVqmZJurM,4t8=$7pg_c(!!r-=pF@qhLM!F+eW,XBpjj[A[##>aJ`0B(cIGVgJ?Rhf-'brk18TBlOIr7$coO]B%Rq[IuHj;WVssa&Gtb+cRm:8mW)>9aLFB"fa*M#H-N^%Gn+D%X>>];PondH<=@H%3pUZ%#5eRKLTU8#T>CHl[$4d[%nrh#"meS4DA%SK(bM)&:rP]T[lFL'-+sN=3()G#&>6EbL0KpJ/mtU]oTIH`FF-l7\PLs[\=h\:q1)\8g_M>,7+B)Sgq)Q#K"i(?"jR'5WBi;KiX+%^'d1$89m,dX+1Vc+fcYDhi"B3h,8DAfq^#@?&.rpRT$"0.#*5GSPE=At/u%_a@MpBcWsjn#,&k$Gbi%%uJS46TW\k576Zrm4#WnVqS$df"Xuej>%a#5NY4E8;Xf(+DX>=C4:j]7gb2=UPh1lJ9^bMMI(a&MlS=KscrP@W974i::Y,7X*hPJA,cgdFDGMfB1J'W\Z3>1.nBLm6Kc'K:PJ?*XB7nPQ>]3:#MV6H;I4--PX-[N61+"^BHLn?PO^;d9. 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