SOLVING NONLINEAR OPTIMIZATION PROBLEMS BY MEANS OF THE NETWORK PROGRAMMING METHOD

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1 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) SOLVING NONLINEAR OPTIMIZATION PROBLEMS BY MEANS O THE NETWORK PROGRAMMING METHOD Vladr N. BURKOV Pro., Inttute o Control Scence o V.A. Trapeznkov, Ruan Acadey o Scence, Mocow, Rua E-al: vla7@k.ru Irna V. BURKOVA PD Canddate, Inttute o Control Scence o V.A. Trapeznkov, Ruan Acadey o Scence, Mocow, Rua E-al: rur7@gal.co Atract: We ugget a new approac to olve dcrete optzaton prole, aed on te polty o preentng a uncton a a uperpoton o pler uncton. Suc a uperpoton can e ealy repreented n te or o a network or wc te nput correpond to varale, nteredate node to uncton enterng te uperpoton, and n te nal node te uncton calculated. Due to uc repreentaton te etod a een called te etod o network prograng (n partcular, dcotoc). Te network prograng etod appled or olvng nonlnear optzaton prole. Te concept o a dual prole pleented. It proved tat te dual prole a conve prograng prole. Neceary and ucent optalty condton or a dual prole o nteger lnear prograng are developed. Key word: network prograng nonlnear optzaton dual prole nteger lnear prograng. Introducton Prole o nonlnear optzaton (n partcular, dcrete optzaton) reer to te cla o o-called NP-dcult prole or wc no eectve etod o eact oluton do et. Soe general approace are avalale, aong oter te ranc and ound etod and te etod o dynac prograng []. Unortunately, te dynac prograng etod applcale only to a narrow cla o prole. Te ecency o te ranc and ound etod depend eentally on accuracy o te upper and lower etate (ound). 77

2 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) To ae toe etate te etod o ultpler o Lagrange [] developed. Tee etod are known ro te pat 60-, and nce ten ore tan tey ave not een proved gncantly. In 004 V.N.Burkov and I.V.Burkova uggeted a new approac to olve dcrete optzaton prole, aed on te polty o preentng a uncton a a uperpoton o pler uncton. Suc a uperpoton can e ealy repreented n te or o a network or wc te nput correpond to varale, nteredate node to uncton enterng te uperpoton, and n te nal node te uncton calculated. Due to uc repreentaton te etod a een called te etod o network prograng [] (n partcular, dcotoc). T etod applcale to cae wen te goal uncton and retrcton uncton otan dentcal network tructure. or uc cae network node optzaton prole, pler tan te pregven one, are olved. Te prole' oluton or te nal node preent te upper (or lower) etate or te gven prole. or te cae wen te network tructure a tree, te oluton ecoe an eact one. Te Bellan' dynac prograng etod or wc te network tructure copre tree rance, ecoe tu a partcular cae o te ore generalzed propoed approac. A varety o prole or wc te dynac prograng etod napplcale, ave een olved y te network prograng etod. In te preent paper te network prograng etod [] appled to nonlnear prograng prole. Te concept o a dual prole, or wc one o te eale (ut uually non-optal) oluton otaned, uggeted y ean o ultpler o Lagrange. It proved tat te dual prole a conve prograng prole. Neceary and ucent optalty condton or a dual prole o nteger lnear prograng are developed.. Te Network or o a Nonlnear Prograng Prole Let' conder a prole o nonlnear prograng - to deterne,, n, atyng { } ( ) a () uect to ( ),,, () X. () On gure te network repreentaton o retrcton (-) gven. Here X denote te -t retrcton (),,. 78

3 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) X I X X X X gure. Network repreentaton o retrcton In order to apply te network prograng etod we ave to repreent te goal uncton wt te ae network tructure. or t purpoe we wll preent (х) n te or ( ) ( ) ( ), (4) were (х) tand or uncton wc delver oluton or te elow prole (5-6). In eac verte o te network tructure everal optzaton u-prole wt one retrcton are olved. Te rt u-prole are a ollow: ( ), ( ). a wle te () -t u-prole look a ollow: ( ) a ( ) ( ) X a. (6) X Denote () te value o te goal uncton or te optal oluton o te -t uprole. Teore. Lnear odel ( ) ( ) ( ) (7) delver te upper etate or a pre-gven prole. Proo. All eale oluton (-) are eale or all u-prole (5-6), and any eale oluton х ate ( ) ( ). Tereore () () or any eale х.. Te Dual Prole It ovou to ugget te prole o deternng uncton (х), (5),, wc nze te upper etate (7). T prole, n eence, a generalzed dual prole or te ntal prole o nonlnear prograng. Te reaon or t are a ollow. rt, a own elow (ee Eaple ), one o te eale oluton o te generalzed dual prole 7

4 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) 80 a na o uncton o Lagrange. Note tat deternng te na Lagrange uncton oten called te dual one or te prole o nonlnear prograng. Second, or a prole o lnear prograng wtout an nteger oluton te generalzed dual prole a uual dual prole o lnear prograng (ee Secton 4). Teore. uncton () a conve one. Proo. Let (х) and (х) e two oluton o a dual prole. Conder te oluton ( ) ( ) 0,. We otan ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). a a a a a a X X X Te nequalty te ro te evdent reaon tat te au o te u le or equal to te u o aa. Tu, te dual prole a conve prograng prole. Eaple. Conder one o te eale oluton o te dual prole, naely,, ), ( ) (. Te rt u-prole are a ollow: a ) ( uect to ( ). Evdently ( ),,. By ean o t aerton togeter wt (7), we nally otan ( ) ( ) ( ) ( ) ( ) L X X, a a. (8) Mazng te rgt part (8) on notng ut te etod o ultpler o Lagrange. Tu, te etod o ultpler o Lagrange provde a eale oluton o te dual prole (wc, generally peakng, ay e not an optal one). 4. Upon One Integer Lnear Prograng Prole Conder an nteger lnear prograng prole a ollow: deterne an nteger nonnegatve vector х, to aze ( ) n c C () uect to,, a n. (0)

5 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) Take te lat retrcton n (0) a te et Х. Su-dvde eac value on partal value a ollow:, c,,, c, n. () Solve () u-prole a ollow: deterne an nteger nonnegatve vector, to aze ( ) S. uect to n a (). () Denote y () te value S () provdng te optal oluton or te -t uprole. Accordng to Teore () ( ) () (4) an upper etate or С(х): () C(). Te dual prole: deterne{,, n,, }, nzng (4). Note tat cancellng te requreent o ntegralty reult n tranorng prole (4) to a coon dual lnear prograng prole []. To prove t acceon conder prole (-0) wtout te ntegralty requreent. In t cae te etaton prole are ealy olved, naely ( ) a. a Denote y a,,. a Tu, te upper etate or te oectve o te ntal prole look a ollow: Ф ( y) y. (5) Snce a y, relaton () traner to a y c,, n. Te dual prole to nze (5) uect to (6). T a coon dual lnear prograng prole. Set,, n,. A outlned aove, te prole ol down to te, etod o ultpler o Lagrange a ollow: deterne vector, nzng a c a. (7) X (6) 8

6 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) Conder neceary and ucent condton to otan te optal oluton o te dual prole. Let e a eale oluton. Denote Р ( ) te et o optal oluton or () u-prole (-),,. Teore. Te neceary and ucent condton to otan te optal oluton te nalty to olve nequalty a < 0 ( ) y P (8) uect to y 0,, n. () Proo. Denote y y all ncreent o. We wll prove tat relaton () te ro (). Indeed, t ol down ro () tat ( y ) c a nd c old. Te latter provde (). Te ncreent o value ( ), ovouly, equal Δ a, ( ) y P wle te total ncreent ate Δ Δ. Snce te optal oluton, cannot e negatve. Nuercal Eaple. 0,, a, (0) 6 4 5, () () Apply te etod o ultpler o Lagrange,.e. deterne te nu o uncton a[ ( 0 6) ( 8 ) ( 6 ) ( 7 5) 4 ], X were Х deterned y (). Wt pre-et t a one-denonal knapack prole. In cae wen te dependence o te rgt part o (ee retrcton ()) ro n unknown, t prole turn to e NP-dcult [4]. However n practce, eter doe not depend on n, or a lnear uncton o n. In uc cae, or nteger paraeter, te prole ecently olved y ean o eter dynac or dcotoc prograng. Te deterned optal value 0, wt te upper etate 0. T level 0 correpond to te ollowng value,,4,, : a c 0 7 a 0 4 a a Let' apply te network prograng etod. Step. Deterne neceary optalty condton or oluton (). Conder:. () 8

7 P P ( ) {(,,,0 ) (,0,0, )} ( ) (,,0, ) ( 0,,,0 ) { }. Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) Snce у у 0, denote у у -у. In t cae relaton (8-) can e repreented a a( y y y y y4 ) < n( y y y4 y y ). One o te oluton or toe relaton a ollow: y y y y 0 0. ε ε ε ε 4 > Set ε 5 6 nce t value reult n a new oluton o te econd u-prole. We otan: P P ( ) {(,0,0, ) (,,,0 )} 8 ( ) {(,,0, ) ( 0,,,0 ) (,0,,0 )} Step. Conder optalty condton a( y y y y y4 ) < n( y y y4 y y y y ). It can e well-recognzed tat t nequalty a no eale oluton. Indeed, condton у у у < у у у4 reult n у < у 4, wle condton у у 4 < у у lead to a contradctve у 4 < у. Hence, te optal oluton o te dual prole otaned. Te deterned upper etate ay e ued n te ranc and ound etod. Start rancng wt varale х. I х ten te oluton o te correpondng dual prole reult n te ae etate (х ) 0 /. In cae х 0 te otaned etate (х 0) 4. Cooe value х and undertake rancng or varale х. х reult n a eale etate (х, х ) 8. х 0 reult n anoter eale etate (х, х 0) 7. Tu, te optal oluton х, х, х 0, х 4 0, С ах Concluon Te uggeted approac provde a generalzed etod to deterne etate or a road cla o nonlnear prograng prole. T approac enale ung new algort to olve a varety o prole, wt te coputng coplety eng le, tan tat wen ung clacal algort (te knapack prole [], te aal low prole [5], te "tone" prole [], etc.). urter reearc a to e undertaken to etate te coputng coplety o te network prograng etod or varou prole o nonlnear prograng. Reerence. Burkov, V.N., Zaloznev, A.J. and Novkov, D.A. Grap Teory n Managng Organzatonal Syte, Snteg, Mocow, 00 (n Ruan). Burkov, V.N. and Burkova, I.V. Network prograng etod, Manageent Prole,, 005, pp. - (n Ruan). Burkov, V.N. and Burkova, I.V. Metod o dcotozng prograng, Inttute o Control Scence, te Ruan Acadey o Scence, 004, pp (n Ruan) 8

8 Internatonal Sypou on Stocatc Model n Relalty Engneerng, Le Scence and Operaton Manageent (SMRLO'0) 4. Gary, М. and Jonon, D. Coputer and Dcultly-Solved Prole, Mr, Mocow, 8 (n Ruan) 5. Burkov, V.N. (Ed.), Mateatcal ackground o proect anageent, Manual, Vaya Skola, Mocow, 005, pp. -6 (n Ruan) 84