Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 35 
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fu n c tio n s , flow can be diverted from i to j , or a t le a s t the queue > o f one l i n k included in j can be increased, such th a t e^>0 and c ^ . ( e , x ) > Cj ( e ' , x ' ). Thus (e, x) i s not u s e ro p tim izin g. UserOptimized T r a f f ic Assignment Problem: Given a (capacity constrained) tra n s p o rta tio n network, T = (G ,d ,u ,y ), determine a fe a s ib le assignment (e*,A *) which s a t is f ie s the e q u ilib riu m co n d itio n s. The fe a s ib le flow pattern f * determined by e *, is termed a useroptimized flow p a tte rn . 3.2 CAPACITYCONSTRAINED, USEROPTIMIZED TRAFFIC ASSIGNMENT In t h is s e c tio n , we s ta te several theorems p e rta in in g to existance and uniqueness and e s ta b lis h an important connection to an ^'associated*' systemoptimized t r a f f i c assignment problem th a t allows one to employ standard lin e a r and nonlinear programming techniques. Given T, we define an associated capacity constrained transporta tio n network, t = (G ,u ,d ,Y ), by X 7 S T i ( y ) d y ; , X o 0 ( 12) X g J Y ^ ( 0 ) ; y ^ 0 X = 0 I t is e a s ily established under Assumption 1, from p ro p e rtie s o f d if f e r e n t ia b le fu n c tio n s , th a t x y j ( x ) is convex ( e . g . . Reference 28, r» p. 116). Then i t follow s th a t the to ta l cost fu n c tio n a l o f T, S ^ ( . ) , i s convex. Furthermore, from th is r e s u lt and the d e f in it io n o f co nvexity, i t e a s ily fo llo w s th a t S2 ( . ) is convex. A l s o , i t is E a s ily seen th a t the set o f fe a s ib le flow assignments e Is convex. 27
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Title  Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 35 
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Full text  fu n c tio n s , flow can be diverted from i to j , or a t le a s t the queue > o f one l i n k included in j can be increased, such th a t e^>0 and c ^ . ( e , x ) > Cj ( e ' , x ' ). Thus (e, x) i s not u s e ro p tim izin g. UserOptimized T r a f f ic Assignment Problem: Given a (capacity constrained) tra n s p o rta tio n network, T = (G ,d ,u ,y ), determine a fe a s ib le assignment (e*,A *) which s a t is f ie s the e q u ilib riu m co n d itio n s. The fe a s ib le flow pattern f * determined by e *, is termed a useroptimized flow p a tte rn . 3.2 CAPACITYCONSTRAINED, USEROPTIMIZED TRAFFIC ASSIGNMENT In t h is s e c tio n , we s ta te several theorems p e rta in in g to existance and uniqueness and e s ta b lis h an important connection to an ^'associated*' systemoptimized t r a f f i c assignment problem th a t allows one to employ standard lin e a r and nonlinear programming techniques. Given T, we define an associated capacity constrained transporta tio n network, t = (G ,u ,d ,Y ), by X 7 S T i ( y ) d y ; , X o 0 ( 12) X g J Y ^ ( 0 ) ; y ^ 0 X = 0 I t is e a s ily established under Assumption 1, from p ro p e rtie s o f d if f e r e n t ia b le fu n c tio n s , th a t x y j ( x ) is convex ( e . g . . Reference 28, r» p. 116). Then i t follow s th a t the to ta l cost fu n c tio n a l o f T, S ^ ( . ) , i s convex. Furthermore, from th is r e s u lt and the d e f in it io n o f co nvexity, i t e a s ily fo llo w s th a t S2 ( . ) is convex. A l s o , i t is E a s ily seen th a t the set o f fe a s ib le flow assignments e Is convex. 27 