Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 38 
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As in the preceding discussion o f convexity, under Assumption 2, i t is easy to e s ta b lis h th a t S^( . ) i s s t r i c t l y convex over the convex set o f fe a s ib le flow patterns f . Then Theorem 3 fo llo w s from convex optimization theory 29 ’ 30 and Theorem 2. Note however, th a t S2 ( . ) is s t i l l only convex, as two or more values o f e may determine the same f . Theorem 3: Under Assumption 2, i f there is a fe a s ib le flow p a tte rn f fo r network T, then t h e r e e x i s t s a unique u s e r  o p t im i z e d flow p a t t e r n f * . Comment: This theorem does not hold f o r constant l in k u n it costs because then S^( . ) i s merely convex. Al so, as more than one usero p tim izin g flow assignment e* may determine f * , e* is not necessarily unique under Assumption 2. Under the preceding assumptions, the vector o f useroptimized queues A* is not necessarily unique. A reason is th a t as the rows o f the A m atrix are g e n e ra lly lin e a r ly dependent, the in e q u a lity c o n s tra in ts . Equation (13b), are redundant. Therefore, the uniqueness ! of A* depends on the useroptimizing assignment e*. 29 But, even i f the a c tiv e c o n s tra in ts are independent, since e* is not necessarily unique, there may e x is t d i s t i n c t unique values o f A* f o r each e*. The uniqueness appears to depend on the p a r t ic u la r s tru c tu re o f the problerr in vo lve d , as the example fo llo w in g shows. This issue has not been resolved; however we are able to give a s u f f i c i e n t c o n d itio n f o r uniqueness, which is Theorem 4, fo llo w in g . The r e s u lt o f the theorem i s/ an instance o f the usual co n d itio n f o r uniqueness o f Lagrange 30
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Title  Traffic assignment for capacitated transportation networks including queueing  with application to freeway corridor control  Page 38 
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Full text  As in the preceding discussion o f convexity, under Assumption 2, i t is easy to e s ta b lis h th a t S^( . ) i s s t r i c t l y convex over the convex set o f fe a s ib le flow patterns f . Then Theorem 3 fo llo w s from convex optimization theory 29 ’ 30 and Theorem 2. Note however, th a t S2 ( . ) is s t i l l only convex, as two or more values o f e may determine the same f . Theorem 3: Under Assumption 2, i f there is a fe a s ib le flow p a tte rn f fo r network T, then t h e r e e x i s t s a unique u s e r  o p t im i z e d flow p a t t e r n f * . Comment: This theorem does not hold f o r constant l in k u n it costs because then S^( . ) i s merely convex. Al so, as more than one usero p tim izin g flow assignment e* may determine f * , e* is not necessarily unique under Assumption 2. Under the preceding assumptions, the vector o f useroptimized queues A* is not necessarily unique. A reason is th a t as the rows o f the A m atrix are g e n e ra lly lin e a r ly dependent, the in e q u a lity c o n s tra in ts . Equation (13b), are redundant. Therefore, the uniqueness ! of A* depends on the useroptimizing assignment e*. 29 But, even i f the a c tiv e c o n s tra in ts are independent, since e* is not necessarily unique, there may e x is t d i s t i n c t unique values o f A* f o r each e*. The uniqueness appears to depend on the p a r t ic u la r s tru c tu re o f the problerr in vo lve d , as the example fo llo w in g shows. This issue has not been resolved; however we are able to give a s u f f i c i e n t c o n d itio n f o r uniqueness, which is Theorem 4, fo llo w in g . The r e s u lt o f the theorem i s/ an instance o f the usual co n d itio n f o r uniqueness o f Lagrange 30 