Rounded Ie Bounded Branch And Bound Method for Mixed Integer Nonlinear Progr
Rounded Ie Bounded Branch And Bound Method for Mixed Integer Nonlinear Progr
Jacob Akoka
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(T) > {4-a) 1 — yj^(T) ^ if Xj^{T) = a^^ (4-b) Vi^(T) 1 if x^CV) = h^ (4-c) \l^(T) =0 if aj^ , 1 I ox, / k k 1=1 > k/ x=x(T) m y A. (T) h. (x(T)) = {4-f) i=l ^ m } 1 i ()) = (}) + J] A. H. Is convex. Therefore: i=l -^'-(f!) (t)[x{W)] > (t)[x(T)] + [x{U) ^ . ..^. — '^"/x = x (T) m „ ^ m ^ ^[xCU) + E A. {T) h. {x(T)) ;> [x(T)] + E A . (T) h. (x (T) ) 1=1 1=1 +Z pi^^ + Z X. (T)--^^ [ ^ [x(U) - x(T)] KeJ-B[ 3x^ i=l ^ 3x^ Jx=x(T) i--^ + E A. (T) — ^^ t 9x 1=1 K -' "" \ -• x=x(T), {4-g) Using (...1-1) and (4-a), we have: ilh A E X. (T) h. (x{T))£ i=l ^ ^ Using (4-f), we obtain: m E X. (T)h. (X(T))=0 i=l " ^ -18- Using {4-b), (4-c), (4-d) and (4-e), we have: E [ -?^- + ^ ^. (T)-. -^--] [x(U)-x {T)]> dx . , 1 ax, — KeJ-B K 1=1 K T M -^r. (U)] > Hx (T) . E[-|i- . E A. (T) 1^ ]^^;^^^ ^^_^^ p 1-1 p Let J^=* U (T) . ^[|^ +.^ ^i(T) ^^] . (4_i) 3 1=1 B x = x{T) and ((. Y = (t. [x(U)] (4-j) Therefore, we obtain: \ 1
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