Solutions_Ch06 - CHAPTER 6 DUALITY THEORY 6.1-1(a minimize subject to&C#C&C\$ C C&C\$#C C \$C\$ C C C\$(b minimize subject to%C#C#C\$ C#C C\$ C C C\$ C C

# Solutions_Ch06 - CHAPTER 6 DUALITY THEORY 6.1-1(a...

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6-1 CHAPTER 6: DUALITY THEORY 6.1-1. (a) minimize "&C  "#C  %&C " # \$ subject to C  C  &C   "! " # \$ #C  C  \$C   #! " # \$ C ß C ß C   ! " # \$ (b) minimize %C  #C  "#C " # \$ subject to C  #C  C   # " # \$ C  C  C   # " # \$ C  C  \$C   \$ " # \$ C ß C ß C   ! " # \$ 6.1-2. (a) B B B B C " # % \$ Ÿ #! C % ' & % Ÿ %! C # \$ \$ ) Ÿ &! & Ÿ " Ÿ \$ Ÿ % Ÿ " # \$ % " # \$ minimize #!C  %!C  &!C " # \$ subject to C  %C  #C   & " # \$ #C  'C  \$C   " " # \$ %C  &C  \$C   \$ " # \$ \$C  %C  )C   % " # \$ C ß C ß C   ! " # \$ (b) The dual problem has no feasible solution. 6.1-3. (a) Apply the simplex method to the dual of the problem, since the dual has fewer constraints (not including nonnegativity constraints). We expect that the simplex method will go through fewer basic feasible solutions. (b) Apply the simplex method to the primal problem, since it has fewer constraints (not including nonnegativity constraints). We expect that the simplex method will go through fewer basic feasible solutions. 6.1-4. (a) minimize "#C  C " # subject to C  C   " " # C  C   # " # #C  C   " " # C ß C   ! " # (b) It is clear from the dual problem that is the optimal dual solution. By ÐC ß C Ñ œ Ð!ß !Ñ " # strong duality, . ^ œ ! Ÿ !