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SIMULATION AND THE MONTE CARLO METHOD Episode 3

Tham khảo tài liệu 'simulation and the monte carlo method episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 40 PRELIMINARIES 2. Solve for fixed A and 3 min p A 3 1-83 p by solving Vp p A 3 0 which gives the set of equations VPt p A 0 In ụ 1 - 52 Xi Si k 3 0 k l . r . qk i i Denote the optimal solution and the optimal function value obtained from the program 1.83 as p A 3 and c A 3 respectively. The latter is the Lagrange dual function. We have pt A 3 Qfcexp i- 3 -1 52 1 xk j fc l . r. 1.84 i 1 Since the sum of the Pk must be 1 we obtain 52 7fcexp i-l Ai Xfc j . 1.85 k i 1 1 Substituting p A 3 back into the Lagrangian gives A 3 1 52 A 7i 3 . 1.86 i l 3. Solve the dual program max r A 3 . 1.87 A Since 3 and A are related via 1.85 solving 1.87 can be done by substituting the corresponding 3 A into 1.86 and optimizing the resulting function D A -1 4- 2 Ai7i - In 2 exp -l At Si xfc I . 1.88 i 1 lfc l Since jD A is continuously differentiable and concave with respect to A we can derive the optimal solution A by solving VÀD A 0 1.89 which can be written componentwise in the following explicit form _ EU15 x Mexp -i EJL1 AjSj xfc VĂJD A 7t------- V EL1 9k exp I -1 Xj Sj xfc I _ 1.90 _ Eq s. X exp -l JnL1AJSJ X 7i Eq exp -1 7 1 Xj Sj X PROBLEMS 41 for j 1 . m. The optimal vector A AĨ . can be found by solving 1.90 numerically. Note that if the primal program has a nonempty interior optimal solution then the dual program has an optimal solution A . 4. Finally substitute A A and 3 Z A back into 1.84 to obtain the solution to the original MinxEnt program. It is important to note that we do not need to explicitly impose the conditions Pi 0 i 1 . n because the quantities Pi in 1.84 are automatically strictly positive. This is a crucial property of the CE distance see also 2 . It is instructive see Problem 1.37 to verify how adding the nonnegativity constraints affects the above procedure. When inequality constraints Ep Si X 7i are used in 1.80 instead of equality constraints the solution procedure remains almost the same. The only difference is that the .