TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 28

CHAPTER 55 Numerical Minimization. Literature: [Thi88, p. 199–219] or [KG80, pp. 425–475]. Regarding numerical methods, the books are classics, and they are available on-line for free at Assume θ → f (θ) is a scalar function of a vector argument | CHAPTER 55 Numerical Minimization Literature Thi88 p. 199-219 or KG80 pp. 425-475 . Regarding numerical methods the books are classics and they are available on-line for free at numerical Assume 0 f 0 is a scalar function of a vector argument with continuous first and second derivatives which has a global minimum . there is an argument 0 with f 0 f 0 for all 0. The numerical methods to find this minimum argument are usually recursive the computer is given a starting value 00 uses it to compute 01 then it uses 01 to compute 02 and so on constructing a sequence 01 02 . that converges towards 1207 1208 55. NUMERICAL MINIMIZATION a minimum argument. If convergence occurs this minimum is usually a local minimum and often one is not sure whether there is not another better local minimum somewhere else. At every step the computer makes two decisions which can be symbolized as 0i i 0i aidi. Here di a vector is the step direction and ai a scalar is the step size. The choice of the step direction is the main characteristic of the program. Most programs notable exception simulated annealing always choose directions at every step along which the objective function slopes downward so that one will get lower values of the objective function for small increments in that direction. The step size is then chosen such that the objective function actually decreases. In elaborate cases the step size is chosen to be that traveling distance in the step direction which gives the best improvement in the objective function but it is not always efficient to spend this much time on the step size. Let us take a closer look how to determine the step direction. If g g 0i T is the Jacobian of f at 0i . the row vector consisting of the partial derivatives of f then the objective function will slope down along direction di if the scalar product gidi is negative. In determining the step direction the following fact is useful All vectors di for which gj di 0 can be .

• Kế toán - Kiểm toán
• thống kê học
• kinh tế lượng
• kinh tế học
• toán kinh tế
• giáo trình kinh tế
• Phương pháp thống kê
• Phương pháp chỉ số
• Tài liệu thống kê
• Tài liệu kiểm toán