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Advanced Techniques in Optimization Methods for Communication Networks

Explore cubic interpolation and direct root methods for optimizing communication networks. Learn about minimizing step lengths, normalization, and convergence criteria in network optimization processes. Detailed examples provided.

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Advanced Techniques in Optimization Methods for Communication Networks

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  1. Al-Mustansiriyah University College of Engineering Electrical Engineering Department Ph.D.courses /Comm. Eng./2018-2019 Technical College / Al-Najaf Communications Techniques Eng. Dpt. Technical Engineering College / Al-Najaf Communications Techniques Eng. Dpt. Computer Networks-4th Class-2011/2012 Computer Networks-4thClass-2015/2016 Optimization Lect.4: One Dimension Minimization Methods Mon. 15/10/2018

  2. Al-Mustansiriya University College of Engineering Electrical Engineering Department PhD courses /Comm. Eng./2018-2019 Technical College / Al-Najaf Communications Techniques Eng. Dpt. Technical Engineering College / Al-Najaf Communications Techniques Eng. Dpt. Computer Networks-4th Class-2011/2012 Computer Networks-4thClass-2015/2016 Main Topics • Cubic Interpolation Method • Direct Root Method

  3. c Cubic Interpolation Method • Linear interpolation is the simplest method of getting values at positions in between the data points. • Quadratic interpolation provide a smooth transition between adjacent segments. Linear Quadratic Cubic

  4. c Cubic Interpolation Method • Cubic interpolation offers true continuity between the segments. • We need sufficient condition to have a regular curve, without the sharp edges. • We do that by using the derivative of the function which satisfy the condition, the slope on the leftis the same as the one on its right • The cubic interpolation method finds the minimizing step length in four stages Stage4 Stage1 Stage2 Stage3 Bounds on Normalization Refitting Approximation

  5. c Cubic Interpolation Method Stage1: Normalization Recall that Where Note:This stage is not required if the one-dimensional minimization Stage2: To establish lower and upper bounds on the optimal step size , we need to find two points A and B at which the slope has different signs.

  6. c Stage 3: Approximation 𝑇𝑜 𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠 𝑎, 𝑏, 𝑐 and d 𝑖𝑛 Evaluate at points named A and B C1

  7. c C1 Stage 3: Approximation C2

  8. c C2 The necessary condition for the minimum of is Stage 3: Approximation The sufficiency condition for the minimum of leads to the relation By substituting the expressions for we obtain: End

  9. c Stage 4: The value of found in stage 3 is the true minimum of and may not be close to the minimum of Hence the following convergence criteria can be used before choosing : If the criteria stated is not satisfied, a new cubic equation is used

  10. c Example// Find the minimum of by the cubic interpolation method. Sol. Since this problem has not arisen during a multivariable optimization process, we can skip stage 1. To find at which is nonnegative, we start with = 0.4 and We evaluate the derivative at , 2, 4. This gives

  11. c Thus we find that * Iteration 1 To find the value of and to test the convergence criteria, we first compute as

  12. c Hence By discarding the negative value, we have Convergence criterion: If is close to the true minimum, , then should be approximately zero. Since

  13. c Since this is not small, we go to the next iteration or refitting. As and Thus

  14. c

  15. c

  16. c Direct Root Methods * The direct root methods seek to find the root (or solution) of the equation, = 0. * The necessary condition for to have a minimum of is that = 0. Three root-finding methods • The Newton method • The quasi-Newton method • and the Secant methods

  17. c The Newton Method We consider the quadratic approximation of the function f (λ) at using the Taylor’s series expansion: By setting the derivative of Eq. 3 equal to zero for the minimum of ), we obtain If denotes an approximation to the minimum of Eq. 4 can be rearranged to obtain an improved approximation as

  18. c The Newton Method The iterative process given by Eq. 5 can be assumed to have converged when the derivative, , is close to zero: Example// Find the minimum of the function using the Newton method with the starting point . Use

  19. c The Newton Method Sol. The first and second derivatives of the function are given by

  20. c The Newton Method

  21. c The Newton Method

  22. c Thank You Any Questions?

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