The following references can be refered for more detailed information about steepest descent. 1. A. Ravindran,...

 

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The following references can be refered for more detailed information about steepest descent. 1. A. Ravindran, K.M. Ragsdell, G.V. Reklaltis,"Engineering Optimization: Methods and Appli- cations", Second Edition (Chapter 3.3.1 Cauchy's Method) 2. David G. Lunberger, Yinyu Ye, "Linear and Nonlinear Programming", Third Edition, Springer (Chapter 8.1 Golden Section, Chapter 8.6 Steepest Descent) The pseudocode of Steepest Descent algorithm is given as follows: Algorithm 2 Steepest Descent 1: procedure STEEPEST DESCENT 2: Step 1: Initialization: 3: 4: & 6: 7: Initialize Xo = 0 and N = 150(marimum iteration number) Determine the values of the stopping criterias 1-10-10, 2-10-10, 3= 10-10 Set k = 1 Step 2: Calculate the gradient vector f(x) at x Calculate the direction vector p = f(x) Minimize f(x + p) with respect to n via golden section. Hint: You can set n = 1. 10: Update X+1=Xk+ 1/Pk 11: Step 3: Termination 12: Calculate Aff(x+1)-f(x) and Ax=Xk+1-Xk 13: if Af & then, terminate algorithm since the value of the objective function does not change if ||Ax| 2 then, terminate algorithm since the value of the variable does not change if Vf(x)| ≤ 3 then, terminate algorithm since the value of the variable does not change k+k+1.. 8: 9. 14: 15: 16: 17: if k= N then, terminate algorithm owing to reaching marimum iteration number. 18: Step 4: Obtain solution: Implement the above code using Matlab m-file enviroment. [15-marks] Solve the given problem in (2) via implemented Matlab codes, and show the following figures. [10-marks] Figure 1: Convergence of the varibles to their optimum solutions as subfigures(w versus k w versus k). Figure 2: Convergence of the objective function and it's derivative(f(w) versus k. Vf(w) versus k). Figure 3: Convergence of variables to solution point (w₂ versus w₁). • Prepare a report containing Matlab m-file codes, the above figures and information about steepest descent algorithm [35-marks]. f(w): w+w+1.6364w1w2 2 0.8182w1 0.354w2 +0.1250 The sample figure is illustrated in Figure 2 for implementation and report. You may need to know 1.5 0.5 O -0.5 -11 Sample 0 5 10 Sample 15 Sample 20 25 k W₂[k] 30 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 35 40 45 w,[k] W₂[k] 0.5 50 0.2 0.1 o -0.1 -0.2! -0.3- -0.4 w,[k] 0 Sample Sample Sample 5 10 Sample Sample 15 20 w₂[k] vs w₁ [k] 1.5 25 30 Figure 2: Sample Figure for Implementation and Report. Sams-le 35 40 ·[k] ----[k] 45 Zero(0) 50 the following commands in order to deploy in Matlab environment. . Matlab Commands: fprintf, feval, inline, norm, ceil etc. • Learn the if, while, switch and for statements. • Learn how to use latex as an interpreter in Matlab so as to utilize Greek symbols in Figures such as "V The following references can be refered for more detailed information about steepest descent. 1. A. Ravindran, K.M. Ragsdell, G.V. Reklaltis,"Engineering Optimization: Methods and Appli- cations", Second Edition (Chapter 3.3.1 Cauchy's Method) 2. David G. Lunberger, Yinyu Ye, "Linear and Nonlinear Programming", Third Edition, Springer (Chapter 8.1 Golden Section, Chapter 8.6 Steepest Descent) The pseudocode of Steepest Descent algorithm is given as follows: Algorithm 2 Steepest Descent 1: procedure STEEPEST DESCENT 2: Step 1: Initialization: 3: 4: & 6: 7: Initialize Xo = 0 and N = 150(marimum iteration number) Determine the values of the stopping criterias 1-10-10, 2-10-10, 3= 10-10 Set k = 1 Step 2: Calculate the gradient vector f(x) at x Calculate the direction vector p = f(x) Minimize f(x + p) with respect to n via golden section. Hint: You can set n = 1. 10: Update X+1=Xk+ 1/Pk 11: Step 3: Termination 12: Calculate Aff(x+1)-f(x) and Ax=Xk+1-Xk 13: if Af & then, terminate algorithm since the value of the objective function does not change if ||Ax| 2 then, terminate algorithm since the value of the variable does not change if Vf(x)| ≤ 3 then, terminate algorithm since the value of the variable does not change k+k+1.. 8: 9. 14: 15: 16: 17: if k= N then, terminate algorithm owing to reaching marimum iteration number. 18: Step 4: Obtain solution: Implement the above code using Matlab m-file enviroment. [15-marks] Solve the given problem in (2) via implemented Matlab codes, and show the following figures. [10-marks] Figure 1: Convergence of the varibles to their optimum solutions as subfigures(w versus k w versus k). Figure 2: Convergence of the objective function and it's derivative(f(w) versus k. Vf(w) versus k). Figure 3: Convergence of variables to solution point (w₂ versus w₁). • Prepare a report containing Matlab m-file codes, the above figures and information about steepest descent algorithm [35-marks]. f(w): w+w+1.6364w1w2 2 0.8182w1 0.354w2 +0.1250 The sample figure is illustrated in Figure 2 for implementation and report. You may need to know 1.5 0.5 O -0.5 -11 Sample 0 5 10 Sample 15 Sample 20 25 k W₂[k] 30 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 35 40 45 w,[k] W₂[k] 0.5 50 0.2 0.1 o -0.1 -0.2! -0.3- -0.4 w,[k] 0 Sample Sample Sample 5 10 Sample Sample 15 20 w₂[k] vs w₁ [k] 1.5 25 30 Figure 2: Sample Figure for Implementation and Report. Sams-le 35 40 ·[k] ----[k] 45 Zero(0) 50 the following commands in order to deploy in Matlab environment. . Matlab Commands: fprintf, feval, inline, norm, ceil etc. • Learn the if, while, switch and for statements. • Learn how to use latex as an interpreter in Matlab so as to utilize Greek symbols in Figures such as "V

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