Introduction to Unconstrained Optimization - Scilab

More documents

Recommendations

Info

-->x = [1 1]; -->[ f , g , H ] = rosenbrock ( x ); -->D = spec ( H ) D = 0.3993608 1001.6006 We see that both eigenvalues are strictly positive, although the second is much larger than the first in magnitude. Hence, the matrix H(x) is positive definite for x = (1, 1) T . Let us check that it is indefinite at the point x = (0, 1) T . -->x = [0 1]; -->[ f , g , H ] = rosenbrock ( x ); -->D = spec ( H ) D = - 398. 200. We now see that the first eigenvalue is negative while the second is positive. Hence, the matrix H(x) is indefinite for x = (0, 1) T . Let us make a random walk in the interval [−2, 2] × [−1, 2] and check that many points are associated with an indefinite Hessian matrix. In the following script, we use the rand function to perform a random walk in the required interval. To make so that the script allways performs the same computation, we initialize the seed of the random number generator to zero. Then we define the lower and the upper bound of the simulation. Inside the loop, we generate a uniform random vector t in the interval [0, 1] 2 and scale it with the low and upp arrays in order to produce points in the target interval. We use particular formatting rules for the mprintf function in order to get the numbers aligned. rand (" seed " ,0); ntrials = 10; low = [ -2 -1] ’; upp = [2 2] ’; for it = 1 : ntrials t = rand (2 ,1); x = ( upp - low ) .* t + low ; [ f , g , H ] = rosenbrock ( x ); D = spec ( H ); mprintf ("( %3d ) x=[%+5f %+5f], D =[% +7.1 f % +7.1 f]\n" ,.. it ,x(1) ,x(2) ,D(1) ,D (2)) end The previous script produces the following output. ( 1) x =[ -1.154701 +1.268132] , D =[ +4.4 +1290.4] ( 2) x =[ -1.999115 -0.009019] , D =[ +65.0 +4936.4] ( 3) x =[+0.661524 +0.885175] , D =[ -78.4 +451.5] ( 4) x =[+1.398981 +1.057193] , D =[ +34.6 +2093.1] ( 5) x =[+1.512866 -0.794878] , D =[ +77.5 +3189.0] ( 6) x =[+0.243394 +0.987071] , D =[ -339.3 +217.6] ( 7) x =[+0.905403 -0.404457] , D =[ +77.4 +1270.1] ( 8) x =[+0.177029 -0.303776] , D =[ +107.1 +254.0] ( 9) x =[ -1.075105 -0.350610] , D =[ +73.0 +1656.3] ( 10) x =[+1.533555 +0.957540] , D =[ +43.1 +2598.0] By increasing the number of trials, we see that the first eigenvalue is often negative. References [1] Maxima. http://maxima.sourceforge.net/. 40
[2] Grégoire Allaire. Analyse numérique et optimisation. Éditions de l’école polytechnique, 2005. [3] Dimitry Bertsekas. Convex Analysis and Optimization. Athena Scientific, Belmont, Massachusetts, 2003. [4] Fréderic Bonnans, Claude Lemaréchal, Charles Gilbert, and Claudia Sagaztizábal. Numerical Optimization. Springer, 2002. [5] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [6] P. E. Gill, W. Murray, and M. H. Wright. Practical optimization. Academic Press, London, 1981. [7] Stephen J. Wright Jorge Nocedal. Numerical Optimization. Springer, 1999. [8] C.T. Kelley. Iterative Methods For Optimization. Springer, 1999. [9] G. Luenberger, David. Linear and nonlinear programming, Second Edition. Addison-Wesley, 1984. [10] Maple. Maplesoft. http://www.maplesoft.com. [11] Wolfram Research. Mathematica. http://www.wolfram.com/products/ mathematica. [12] Wolfram Research. Wolfram alpha. http://www.wolframalpha.com. [13] H. H. Rosenbrock. An automatic method for finding the greatest or least value of a function. The Computer Journal, 3(3):175–184, March 1960. 41
Page 1 and 2: Introduction to Unconstrained Optim
Page 3 and 4: Copyright c○ 2008-2010 - Michael
Page 5 and 6: Optimization Discrete Parameters Co
Page 7 and 8: A subclass of optimization problems
Page 9 and 10: -->[ f , g , H ] = rosenbrock ( x0
Page 11 and 12: 3000 2500 2000 Z 1500 1000 500 0 2.
Page 13 and 14: f(x) strong local optimum weak loca
Page 15 and 16: The previous definition intuitively
Page 17 and 18: x 1 x 2 x 1 x 2 Figure 7: Convex se
Page 19 and 20: 0 0 Figure 11: Conic hulls f(x) f(y
Page 21 and 22: T f(x)+g(x) (y-x) f(x) x Figure 15:
Page 23 and 24: By hypothesis x = θy 1 + (1 − θ
Page 25 and 26: Proof. We assume that the Hessian m
Page 27 and 28: 5 4 3 2 1 Z 0 -1 -2 -3 -4 -5 5 4 3
Page 29 and 30: 2.0 1.5 1.0 20 0.5 10 0.0 5 2 2 5 -
Page 31 and 32: φ(α) α 2.0 1.5 -20 φ(α) -10 1.
Page 33 and 34: 6 20 4 2 0 -5 0 5 -2 -4 20 -6 -10 -
Page 35 and 36: 3 Answers to exercises 3.1 Answers
Page 37 and 38: C x 2 x 2 x 1 x 3 x 3 Figure 28: Co
Page 39: for all x, y ∈ C, then f is conve

Introduction to Unconstrained Optimization - Scilab

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?