<Text-field style="Heading 1" layout="Heading 1">Fox' Gradient Search with Adaptive Stepsize </Text-field> by Dr. William P. Fox (WFox@fmarion.edu) and Dr. William H. Richardson (WRichardson@fmarion.edu) Dept. of Math., Francis Marion University, Florence. Adaptation and minor improvements to Maple18 (Maple 2015) by Dr. Henk Koppelaar (Koppelaar.Henk@GMail.com) This worksheet solves nonlinear optimization problems by the method of steepest ascent. Given a function 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 and a current point 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, the search direction is taken to be the gradient of 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 at LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXJvd0dGJDYoLUkjbWlHRiQ2JVEnYHhfXzBgRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GOTYtUSJ+RidGPEY+L0ZCRkBGQ0ZFRkdGSUZLRk0vRlFGT0ZTLUYvNiVRJ2B5X18wYEYnRjJGNUY8Rjw=. The step length is computed by line search, i.e. as the step length that maximizes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY+Ni1RIn5GJ0ZBRkMvRkdGRUZIRkpGTEZORlBGUi9GVkZULUYsNiVRInlGJ0YvRjJGQUZBRkE= along the gradient direction. Algorithm for the Gradient Method of Steepest Ascent : Step 1. Enter the function to be maximized f, the maximum number of iterations allowed n, starting point 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, and an accuracy tolerance t. Step 2. Choose \316\273 to maximize, 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 Step+ 3. Move to next point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkobWZlbmNlZEdGJDYkLUYjNistSSNtaUdGJDYlUSlgeF9faysxYEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjs2LVEifkYnRj5GQC9GREZCRkVGR0ZJRktGTUZPL0ZTRlFGVS1GMTYlUSlgeV9faysxYEYnRjRGNy1GMTYjUSFGJy8lK2V4ZWN1dGFibGVHRjYvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRj5GPi1GOzYtUSI9RidGPkZARlhGRUZHRklGS0ZNL0ZQUSwwLjI3Nzc3NzhlbUYnL0ZTRmNvRlUtRiw2JC1GIzYrLUYxNiVRJ2B4X19rYEYnRjRGN0Y6RlVGVS1GMTYlUSdgeV9fa2BGJ0Y0RjdGZ25Gam5GXG9GPkY+LUY7Ni1RIitGJ0Y+RkBGWEZFRkdGSUZLRk0vRlBRLDAuMjIyMjIyMmVtRicvRlNGY3AtRjE2JVEnJiM5NTU7RicvRjVGQkY+LUY7Ni1RJyZzZG90O0YnRj5GQEZYRkVGR0ZJRktGTUZPRllGVS1GMTYlUSVHcmFkRidGNEY3LUYsNiQtRiM2Jy1GMTYlUSJmRidGNEY3RmVvRmpuRlxvRj5GPkZqbkZcb0Y+ Step 4 . If 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, return to Step 2. Otherwise, we can terminate. We may also terminate if the number of points found by iterating is greater than the number n of allowed iterations (in Step 1). 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---------------------------------------------------------------------------- Starting point ( 0.6000, 4.0000) Iter Gradient Vector G magnitude of G x[k] Step Length 1, ( 6.8000, -12.8000) 14.4940 ( 0.6000, 4.0000), 0.1917 2, ( -0.7138, -0.3792) 0.8083 ( 1.9034, 1.5465), 1.2772 3, ( 0.1409, -0.2652) 0.3004 ( 0.9917, 1.0622), 0.1917 4, ( -0.0148, -0.0079) 0.0168 ( 1.0187, 1.0113) ---------------------------------------------------------------------------- Approximate Solution: ( 1.0187, 1.0113) Maximum Functional Value: 0.9998 Number gradient evaluations: 5 Number function evaluations: 4 ---------------------------------------------------------------------------- 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 ---------------------------------------------------------------------------- Starting point ( 1.0000, 1.0000) Iter Gradient Vector G magnitude of G x[k] Step Length 1, ( 47.0000, 105.0000) 115.0400 ( 1.0000, 1.0000), 0.0380 2, ( 32.7185, -14.6454) 35.8470 ( 2.7852, 4.9882), 0.0857 3, ( 10.2936, 22.9964) 25.1950 ( 5.5883, 3.7335), 0.0380 4, ( 7.1658, -3.2075) 7.8509 ( 5.9793, 4.6069), 0.0857 5, ( 2.2544, 5.0365) 5.5180 ( 6.5932, 4.3321), 0.0380 6, ( 1.5694, -0.7025) 1.7195 ( 6.6788, 4.5234), 0.0857 7, ( 0.4938, 1.1031) 1.2085 ( 6.8133, 4.4632), 0.0380 8, ( 0.3437, -0.1539) 0.3766 ( 6.8320, 4.5051), 0.0857 9, ( 0.1081, 0.2416) 0.2647 ( 6.8615, 4.4919), 0.0380 10, ( 0.0753, -0.0337) 0.0825 ( 6.8656, 4.5011), 0.0857 11, ( 0.0237, 0.0529) 0.0580 ( 6.8720, 4.4982), 0.0380 12, ( 0.0165, -0.0074) 0.0181 ( 6.8729, 4.5002), 0.0857 13, ( 0.0052, 0.0116) 0.0127 ( 6.8744, 4.4996), 0.0380 14, ( 0.0036, -0.0016) 0.0040 ( 6.8745, 4.5001), 0.0857 15, ( 0.0011, 0.0025) 0.0028 ( 6.8749, 4.4999), 0.0380 16, ( 0.0008, -0.0004) 0.0009 ( 6.8749, 4.5000) ---------------------------------------------------------------------------- Approximate Solution: ( 6.8749, 4.5000) Maximum Functional Value: 392.8125 Number gradient evaluations: 17 Number function evaluations: 16 ---------------------------------------------------------------------------- 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 ---------------------------------------------------------------------------- Starting point ( 0.5000, 0.5000) Iter Gradient Vector G magnitude of G x[k] Step Length 1, ( -0.6487, 1.3513) 1.4989 ( 0.5000, 0.5000), 0.2874 2, ( 0.4084, 0.1961) 0.4530 ( 0.3136, 0.8883), 1.7041 3, ( -0.2993, 0.6235) 0.6917 ( 1.0095, 1.2224), 0.2001 4, ( 0.1097, 0.0526) 0.1216 ( 0.9496, 1.3472), 0.7345 5, ( -0.0298, 0.0621) 0.0689 ( 1.0302, 1.3859), 0.1868 6, ( 0.0090, 0.0043) 0.0099 ( 1.0246, 1.3975) ---------------------------------------------------------------------------- Approximate Solution: ( 1.0246, 1.3975) Maximum Functional Value: 8.8277 Number gradient evaluations: 7 Number function evaluations: 6 ----------------------------------------------------------------------------