Derivative free optimization - Revision history

Fall2024 Wiki Team7: /* Medical */

2024-12-15T19:02:51Z

Medical

← Older revision		Revision as of 15:02, 15 December 2024
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	J Cancer 1990 May;61(5):788. PMID: 2297481; PMCID: PMC1971319.		J Cancer 1990 May;61(5):788. PMID: 2297481; PMCID: PMC1971319.

	<nowiki>https://pmc.ncbi.nlm.nih.gov/articles/PMC1971319/</nowiki></ref> Nealder-Mead is a common optimizer option in common tools such as Excel, Python (~~Scipy~~) and ~~Matlab~~.		<nowiki>https://pmc.ncbi.nlm.nih.gov/articles/PMC1971319/</nowiki></ref> Nealder-Mead is a common optimizer option in common tools such as Excel, Python (SciPy)<ref>Virtanen, Pauli, et al. ''SciPy: Open Source Scientific Tools for Python.'' SciPy, 2020, <nowiki>https://docs.scipy.org/doc/scipy/reference/optimize.minimize-neldermead.html</nowiki>. Accessed 15 Dec. 2024.</ref> and MATLAB.

	== Conclusion ==		== Conclusion ==

Fall2024 Wiki Team7: /* Example 1: Golden Section Search */

2024-12-15T18:45:53Z

Example 1: Golden Section Search

← Older revision		Revision as of 14:45, 15 December 2024
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	::Repeat this process, recalculating <math>x_1</math> and <math>x_2</math> and updating the interval until the difference between <math>a</math> and <math>b</math> is less than 1E-4 .		::Repeat this process, recalculating <math>x_1</math> and <math>x_2</math> and updating the interval until the difference between <math>a</math> and <math>b</math> is less than 1E-4 .

	::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) - 1</math>.		::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) = 1</math>.
	::		::
	::		::

Fall2024 Wiki Team7: /* Example 2: Successive Parabolic Interpolation */

2024-12-15T18:44:16Z

Example 2: Successive Parabolic Interpolation

← Older revision		Revision as of 14:44, 15 December 2024
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	::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) - 1</math>.		::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) - 1</math>.

	~~::Two images have been generated to describe this numerical example through first iteration described below.~~
	::		::
	::		::

Fall2024 Wiki Team7: Updated "Derivative-Free Optimization" and DFO references in text body.

2024-12-15T02:47:48Z

Updated "Derivative-Free Optimization" and DFO references in text body.

← Older revision		Revision as of 22:47, 14 December 2024
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	== Algorithm Description ==		== Algorithm Description ==
	There are multiple methods of Derivative Free Optimization that are available. While DFO algorithms do not require one to calculate the derivative of the objective function, they rely on the algorithm to narrow the examined range of the objective function over multiple iterations by selecting points along the objective function and determining if the local minimum exists between these points. Two of these methods include Golden Section Search and Successive Parabolic Interpolation. Both operate in a similar fashion by evaluating the relationship between three points selected along the objective function and making decisions in order to narrow the examination window along the objective function where the points are selected for the next iteration. The main difference is how these three points are selected and how they are evaluated.		There are multiple methods of Derivative-Free Optimization that are available. While DFO algorithms do not require one to calculate the derivative of the objective function, they rely on the algorithm to narrow the examined range of the objective function over multiple iterations by selecting points along the objective function and determining if the local minimum exists between these points. Two of these methods include Golden Section Search and Successive Parabolic Interpolation. Both operate in a similar fashion by evaluating the relationship between three points selected along the objective function and making decisions in order to narrow the examination window along the objective function where the points are selected for the next iteration. The main difference is how these three points are selected and how they are evaluated.

	=== Golden Section Search ===		=== Golden Section Search ===
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	=== Oil and Gas ===		=== Oil and Gas ===
	Derivative Free Optimization has a number of real world applications, including in the oil industry since often the cost functions and constraints require simulations of fluid mechanics where the gradient information cannot be obtained efficiently. This is an important problem that needs to be solved since oil and natural gas make up a significant source of the world’s current energy supply, so maximizing efficiency has global implications to the global economy and environments (IEA, 2021)<ref>IEA (2021), ''World Energy Balances: Overview'', IEA, Paris <nowiki>https://www.iea.org/reports/world-energy-balances-overview</nowiki>, Licence: CC BY 4.0</ref>. Specifically, ~~Derivative Free Optimization~~ can be used, among other techniques, to optimize the production of oil. This is done by creating discretized simulations of very complex oil-water systems, using as many known variables as possible like fluid mechanics and seismic effects (Ciaurri, 2011)<ref name=":1">Ciaurri, D.E., Mukerji, T., Durlofsky, L.J. (2011). Derivative-Free Optimization for Oil Field Operations. In: Yang, XS., Koziel, S. (eds) Computational Optimization and Applications in Engineering and Industry. Studies in Computational Intelligence, vol 359. Springer, Berlin, Heidelberg. <nowiki>https://doi.org/10.1007/978-3-642-20986-4_2</nowiki></ref>. Thus, the minimization problem and its constraints become known and a search of the gradients becomes possible. Two such ~~derivative-free optimization~~ methods are the generalized pattern search and the Hookes-Jeeves Direct Search (Hooke, 1961)<ref name=":2">Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. Journal of the ACM 8(2), 212–229 (1961)</ref>. The generalized pattern search is a search algorithm for “derivative-free unconstrained optimization” divided into “search” and “poll” phases (Audet, 2000)<ref>Audet, Charles & Dennis, J.. (2000). Analysis of Generalized Pattern Searches. SIAM Journal on Optimization. 13. 10.1137/S1052623400378742.</ref>. In the search phase, a finite number of points are evaluated to find a minimum on a mesh. The poll phase is then used to poll the neighboring points on the mesh to find a better minimum before the process is repeated using a more refined mesh or an iterated search. The Hookes-Jeeves Direct Search method is a “Compass-based pattern search method” with an exploratory mode and a pattern mode, similar to that of the generalized pattern search (Hooke, 1961)<ref name=":2" />. In the exploratory mode, the local area is searched to refine the optima estimate while the pattern phase is used to identify patterns in the exploratory phase moves, which results in a better guess in the direction of the optima (Hooke, 1961)<ref name=":2" />. Using these search methods to analyze their analysis results, the optimal control scheme to optimize oil production was determined. This was done by simulating using constraints for the water quantity, water to oil ratio, and resulting oil output while they searched for the optimal water pressure to use that minimizes cost (Ciaurri, 2011)<ref name=":1" />. They varied the configurations as well as utilized variations of the generalized pattern search and the Hookes-Jeeves Direct Search to search their gradients and find the optima. This work resulted in finding a feasible and optimal configuration scheme, expected profits, an increase in oil production, a decrease in water consumption, and thus showed how ~~derivative-free~~ computational ~~optimization~~ techniques can be applied to improve industries.		Derivative-Free Optimization (DFO) has a number of real world applications, including in the oil industry since often the cost functions and constraints require simulations of fluid mechanics where the gradient information cannot be obtained efficiently. This is an important problem that needs to be solved since oil and natural gas make up a significant source of the world’s current energy supply, so maximizing efficiency has global implications to the global economy and environments (IEA, 2021)<ref>IEA (2021), ''World Energy Balances: Overview'', IEA, Paris <nowiki>https://www.iea.org/reports/world-energy-balances-overview</nowiki>, Licence: CC BY 4.0</ref>. Specifically, DFO can be used, among other techniques, to optimize the production of oil. This is done by creating discretized simulations of very complex oil-water systems, using as many known variables as possible like fluid mechanics and seismic effects (Ciaurri, 2011)<ref name=":1">Ciaurri, D.E., Mukerji, T., Durlofsky, L.J. (2011). Derivative-Free Optimization for Oil Field Operations. In: Yang, XS., Koziel, S. (eds) Computational Optimization and Applications in Engineering and Industry. Studies in Computational Intelligence, vol 359. Springer, Berlin, Heidelberg. <nowiki>https://doi.org/10.1007/978-3-642-20986-4_2</nowiki></ref>. Thus, the minimization problem and its constraints become known and a search of the gradients becomes possible. Two such DFO methods are the generalized pattern search and the Hookes-Jeeves Direct Search (Hooke, 1961)<ref name=":2">Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. Journal of the ACM 8(2), 212–229 (1961)</ref>. The generalized pattern search is a search algorithm for “derivative-free unconstrained optimization” divided into “search” and “poll” phases (Audet, 2000)<ref>Audet, Charles & Dennis, J.. (2000). Analysis of Generalized Pattern Searches. SIAM Journal on Optimization. 13. 10.1137/S1052623400378742.</ref>. In the search phase, a finite number of points are evaluated to find a minimum on a mesh. The poll phase is then used to poll the neighboring points on the mesh to find a better minimum before the process is repeated using a more refined mesh or an iterated search. The Hookes-Jeeves Direct Search method is a “Compass-based pattern search method” with an exploratory mode and a pattern mode, similar to that of the generalized pattern search (Hooke, 1961)<ref name=":2" />. In the exploratory mode, the local area is searched to refine the optima estimate while the pattern phase is used to identify patterns in the exploratory phase moves, which results in a better guess in the direction of the optima (Hooke, 1961)<ref name=":2" />. Using these search methods to analyze their analysis results, the optimal control scheme to optimize oil production was determined. This was done by simulating using constraints for the water quantity, water to oil ratio, and resulting oil output while they searched for the optimal water pressure to use that minimizes cost (Ciaurri, 2011)<ref name=":1" />. They varied the configurations as well as utilized variations of the generalized pattern search and the Hookes-Jeeves Direct Search to search their gradients and find the optima. This work resulted in finding a feasible and optimal configuration scheme, expected profits, an increase in oil production, a decrease in water consumption, and thus showed how DFO computational techniques can be applied to improve industries.

	=== Engineering ===		=== Engineering ===
	Another real-world application of ~~Derivative Free Optimization~~ is for the minimization of the number of seismic dampers to use to reduce the damage that tall buildings cause to each other during an earthquake in dense urban environments. Such events can put thousands of lives in danger and caused billions of dollars in damage in 2023 (USGS, 2023)<ref>U.S. Geological Survey (USGS). (n.d.). ''New USGS-FEMA study highlights economic earthquake risk in the United States''. April 18, 2023, from <nowiki>https://www.usgs.gov/news/national-news-release/new-usgs-fema-study-highlights-economic-earthquake-risk-united-states</nowiki></ref>. Shock absorbing dampers, originally developed by NASA for the Apollo programs, are used to in building construction to join adjacent buildings together which reduces the chance that buildings collide while increasing the structural stability overall (NASA, 2015)<ref>NASA Spinoff. (2015). ''Shock absorbers save structures and lives during earthquakes''. Retrieved from <nowiki>https://spinoff.nasa.gov/Spinoff2015/ps_2.html</nowiki></ref>. The upfront cost of seismic dampers is extremely high, so it is desirable to minimize the number of dampers that are required to put the building’s risk of damage in an earthquake within acceptable limits. Thus, ~~Derivative Free Optimization~~, and specifically Model-Based ~~Derivative Free Optimization~~, is the optimal choice for solving this kind of problem since the objective function is a “numerical integration” generated through an earthquake simulator that results in the optimizer not having access to the objective function gradients (Audet, 2017)<ref name=":0">Audet, C., & Hare, W. (2017). Derivative-Free and Blackbox Optimization. Springer.</ref>. Other optimization applications in this field, such as in minimizing peak floor accelerations in multi-story buildings when subjected to earthquakes, may also benefit from ~~Derivative Free Optimization~~ as can resolve the issues of noisy or discontinuous objective functions that pop up in these applications (Ortega, 24)<ref>Mario Ortega, Diego Lopez-Garcia, Gaston Fermandois, Optimal properties of hysteretic dampers to minimize peak floor accelerations in multi-story buildings subjected to earthquakes, Structures, Volume 70, 2024,107862, ISSN 2352-0124, <nowiki>https://doi.org/10.1016/j.istruc.2024.107862.(https://www.sciencedirect.com/science/article/pii/S2352012424020150)</nowiki></ref>.		Another real-world application of DFO is for the minimization of the number of seismic dampers to use to reduce the damage that tall buildings cause to each other during an earthquake in dense urban environments. Such events can put thousands of lives in danger and caused billions of dollars in damage in 2023 (USGS, 2023)<ref>U.S. Geological Survey (USGS). (n.d.). ''New USGS-FEMA study highlights economic earthquake risk in the United States''. April 18, 2023, from <nowiki>https://www.usgs.gov/news/national-news-release/new-usgs-fema-study-highlights-economic-earthquake-risk-united-states</nowiki></ref>. Shock absorbing dampers, originally developed by NASA for the Apollo programs, are used to in building construction to join adjacent buildings together which reduces the chance that buildings collide while increasing the structural stability overall (NASA, 2015)<ref>NASA Spinoff. (2015). ''Shock absorbers save structures and lives during earthquakes''. Retrieved from <nowiki>https://spinoff.nasa.gov/Spinoff2015/ps_2.html</nowiki></ref>. The upfront cost of seismic dampers is extremely high, so it is desirable to minimize the number of dampers that are required to put the building’s risk of damage in an earthquake within acceptable limits. Thus, DFO, and specifically Model-Based DFO, is the optimal choice for solving this kind of problem since the objective function is a “numerical integration” generated through an earthquake simulator that results in the optimizer not having access to the objective function gradients (Audet, 2017)<ref name=":0">Audet, C., & Hare, W. (2017). Derivative-Free and Blackbox Optimization. Springer.</ref>. Other optimization applications in this field, such as in minimizing peak floor accelerations in multi-story buildings when subjected to earthquakes, may also benefit from DFO as can resolve the issues of noisy or discontinuous objective functions that pop up in these applications (Ortega, 24)<ref>Mario Ortega, Diego Lopez-Garcia, Gaston Fermandois, Optimal properties of hysteretic dampers to minimize peak floor accelerations in multi-story buildings subjected to earthquakes, Structures, Volume 70, 2024,107862, ISSN 2352-0124, <nowiki>https://doi.org/10.1016/j.istruc.2024.107862.(https://www.sciencedirect.com/science/article/pii/S2352012424020150)</nowiki></ref>.

	=== Medical ===		=== Medical ===
	Using Model-Based ~~Derivative Free Optimization~~ consistently beats other methods in its efficiency of earthquake retrofitting optimization like Genetic Algorithms and Nelder-Mead since, in this application, the objective function appears to be differentiable which makes it an ideal use case (Audet, 2017)<ref name=":0" />. One application that highly relies on ~~derivative-free optimization~~ is the medical experimentation field. In the medical experimentation field ~~derivative free optimization~~ is often needed due to the complexity, noisy, non-differentiable functions or lack of ability to acquire data to obtain functions. An example of this is in cancer treatment where chemotherapy regimens involve 20+ independent variables with vast amounts of interdependent biological responses. Many biological responses are non-linear. Collection of testing combinations of these chemotherapy, dosage amounts, and interval timing between doses are not feasible; therefore there is not an ability to create a response function in order to use gradient optimization. Nelder and Mead (1964) and Box (1964) methods are example ~~derivative-free optimization~~ algorithms that have been used in this chemotherapy optimization problem. These are examples of Direct Search Methods (DSM) that help narrow the field of combinations of variables. A case study of this can be found in Berenbaum MC. “Direct search methods in the optimisation of cancer chemotherapy regimens (1990).”<ref>Berenbaum MC. Direct search methods in the optimisation of cancer chemotherapy		Using Model-Based DFO consistently beats other methods in its efficiency of earthquake retrofitting optimization like Genetic Algorithms and Nelder-Mead since, in this application, the objective function appears to be differentiable which makes it an ideal use case (Audet, 2017)<ref name=":0" />. One application that highly relies on DFO is the medical experimentation field. In the medical experimentation field DFO is often needed due to the complexity, noisy, non-differentiable functions or lack of ability to acquire data to obtain functions. An example of this is in cancer treatment where chemotherapy regimens involve 20+ independent variables with vast amounts of interdependent biological responses. Many biological responses are non-linear. Collection of testing combinations of these chemotherapy, dosage amounts, and interval timing between doses are not feasible; therefore there is not an ability to create a response function in order to use gradient optimization. Nelder and Mead (1964) and Box (1964) methods are example DFO algorithms that have been used in this chemotherapy optimization problem. These are examples of Direct Search Methods (DSM) that help narrow the field of combinations of variables. A case study of this can be found in Berenbaum MC. “Direct search methods in the optimisation of cancer chemotherapy regimens (1990).”<ref>Berenbaum MC. Direct search methods in the optimisation of cancer chemotherapy

	regimens. Br J Cancer. 1990 Jan;61(1):101-9. doi: 10.1038/bjc.1990.22. Erratum in: Br		regimens. Br J Cancer. 1990 Jan;61(1):101-9. doi: 10.1038/bjc.1990.22. Erratum in: Br
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	== Conclusion ==		== Conclusion ==
	Thanks to its robustness, Derivative-Free Optimization has emerged as a useful method of solving complex optimization problems where traditional methods that require derivatives to be available are not practical. This method of optimization is limited by the high computation cost, sensitivity to noise that leads to imperfect results, and the lack of gradient information that leads to slower convergence. However, advancements in computational power and algorithmic design are helping to mitigate these challenges, making DFO methods more efficient and accessible. There are numerous applications of ~~Derivative-Free Optimization~~ including in the oil, architecture, and medical industries where solving optimization problems has implications on global and local economies, the environment, and people’s health.		Thanks to its robustness, Derivative-Free Optimization (DFO) has emerged as a useful method of solving complex optimization problems where traditional methods that require derivatives to be available are not practical. This method of optimization is limited by the high computation cost, sensitivity to noise that leads to imperfect results, and the lack of gradient information that leads to slower convergence. However, advancements in computational power and algorithmic design are helping to mitigate these challenges, making DFO methods more efficient and accessible. There are numerous applications of DFO including in the oil, architecture, and medical industries where solving optimization problems has implications on global and local economies, the environment, and people’s health.

	These examples demonstrate the usefulness that computational optimization has on humanity and its importance in our continuingly complex modern world. As the world continues to grow, environments continue to change, and better data becomes available, the methods of ~~Derivative-Free Optimization~~ will continue to grow with the world while helping humanity solve many of its most important challenges.		These examples demonstrate the usefulness that computational optimization has on humanity and its importance in our continuingly complex modern world. As the world continues to grow, environments continue to change, and better data becomes available, the methods of DFO will continue to grow with the world while helping humanity solve many of its most important challenges.

	== References ==		== References ==

Fall2024 Wiki Team7: /* Numerical Examples */

2024-12-15T00:00:09Z

Numerical Examples

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	=== Example 1: Golden Section Search ===		=== Example 1: Golden Section Search ===
	[[File:Golden ex onepic.png~~\|border~~\|thumb\|599x599px\|Example of Golden Ratio Search on <math>f(x) = (x-2)^2 +1</math>.]]		[[File:Golden ex onepic.png\|thumb\|599x599px\|Example of Golden Ratio Search on <math>f(x) = (x-2)^2 +1</math>.]]
	Suppose there is a function<math>f(x) = (x-2)^2+1		Suppose there is a function<math>f(x) = (x-2)^2+1
	</math>, and the aim is to find its minimum within the interval <math>[a, b] = [0, 5]</math>.		</math>, and the aim is to find its minimum within the interval <math>[a, b] = [0, 5]</math>.
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	::		::
	::		::
	::[[File:Example- Golden Search - Iter0 .png\|thumb\|'''Figure 7.''' Example 1: Golden Section Search - Iteration 0[[File:Iteration 1 .png\|thumb\|'''Figure 8.''' Example 1: Golden Section Search - Iteration 1 \|none]]\|none]]		::

	=== Example 2: Successive Parabolic Interpolation ===		=== Example 2: Successive Parabolic Interpolation ===

Fall2024 Wiki Team7: /* Example 1: Golden Section Search */

2024-12-14T23:59:12Z

Example 1: Golden Section Search

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	=== Golden Section Search ===		=== Golden Section Search ===
	[[File:Golden single pic.png\|thumb\|~~592x592px~~\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.		[[File:Golden single pic.png\|thumb\|600x600px\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.

	'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.		'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.
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	=== Example 1: Golden Section Search ===		=== Example 1: Golden Section Search ===
			[[File:Golden ex onepic.png\|border\|thumb\|599x599px\|Example of Golden Ratio Search on <math>f(x) = (x-2)^2 +1</math>.]]
	Suppose there is a function<math>f(x) = (x-2)^2+1		Suppose there is a function<math>f(x) = (x-2)^2+1
	</math>, and the aim is to find its minimum within the interval <math>[a, b] = [0, 5]</math>.		</math>, and the aim is to find its minimum within the interval <math>[a, b] = [0, 5]</math>.
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	::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) - 1</math>.		::After a few iterations, the interval narrows around <math>x=2</math>, the minimum of the function, where <math>f(x) - 1</math>.

	::Two images have been generated to describe this numerical example through first iteration described below.[[File:Example- Golden Search - Iter0 .png\|thumb\|'''Figure 7.''' Example 1: Golden Section Search - Iteration 0[[File:Iteration 1 .png\|thumb\|'''Figure 8.''' Example 1: Golden Section Search - Iteration 1 \|none]]\|none]]		::Two images have been generated to describe this numerical example through first iteration described below.
			::
			::
			::[[File:Example- Golden Search - Iter0 .png\|thumb\|'''Figure 7.''' Example 1: Golden Section Search - Iteration 0[[File:Iteration 1 .png\|thumb\|'''Figure 8.''' Example 1: Golden Section Search - Iteration 1 \|none]]\|none]]

	=== Example 2: Successive Parabolic Interpolation ===		=== Example 2: Successive Parabolic Interpolation ===

Fall2024 Wiki Team7: /* Golden Section Search */

2024-12-14T23:50:35Z

Golden Section Search

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	=== Golden Section Search ===		=== Golden Section Search ===
	[[File:Golden single pic.png\|thumb\|~~501x501px~~\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.		[[File:Golden single pic.png\|thumb\|592x592px\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.

	'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.		'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.

Fall2024 Wiki Team7: /* Successive Parabolic Interpolation */

2024-12-14T22:49:35Z

Successive Parabolic Interpolation

← Older revision		Revision as of 18:49, 14 December 2024
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	'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>x_1</math> and <math>x_2</math>.		'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>x_1</math> and <math>x_2</math>.
	[[File:Parabolic2.png~~\|none~~\|thumb\|~~767x767px~~\|Flow Chart describing the Successive Parabolic Interpolation Algorithm]]		[[File:Parabolic2.png\|thumb\|1077x1077px\|Flow Chart describing the Successive Parabolic Interpolation Algorithm\|center]]

	== Numerical Examples ==		== Numerical Examples ==

Fall2024 Wiki Team7: /* Successive Parabolic Interpolation */

2024-12-14T22:47:58Z

Successive Parabolic Interpolation

← Older revision		Revision as of 18:47, 14 December 2024
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	'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>x_1</math> and <math>x_2</math>.		'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>x_1</math> and <math>x_2</math>.
	[[File:~~Parabolic~~.png\|none\|thumb\|767x767px\|~~'''Figure 6.'''~~ Flow Chart describing the Successive Parabolic Interpolation Algorithm]]		[[File:Parabolic2.png\|none\|thumb\|767x767px\|Flow Chart describing the Successive Parabolic Interpolation Algorithm]]

	== Numerical Examples ==		== Numerical Examples ==

Fall2024 Wiki Team7: /* Golden Section Search */

2024-12-14T22:44:16Z

Golden Section Search

← Older revision		Revision as of 18:44, 14 December 2024
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	=== Golden Section Search ===		=== Golden Section Search ===
	[[File:Golden single pic.png\|thumb\|501x501px\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]		[[File:Golden single pic.png\|thumb\|501x501px\|Example of Golden Search Algorithm through three iterations on function <math>y = (x-1)^3-x+2</math>]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.
	~~[[File:DFO - GoldenSearch - Iter0.png\|thumb\|'''Figure 1.''' Golden Section Search - Iteration 0 ]][[File:DFO - GoldenSearch - Iter1.png\|thumb\|~~

	~~'''Figure 2.''' Golden Section Search - Iteration 1~~

	~~]][[File:DFO - GoldenSearch - Iter2.png\|thumb\|'''Figure 3.''' Golden Section Search - Iteration 2 ]]~~

	~~[[File:DFO - GoldenSearch - Iter3.png\|thumb\|'''Figure 4.''' Golden Section Search - Iteration 3~~ ]]Golden Section Search gets its name from the ratio of points selected along the objective function which aligns closely to the Golden Ratio.

	'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.		'''Step 1:''' Identify a function, <math display="inline">f(x)</math>, with the objective to find the minimum value over a given range.
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	'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>a</math> and <math>b</math>.		'''Note:''' This algorithm will not produce the exact value for the function minimum. It will predict that the minimum will be between <math>a</math> and <math>b</math>.

	~~Four images have been generated to describe the Golden Section Search Algorithm through 3 iterations.~~		[[File:Goldenratio.png\|thumb\|822x822px\|'''Figure 5.''' Flow Chart describing the Golden Ratio Algorithm\|center]]

	~~<gallery widths="240" perrow="2" heights="240">~~
	~~File:Index.php?title=File:DFO - GoldenSearch - Iter0.png\|Golden Search - Iteration 0~~
	~~File:Index.php?title=File:DFO - GoldenSearch - Iter1.png\|Golden Search - Iteration 1~~
	~~File:Index.php?title=File:DFO - GoldenSearch - Iter2.png\|Golden Search - Iteration 2~~
	~~File:Index.php?title=File:DFO - GoldenSearch - Iter3.png\|Golden Search - Iteration 3~~
	~~</gallery>~~[[File:Goldenratio.png\|thumb\|822x822px\|'''Figure 5.''' Flow Chart describing the Golden Ratio Algorithm\|~~none~~]]

	=== Successive Parabolic Interpolation ===		=== Successive Parabolic Interpolation ===