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Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 8

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Optimization of thermal processes2007/2008 Overview of the lecture Indirect search (descent) methods Steepest descent (Cauchy) method −The concept of the method −Elementary example −Practical example: optimal design of three-stage compressor −Possible problems with steepest descent method Conjugate directions methods −Davidon-Fletcher-Powell method

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Optimization of thermal processes2007/2008 General types of iterative methods Direct search methods (discussed on the previous lecture) −only the value of the objective function is required (Gauss-Seidel and Powell’s method are good examples) Indirect (descent) methods −methods of this type require not only the value of the objective function but also values of its derivatives −thus, we need and Gradient of the function Unit vector of i axis ?? descent minimum 1D case

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Optimization of thermal processes2007/2008 Indirect search (descent methods) peak minimum Descent directions 2D case. The gradient is: The gradient points in the direction of steepest ascent. So: is the steepest descent direction.

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Indirect search (descent methods) optimum Optimization of thermal processes2007/2008 gradient Gradient is always perpendicular to the isocontours of the objective function. descent direction The step length may be constant (this is the idea of simple gradient method) or may be found with some one- dimensional optimization technique. All descent methods make use of the gradient vector. But in some of them gradient is only one of the components needed to find the search direction

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Optimization of thermal processes2007/2008 Calculation of the gradient vector To find the gradient we need to calculate the partial derivatives of the objective function. But this may lead to certain problems: −when the function if differentiable, but the calculation of the components of the gradient is either impractical or impossible −although the partial derivatives can be calculated, it requires a lot of computational time −when the gradient is not defined at all points In the first (or the second) case we can use e.g. finite difference formula: Vector whose i -th component has a value 1 and all other componets are zero (unit vector of the i -th axis) Small scalar quantity (choose carefully!)

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Optimization of thermal processes2007/2008 Calculation of the gradient vector The scalar quantity (grid step) x i cannot be: −too large – the truncation error of finite difference formula may be large −too small – numerical round-off error may be unacceptable We can use the central finite difference scheme: which is more accurate, but requires an additional function evaluation. In the third case (when gradient is not defined), we usually have to resort to direct search methods.

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Optimization of thermal processes2007/2008 Steepest descent (Cauchy) method 1.Start with arbitrary initial point X 1. Set the iteration numer as i=1. 2.Find the search direction S i as 3.Determine the optimal step length in the direction and set 4.Test the new point for optimality. If is optimum, stop the process. Otherwise, go to step 5. 5.Set the new iteration number i=i+1 and go to step 2. Steepest descent directionOptimal step length

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Optimization of thermal processes2007/2008 Steepest descent method (example) Minimize starting from the point Iteration 1 gradient gradient at the starting point So, the first search direction is:

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Optimization of thermal processes2007/2008 Steepest descent method (example) Now, we must optimize to find the step length. Using the necessary condition: Is it optimum? Let’s calculate the gradient at this point: We didn’t reach the optimum (non-zero slope)

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Optimize for the step length Optimization of thermal processes2007/2008 Steepest descent method (example) We didn’t reach the optimum (non-zero slope) Proceed to the next iteration. Iteration 2 New search direction Necessary conditionNext point Gradient at the next point

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Optimize for the step length Optimization of thermal processes2007/2008 Steepest descent method (example) We didn’t reach the optimum (non-zero slope) Iteration 3 New search direction Necessary conditionNext point Gradient at the next point This process should be continued until the optimum is found:

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Optimization of thermal processes2007/2008 When to stop? (convergence criteria) 1.When the change in function value in two consecutive iterations is small: 2.When the partial derivatives (components of the gradient) of f are small: 3.When the change in the design vector in two consecutive iterations is small: iterations Near the optimum point the gradient should not differ much from zero.

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Optimization of thermal processes2007/2008 Optimum design of three-stage compressor (steepest descent method) CompressorsHeat exchangers Interstage pressure Gas constant Work input Initial pressure Final pressure Objective: find the values of interstage pressure to minimize work input. The heat exchangers reduce the temperature of the gas to T after compression. Let’s use steepest descent method.

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Optimization of thermal processes2007/2008 Optimum design of three-stage compressor (steepest descent method) It’s convenient to use the following the objective function: The gradient and its components are: Let’s choose the starting point (initial guess):

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Optimization of thermal processes2007/2008 Optimum design of three-stage compressor (steepest descent method) Using this values we calculate the first search direction: The next point we find from the formula: Now we must find optimum step length 1 along direction S 1 to minimize the expression: Derivative calculated at point X 1

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Optimization of thermal processes2007/2008 Optimum design of three-stage compressor (steepest descent method) This means that we are looking for the minimum of the single-variable function: Using the necessary condition we find The next point is:

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Optimization of thermal processes2007/2008 Optimum design of three-stage compressor (steepest descent method) The value of the objective function at this point is: The first iteration has ended. Now we can start the second iteration, calculating the new search direction: Next point New step length And minimize the expression: and so on... Solution:

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Optimization of thermal processes2007/2008 Steepest descent method (possible difficulty) Contours of the objective function Long narrow valey optimum If the minimum is in a long narrow valey, steepest descent method may converge rather slowly The problem stems from the fact, that gradient is only local property of the objective function More clever choice of search directions is possible. It is based on the concept of conjugate directions.

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Optimization of thermal processes2007/2008 Conjugate directions A set of n vectors (directions) is said to be conjugate (more accurately A -conjugate) if: for all Theorem If a quadratic function is minimized sequentially once along each direction of a set of n mutually conjugate directions, the minimum of the function Q will be found at or before the n -th step irrespective of the starting point. Remark: Powell’s method is an example of conjugate directions method.

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Optimization of thermal processes2007/2008 Conjugate directions (example) For instance, suppose we have: Then and Orthogonality condition So, conjugate direction are in this case simply perpendicular. Convergence after two iterations. Contours of Q

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Optimization of thermal processes2007/2008 Conjugate directions (quadratic convergence) Thus, for quadratic functions conjugate directions method converges after n steps (at most) where n is the number of design variables That is really fast but what about other functions? Fortunately, a general nonlinear function can be approximated reasonably well by a quadratic function near its minimum (see the Taylor expansion) Conjugate directions method is expected to speed up the convergence for even general nonlinear objective functions

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Optimization of thermal processes2007/2008 Davidon-Fletcher-Powell method 1.Start with an initial point X 1 and a n n positive definite symmetric matrix [B 1 ] to approximate the inverse of the Hessian matrix of f. Usually, [B 1 ] is taken as the identity matrix [I]. Set the iteration numer as i=1. 2.Compute the gradient of the function,, at point X i, and set 3.Find the optimal step length in the direction S i and set 4.Test the new point for optimality. If is optimal, terminate the iterative process. Otherwise, go to step 5.

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Optimization of thermal processes2007/2008 Davidon-Fletcher-Powell method 5.Update the matrix [B 1 ] as: 6.Set the new iteration number as i=i+1 and go to step 2. where

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Optimization of thermal processes2007/2008 Thank you for your attention

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