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This algorithm generates conjugate directions for a quadratic function using Modified Powell’s Method. It starts at two arbitrary points and a specified direction, then iterates to find Q-conjugate directions for optimization. The process involves generating points, finding the index for the direction with the largest decrease, calculating the pattern direction, updating the search direction set, and repeating until convergence is achieved. This is the first cycle of Modified Powell’s Method for optimization problems.
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Generating of conjugate directions For a quadratic function and two arbitrary points , and a specified direction ,let denotes the minimum point of on the line starting at along and the minimum point on the line starting at along , then the directions and ( - ) are Q-conjugate.
Algorithm of the Modified Powell’s Method • 1. Given a starting point ,for each coordinate direction, generate the points at which the minimum of the objective function occurs, denotes them by , …, . • 2. Find the index corresponding to the direction of the above univariate search which yields the largest decrease from to . • 3. Calculate the pattern direction and determine the point at which the minimization of f along the pattern direction occurs. • 4. Check the following criterion to see if the search direction set needs to be undated. • 5. k->k+1, repeat step 2 to step 4, until the convergence with a specified accuracy is reached.
