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Chapter 09.04 Multidimensional Gradient Method After reading this chapter, you should be able to: 1. Understand how multi-dimensional gradient methods are different from direct search methods 2. Understand the use of first and second derivatives in multi-dimensions 3. Understand how the steepest ascent/descent method works 4. Solve multi-dimensional optimization problems using the steepest ascent/descent method How do gradient methods differ from direct search methods in multi-dimensional optimization? The difference between gradient and direct search methods in multi-dimensional optimization is similar to the difference between these approaches in one-dimensional optimization. Direct search methods are useful when the derivative of the optimization function is not available to effectively guide the search for the optimum. While direct search methods explore the parameter space in a systematic manner, they are not computationally very efficient. On the other hand, gradient methods use information from the derivatives of the optimization function to more effectively guide the search and find optimum solutions much quicker. Newton’s Method When Newton’s Method ( ) was introduced as a one-dimensional optimization method, we discussed the use of the first and second derivative of the optimization function as sources of information to determine if we have reached an optimal point (where the value of the first derivative is zero). If that optimal point is a maximum, the second derivative is negative. If the point is a minimum, the second derivative is positive. 09.04.1

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09.04.2 Chapter 09.04 What are Gradients and Hessians and how are Gradients and Hessians used in multi- dimensional optimization? Gradients and Hessians describe the first and second derivatives of functions, respectively in multiple dimensions and are used frequently in various gradient methods for multi- dimensional optimization. We describe these two concepts of gradients and Hessians next. Gradient: The gradient is a vector operator denoted by (referred to as “del”) which, when applied to a function f , represents its directional derivatives. For example, consider a two dimensional function y x f , which shows elevation above sea level at points x and y . If
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