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Global optimization is a mathematical and computational approach used to find the best possible solution to an optimization problem over a given domain, often in the presence of multiple local optima (suboptimal solutions). In other words, it seeks to find the global minimum or maximum of an objective function, which represents a specific criterion to be optimized, such as minimizing cost, maximizing profit, or optimizing some other desired outcome. The key challenges in global optimization arise from the fact that the objective function can be highly nonlinear, non-convex, and may have multiple peaks or valleys. Local optimization methods, which focus on finding the best solution in the vicinity of an initial guess, can easily get stuck in a local minimum or maximum and may not discover the global optimum. Global optimization methods aim to overcome these challenges by exploring the entire search space systematically or using heuristics to intelligently navigate the domain.

Global Optimization has the following characteristics:

• Global optimization is a broader concept than direct optimization that pertains to finding the global optimum (best possible solution) within a given solution space.
• Global optimization is often used in the non-convex and complex optimization problems where there may be multiple local optima (good solutions) but only one global optimum (the best solution).
• Global optimization methods aim to find the global optimum, which may require exploring various parts of the solution space, including regions far from the initial guess.

While direct optimization methods seek to find solutions in a straightforward manner, global optimization methods are concerned with finding the best solution across the entire solution space, which can be a more challenging task, especially in non-linear and non-convex optimization problems.

In global optimization, we set:

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In this sense, this is a n easy way to compute the derivative of J(θ) with respect to θ.

The Normal Equation is a mathematical formula used in linear regression to find the coefficients (parameters) of a linear model that best fits a given set of data points. Linear regression is a statistical method used to model the relationship between a dependent variable (the target or output) and one or more independent variables (predictors or features) by fitting a linear equation to the observed data.

By solving the Normal Equation, we can obtain the values of the coefficients θ that minimize the sum of squared differences between the predicted values of the dependent variable and the actual observed values. These coefficients define the best-fitting linear model for the given data. While the Normal Equation provides a closed-form solution for linear regression, there are also iterative optimization methods like gradient descent that can be used to find the coefficients, especially when dealing with more complex models or large datasets. Nonetheless, the Normal Equation is a valuable tool for understanding the fundamental principles of linear regression and for solving simple linear regression problems analytically.

When you use the Normal Equation to solve for the coefficients (θ) in linear regression, you are essentially finding the values of θ that correspond to the global minimum of the cost function in a single step. In linear regression, the goal is to find the values of θ that minimize a cost function, often represented as J(θ). This cost function measures the error or the difference between the predicted values (obtained using the linear model with θ) and the actual observed values in your dataset.

To find the values of θ that minimize this cost function, you can use the Normal Equation, which provides an analytical solution. When you solve the Normal Equation, you find the exact values of θ that minimize J(θ) by setting the gradient of J(θ) with respect to θ equal to zero.

The key point is that this solution is obtained directly, without the need for iterative optimization algorithms like gradient descent. Gradient descent, for example, iteratively adjusts the parameters θ to minimize the cost function, which may take many steps to converge to the global minimum. In contrast, the Normal Equation provides a closed-form solution that directly computes the optimal θ values in a single step by finding the point where the gradient is zero.

However, note that the Normal Equation has some limitations:

1. It may not be suitable for very large datasets because of the matrix inversion operation, which can be computationally expensive.
2. It requires that the design matrix (X^T * X) is invertible. In cases where it's not invertible (e.g., due to multicollinearity), you may need to use regularization techniques.

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