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Mathematical optimization is the process of finding the best solution to a problem, often under some constraints. It involves identifying the variables and objective function that describe the problem, specifying the constraints that the solution must satisfy, and then finding the values of the variables that optimize the objective function subject to the constraints.
Optimization problems can be classified into two main types: linear and nonlinear. In linear optimization, the objective function and the constraints are all linear functions of the variables, and the solution can be found using techniques such as linear programming. In nonlinear optimization, the objective function and/or the constraints are nonlinear, and more advanced techniques such as gradient descent or Newton's method may be required to find the optimal solution.
Optimization is used in a wide range of fields, including engineering, economics, finance, and operations research. Some common applications include portfolio optimization, scheduling and routing problems, and machine learning, among others.