Mathematical Optimization, also known as Operations Research, is a scientific method that uses mathematics, statistics, and algorithms to determine the most efficient ways to plan and achieve objectives.
To solve optimization problems, a model is formulated and then optimized based on that model. The results can be applied to real-world decision-making.
The difficult part is modeling real-world problems.
Continuous optimization problems can be classified as non-convex optimization problems, which are problems without clear and easy-to-understand goals.
Research in this field aims to find methods that can solve problems in polynomial time and also explores the use of machine learning techniques.
Some simple optimization problems can be solved using basic high school mathematics, such as finding the maximum or minimum values of a graph. However, there are more advanced methods like simplex method and interior point method that can solve these problems without explicitly drawing the graph.
The shortest path problem, for example, can be represented using a set of 0s and 1s, which makes it an integer programming problem rather than a continuous optimization problem. However, by relaxing the conditions, it can be reduced to a linear programming problem.
Machine learning techniques, such as regression, also fall under the category of mathematical optimization. The gradient descent method, for example, is commonly used to minimize the loss function.
Recently, with the popularity of machine learning, there has been a resurgence of interest in first-order optimization methods, which involve only first-order derivatives.
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