Mathematical Optimization
Mathematical optimization, also known as mathematical programming, is a fundamental area in mathematics, computer science, and operations research that focuses on finding the best solution from a set of available alternatives, based on a given criterion. This involves defining an objective function that needs to be maximized or minimized and possibly subject to a set of constraints.
Optimization problems can be linear or nonlinear, continuous or discrete, and can vary in complexity. The solutions to these problems are critical in various decision-making processes and have applications across numerous domains, including engineering design, economics, logistics, network design, and artificial intelligence, particularly in machine learning models for parameter tuning and model selection.
In machine learning, optimization algorithms are used to minimize a cost function, which is a measure of how well the model performs with respect to its training data and the expected outcomes. For instance, in a linear regression model, the objective might be to minimize the sum of the squared differences between the observed and predicted values, known as the least squares method.
Another example is in logistics and supply chain management, where optimization techniques are used to minimize transportation and warehousing costs while satisfying customer demand and adhering to constraints like vehicle capacity and delivery time windows. These examples illustrate how mathematical optimization underpins the process of making decisions that require the best possible outcome under given circumstances.