I think it was the distinction between upper bound and least upper bound that I was looking for.andrewkirk said:It is equally true, but not as useful. 2 is merely an upper bound, whereas 1 is a least upper bound. In fact it is a maximum, that is achieved when ##x=y##. Arithmetic and Geometric Means are identical when all data are the same.
Optimization problems are mathematical problems that involve finding the best possible solution for a given objective, while satisfying a set of constraints. These problems are commonly used in various fields such as engineering, economics, and computer science to optimize the performance of systems or processes.
There are several types of optimization problems, including linear programming, nonlinear programming, integer programming, dynamic programming, and stochastic programming. Each type has its own set of techniques and algorithms for solving it, depending on the specific problem at hand.
Optimization problems are typically solved using mathematical techniques such as gradient descent, simplex method, branch and bound, or genetic algorithms. These techniques involve finding the optimal solution by iteratively adjusting the variables in the problem until the best solution is reached.
Optimization problems have a wide range of applications in real-world scenarios. They are used to optimize supply chains, production processes, transportation routes, resource allocation, financial portfolios, and many other systems. They are also used in machine learning and artificial intelligence to improve the performance of algorithms.
Yes, there are several challenges in solving optimization problems. These include dealing with complex and high-dimensional problems, finding efficient and accurate algorithms, and handling uncertainty and variability in real-world scenarios. Additionally, it can be difficult to determine the optimal solution when multiple solutions have similar performance.