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Optimization Theory - A Modern face of Applied Mathematics

1. Optimization Theory: A Historical Rundown Optimization is essentially the art, science and mathematics of choosing the best among a given set of finite or infinite alternatives. Though currently optimization is an interdisciplinary subject cutting through the boundaries of mathematics, economics, engineering, natural sciences, and many other fields of human endeavour it had its root in antiquity. In fact the oldest story that goes with the history of optimization concerns an ancient princess named Dido. She was fleeing from the prosecution of her brother and a piece of land on the Mediterranean coast caught her fancy. She made a deal with the local leader. She requested him to cut a bull's hide into thin strips and and tie them up and enclose as much land as one can with it. In modern day language the problem mathematically is as follows Among all closed curves of a given length find the one that encloses maximum area. This is called the Isoperimetric problem. This problem is now mentioned in a regular fashion in any course in the Calculus of Variations. However, most problems of antiquity came from geometry and since there were no general methods to solve such problems, each one of them was solved by very different approaches. An interesting collection of such problems can be found in the book titled Stories about Maxima and Minima by V. M. Tikhomirov. Beginning of the modern methods of optimization can be traced back to the growth of the Calculus of Variations. In 1696 Johann Bernoulli proposed the famous Brachistochrone problem (Figure 1) which says that find the curve along which a particle moving from one point to another in a vertical plane does so in minimum time. This problem precisely gave rise to the Calculus Figure 1: Brachistochrone curve.