Now that we have a basic quiver of optimization methods we need something to use them on. So far we have been limited to functions of varying complexity as well as difficulty. At this point we need to formally go over what is optimization and why do we optimize.
Optimization takes many forms but most often when we talk about optimization we are talking about a cost (or objective) function (wether this actually refers to money is another story). We set up our problems in such a way that all our information is contained in our cost function. From a more general standpoint, we are trying to either maximize our minimize our function while subject to various constraints or rules. We will use the classic diet problem as an example. say we are trying to minimize the amount of money we spend on food while ensuring that we hit all our daily nutritional values. We have a lots foods to choose from all with there own nutritional value and cost. From purely a mathematical standpoint there is a combination of food that will reach all of our daily values as well as cost us the least amount of money. This is our optimal solution. However, sometimes our problems are not always as clear cut as the diet problem. Our question or problem could have different sets of rules. Certain variables could be dependent on each other. Also we could have problems that rely on a choice of yes or no. All these have to fit into our optimization problem. However there are general consistencies amongst most of our optimization problems. Each problem is trying to either minimize or maximize a function. Each problem is also subject to certain parameters which must be followed. Keeping these two things in mind, you can turn almost any real world problem into a rigorous mathematical problem. But why do we even want to optimize? Well one reason is is efficiency. The first optimization method was created during the world wars. Our strategists needed to know the most efficient way to supply our troops. This became a big issue as if the troops are not adequately supplied they had less of a chance of winning. The first optimization methods sought to remedy this. These methods ensured that all troops where adequately supplied with the least amount of waste .After the wars more complex and interesting optimization methods began to develop. Optimization expanded to even more real life applications. Even though the subject matter changes they are all rooted in the desire to carry out objectives efficiently with the guaranteed least amount of waste. Another reason which is closely tied to efficiency is marginal improvement. When we find the optimal conditions for our objective we know safely that changing of any of our parameters will give us no marginal improvement. This lets us safely assess risk and ensure we are making the right decision and do both with confidence. Optimization is an incredibly interesting field and one that I am incredibly passionate about. The applications range from scheduling to determining which subway line to take to get to your destination. Optimization can be used in most of our daily lives. Next time you commute to work ask yourself if you are take the most optimal route. Our you minimizing time or distance? When you sit down at restaurant look around at all the tables. Are they arranged in a way that optimized maximum occupancy while allowing for freedom of movement for the wait staff? The more you look around the more you will see all the possibilities for optimization and hopefully foster a passion of your own. But before you try to optimize the next burger joint you walk into we need to know the most important thing about optimization problems; how to set them up.-Marcello