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Finding the Optimum


The selecting of best element among the available alternatives is refereed as the mathematical optimization. If we consider the most simplest case of finding the optimum problem arouse out of function which contains its minimization or maximization of the symmetrical value chosen from the allowed Set after computing the value of the function. A large sector of applied mathematics is used in the formulation of the techniques this is just the generalization of optimum theory. Now let us understand this with the help of the example:

A function f: S→R from a set ‘S’ to the real Numbers. This type of formulation is the optimum problems. Many times the theoretical problem can be molded in this framework so it can be easy to understand. By the use of this technique energy minimization takes place while working in the field of physics and computer vision.

The set ‘S’ is a subset of the Euclidean space. The function ‘f’ is known as the objective function, whereas in this cost function energy minimization and utility function is energy maximization. The feasible solution is that which helps in sorting the work effectively and efficiently. Problem of optimization are also sometimes expressed with the special notation. There is also multi objective optimization this means when more than one objective is added then it creates the complexities in the organization. For instance to meet the optimization of the structural design it must require one stiff design and one light design and also can be more design some with the stiffness and some are of light part, to resolve the part of these two objective the trade-off have its existence. In other words the description of this part rely on the multi objective optimization signals that what is missing and what are its requirement so that we can easily find the optimum.

Finding the Best Tree

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Finding the best Tree is an approach that is followed in problems of optimization in Math. Optimization means to choose value of any variable such that it being maximum or minimum will have the most suitable solution for the problem. General Calculus methods are useful for this purpose. These problems may become difficult at times when an optimized solution is hard to be found. Optimization procedure can be of several types like:

1. Optimizing the volume and amount of certain 3 – Dimensional figures and other objects in order to lower fabrication budgets. These problems can be solved in 2 – Dimensions also.

To understand such problems let us consider an example: Find out the unknowns i.e. the variables in your equation and Set their values accordingly to write an equation. Optimization of value of one variable is needed to be done in order to get the desired value of another variable.

For Example we have an equation: 2 B = 8 A2 + 12 A,

Separate one of the variables (A and B). We generally consider the variable that lies on y- axis. So, we get:

B = 4 A2 + 6 A,

Next step is to find the derivative of equation we got in previous step as follows:

D(B) /D(A) = 8 A + 6,

Equating the derivative with 0 we get:

0 = 8 A + 6,

On solving above equality we get A = - (3 /4),

Substitute the value of known quantity i.e. A in the original equation to get the corresponding optimized value of B. Thus we are able to find the solution for all the variables in the equation as A = - (3 /4) and B = - (9 /4).