A New Solution for Maxterm Problem in Trigonometric Functions by Simulated Annealing Algorithm

2.1. Hill Climbing

Hill climbing is a method through which one can get the best answer to a problem by finding an answer which is optimal enough.

The method is used to find several equivalent answers and select the best answer (Vaughan, 2000). In this method the following hypotheses are required.

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  Finite Sum of Values: Finite sum of values is a set which describes several values to be used in the function which are used to find target function values.

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  Target Function: It is a function which allocates a clear value to every value in the s. the purpose is to find the best answer provided through the function. In this method, first one of the members of s set is selected randomly and calculates the function value based on the selected members then other members from among the set members are selected for later steps which are adjacent to former member. In other words Function values possess more optimal answer compared to former member.

The optimal answer can be used to find minimum or maximum value of the function. Depending on the optimization condition and whether maximum or minimum is required (Vaughan, 2000; Russell & Norvig, 1995).The Algorithm semi-code is as following (Vaughan, 2000) in Box 1.

Generate a solution (s’)
Best = S'
Loop
S = Best
S' = Neighbors (S)
Best = SelectBest (S')
IF there is no changes in Best solution THEN
Jump to new state in state space
Until stop criterion satisfied

The shortcoming of the method is that if the function has a large local minimum and maximum it will get stuck in a local Maxterm and will not be able to get out of it therefore the method cannot always provide the best answer because of a fore mentioned problem [1, 3].

2.2. Gradient Ascent

This is much like the hill-climbing method which is run in a sample contiguous space. Changes will take place like Formula (1) (Russell & Norvig, 1995; Dabney & Barto, 2003):

Figure1 illustrates a sample change trough the aforementioned method.

Figure 1.

Sample of gradient accent method

Figure 2 shows some steps of the mentioned method.

Figure 2.

Sample of gradient accent method

This method also like hill-climbing has the probability of getting stuck in a local Maxterm and is probable to get stuck in a local maximum or minimum be sides this methods has shortcomings like slow convergence and to remove the problem Gradient Ascent Method is inevitable (Dabney & Barto, 2003).

In order to solve the local optimization problem and to find minimum point of a function Simulated Annealing is used which will be briefly discussed.