Solving one-dimensional unconstrained global optimization problem using parameter free filled function method

It is generally known that almost all filled function methods for one-dimensional unconstrained global optimization problems have computational weaknesses. This paper introduces a relatively new parameter free filled function, which creates a non-ascending bridge from any local isolated minimizer to other first local isolated minimizer with lower or equal function value. The algorithm’s unprecedented function can be used to determine all extreme and inflection points between the two considered consecutive local isolated minimizers. The proposed method never fails to carry out its job. The results of the several testing examples have shown the capability and efficiency of this algorithm while at the same time, proving that the computational weaknesses of the filled function methods can be overcomed.

Suppose that * ( = 0, … , ) are isolated minimizers of Four different parametric filled functions at * are defined by are proposed in [4,26,29,22] respectively where parameter is defined by Unfortunately, the existing filled function methods [23,[26][27][28][29] can not solve the global optimization problems since: a. cannot assure the existence of a better local minimizer in a lower basin [29,30]; b. require the assumption that has only a finite number of local minimizer which have different function values, i.e., ( 1 * ) ≠ ( 2 * ) if 1 * ≠ 2 * ; c. difficult to adjust an appropriate parameter to satisfy the conditions of filled function; d. iteratively updated the parameter; e. can only obtain one global optimizer, and f. contain exponential or logarithmic expressions in their forms which make a large amount of computation. For filled function [24], its two parameters, one of which relies on the diameter of a bounded closed domain which contains all global minimizers, and the other on Lipschitz constant of respectively. The parameter free filled function (PFFF) was initially introduced in [30][31][32][33][34][35]. A PFFF proposed by Ma et al. [34] is where this method also has weaknesses as the others.
has only a finite number of extreme and inflection points in , and ( ) ≠ 0 for ≥ 3 where is an inflection point of . A3. (1) , (2) and ( A7. There exists only one between two consecutive minimizer and maximizer of ( ).
The reason why we need to solve one-dimensional multimodal function is described in many references cited in [20]. The needed is appeared in scientific and engineering applications especially in electrical engineering optimization problem. One of the important issues in global optimization is "the region of attraction" where its detail explanation can be seen in [3]. This paper is organized as follows. Section 2 describes the IYRH's function. Section 3 describes how to find all extreme and inflection points using the IYRH's function. Section 4 discusses the relationship between and IYRH's function. In Section 5, the idea of curvature is described. Section 6 contains the convergence theorem. The numerical results of IYRH's algorithm will be presented in section 7. Comparison and discussion will be given in section 8. Section 9 contains the conclusion and the brief explanation on how this one-dimensional case can be extended to n-dimensional case.

Computation of the Inflection Points
Based on the discussion in Section 3, we have proved the following theorems: Theorem 4: If the hypotheses of Theorem 1 are valid, then the solution of (2) ( , * ) = 0 becomes the critical point of ( ). Theorem 5: If the hypotheses of Theorem 1 are valid, then the critical point of (2) ( , * ) becomes the inflection points of ( ).

Numerical Results
The test examples are listed in Tables 1-3. In Table 1 where , ( ), , * and * denote the number of function, the expression of the objective function, the domain, global minimum value and global minimizer respectively. The numerical results will be presented to compare the capability of the IYRH's method with two-parameter filled function methods [4,26,29], one-parameter filled function methods [6,22,28], and the PFFF method [34]. We also present the results for observing the sensitivity of IYRH's method due to different initial points. Therefore, the presentation is arranging into four categories. For first category, the results of Table 4 shows that IYRH's algorithm can solve the global optimization problems listed in Table 1. In Table 4 In inner iteration, there are several cases that , ( ) equals to + and +1 * as shown in Table 4 for = 2, 3 of example 3 ( = 3), = 3 of example 8 ( = 8), = 1 of example 12 ( = 12) and = 0 of example 17 ( = 17). For second category, Tables 5-8 compare IYRH's algorithm with New algorithm [42], the direct method [42] and Lagrange interpolation [43], using test functions in Table 1 [44] and 100 one-dimensional randomized test functions [45]. Table 5 shows the relative errors [42] of global minimum values and global minimizers of functions as shown in Table 1 obtained by IYRH's algorithm is better than Lagrange interpolant on 81 Chebyshev nodes [43]. Table 6 shows that a "fortune effect" does not happened to IYRH's algorithm when it is applied to the example given in Table 3 for = 67 and * is chosen randomly and differently where its graph is shown in Figure 3.    For third category, Table 7 Table 2. For the last category, the results presented in Table 8 is used to observe the sensitivity of IYRH's algorithm due to three initial points using example 3 from Table 2. It is clear that IYRH's function can be used to solve the global optimization problems from any initial point.   The meaning of the abbreviations used in Table 8

Comparison and Discussion
The graphical comparison between IYRH's filled function method with other best current filled function methods (1)-(4) included tunneling and bridging methods, will be presented.

Comparison with the Tunneling Method [8]
The weakness of tunneling method [8] ( , Γ) = ( ( ) − ( 1 * ))/[( − 1 * ) Γ ( − 1 * )] appeared when Newton's method is used since the non-convexity problem. Fortunately, IYRH's filled function can be utilized (Theorem 6 and Theorem 7) using the radius of curvature applied to Newton's method to find the root of non-convex problems. For example 7 [8], the tunneling method can only obtain the global minimizer, whereas IYRH's algorithm can obtain the entire extreme and inflection points in considered domain.

Comparison with the Two-Parameter Filled Function
The graph of ( ) = cos(3 /5) cos(2 ) + sin( ) (0.5 ≤ ≤ 12) is given in Figure 4 (a) and the graph of ( , 1 * ) ( ∈ [1.34096,2.73151]) of is given in Figure 4  and . Therefore becomes inefficient, (b) discontinuous at a point ′ ∈ (2,3). This condition makes the minimizing difficult, (c) according to (3), they actually have four parameters to be adjusted (see [29]) and (d) (see (5)) contain a parameter > 0, which is also difficult to be adjusted, and it is clear that is not a one-parameter filled function.

Comparison to the Parameter Free Filled Function
Ma et al. [34] suggest a PFFF (6) at * of . When applied to Example 9 in Table 1 at * = 5.36225, it yields a graph as in Figure 6. It is clear that, is discontinuous at + , and almost flat for such that ( ) < ( * ) whereas is continuous as shown in Figure 7.  Table 1 Figure 7. Example 9 in Table 1 8.

Conclusion
This article introduces a new IYRH's method which absolutely different from other filled function methods, in finding all extreme and inflection points of : ⊆ ⟶ According to the results listed in Tables 4-8, this method never fail compute all those points. Thus, this method is an efficient and reliable method for solving the global optimization problems numerically and analytically. Therefore, the IYRH's method is far more advanced and superior than other most of the filled function methods published in the literature.