A global optimization using interval arithmetic

In order to establish an algorithm for bounding the global minimizers of a twice continuously differentiable function 1 R R : f n → in a given box ( a compact interval in Rn ) using interval arithmetic, which is superior in the sense of less time needed compared to other method ([2]), in this paper we will attempt to describe as much as possible the ideas which presents and fulfill the explicitly mentioned algorithm. | Interval mathematics | Symmetric operator | Lagrange multiplier | Gauss algorithm |


Introduction
Hansen [2] has described an algorithm, H, for bounding the global minimizers of a twice continuously differentiable function 1 R R : f n → in a box that is a parallelepiped with sides parallel to the coordinate axes.
In the algorithm H, an interval form of Newton's method for bounding critical points of f, together with monotonicity and convexity tests and a continually-updated upper bound on the least value of f in the box are used to delete sub-boxes which cannot contain global minimizers of f.
Robinson [14] has described a technique for bounding a Kuhn-Tucker (KT) point centred about an estimate of Shearer and Wolfe [15] have described some computable existence and uniqueness tests for solutions of systems of nonlinear algebraic equations, and have also described an improved form of the Krawczyk-Moore algorithm, KMSW [16], and an improved form of the Alefeld-Platzoder algorithm, MAP [17].
Ismail [6] has described how some of the ideas of Hansen [2], and Shearer [15,16] have been used in an algorithm, HM, for computing and bounding the global minimizer(s) of a twice continuously differentiable The purpose of this paper is to describe how some of the ideas of Hansen [2], Ismail [4,5,6,7,8,9], Robinson [14], and Shearer and Wolfe [15][16 [17] have been used in an algorithm named MI, for bounding the global

Notation
An interval number denoted by x , is defined by where I x and S x are called infimum and supremum, respectively.An nx1 interval vector ( a box) are given sets of nx1 real vectors and nxn real matrices respectively, then , and

The Global Optimization Problem
and where , and

Given a box
containing the KT points for problem P, the algorithm MI deletes sub-boxes of z ˆ which do not contain KT points corresponding to global minimizers of f in x ˆ. , we obtain (for detail see [5]) The formulae (3.8) -(3.13) are used in MI to determine u ˆ.

Constructing Sub-boxes Which Might Contain KT Points
If each side of a box is divided into two parts, then this could give rise to n 2 sub-boxes.Therefore, in order to prevent the generation of too many sub-boxes, Hansen [2] has suggested that only one side of the box with largest width is divided into two parts.
In [5], since we can avoid the disadvantage mentioned by Hansen [2], Ismail has shown how to derive a method for obtaining and computing n 2 sub-boxes As explained in [5], let

Ideas due to Hansen
The monotonicity and convexity tests which are described in Section 9 and 5 of [2] , and HM [6] respectively, are used in MI.
The interval form of Hansen's method which is used in H [2], HM [6], and also in MI has the form where using, in particular, the so-called quadratic method which is described in Section 7 of [2].

The Symmetric Operator Test
, and . The procedure (7.11) may be used to update f by iterating until, for some , in which case f has effectively ceased to change, or until for some z probably corresponds to a maximizer or to a saddle point of f.If x satisfies (7.12) and at least quadratically.Thus MAP may be used in MI with to compute arbitrarily sharp bounds on KT points.

Bisection and Selection Rules
Suppose that the Symmetric Operator Test cannot guarantee the existence of a KT point .In this case bisection might not be beneficial because it could occur an indefinite number of times and produce an excessively large number of boxes.
In practice, however, it is found to be desirable to check whether , and to avoid the excessively large number of bisections which could occur by bisecting along the coordinate direction j only if then x is a sufficiently sharp bound on * x .

Numerical Examples
The algorithm MI, which incorporates the ideas which are mentioned in the preceding sections, and the algorithm The function f has two global minimizers in the interior of x ˆ.
The function f has one global minimizer

Discussion and Conclusion
In the Triplex ([1] [11]) implementations of both H and MI it is possible to ignore the boundary of x ˆ.This is useful if it is known that ) x înt( x * ∈ because it usually leads to less computational labour if it is known that the boundary of x ˆ can be deleted.
result from Nickel[13] to obtain a computable interval-arithmetic test for the existence of a unique KT point box.Let the nonlinear programming problem P be defined by

.
The global optimization problem which is solved by MI is equivalent to determining

1 =
be the n-digit binary integer corresponding to the decimal integer i .For n of x ˆ and strict complementary slackness does not hold at * x .
H, have been implemented.Convergence is considered to have occurred when each global minimizer

Table 13 .
1 contains the CPU times, in seconds, corresponding to examples 12.1 and 12.2 when it is known that

Table 13 .
1 : Computational time in seconds

Table 13 .
2 contains the CPU times, in seconds, corresponding to Example 12.3 for various values of n.For Example 12.3 for all values of n, both H and MI are able to bound the global minimizer even when it is Computational experience with the Triplex implementations of H and MI indicates that, with few exceptions, MI requires fewer evaluations of f , ' f , and ' ' f than does H.This might account for the increasing superiority of MI over H as n increases in Example 12.3, as shown in Table 13.2.