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ON THE SUBPROBLEM OF THE CUTTING ANGLE METHOD OF
GLOBAL OPTIMIZATION Burak ORDIN Ege
University Faculty of Science Dept.of
Mathematics, 35100, Bornova, Izmir, Turkey.
bordin_at_sci.ege.edu.tr
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1. Introduction
Cutting Angle Method (CAM) was presented in
a series of papers 1-2 and the book 10, as
an application of the theory of abstract
convexity.
CAM has been developed for solving a broad class
of Global Optimization problems.
This is an iterative method and in each iteration
of the CAM a Subproblem has to be solved, which
is in turn, generally, a global optimization
problem 1, 2, 10.
The solving of the subproblem is the crucial step
in the CAM. Developed algorithms for solving of
the subproblem calculate the set of all local
minima and then find the global minimum among
them.
Unfortunately, the number of local minima of the
objective function increases sharply as the
number of support vectors increases and solving
the subproblem becomes time- consuming.
Therefore the development of good algorithms for
solving this problem is very important 2.
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In this work,
The Subproblem is transformed to a certain (0-1)
problem which is called as Dominating Subset
with Minimal Weight (DSMW) problem.
The mathematical model and the economical
interpretation of the DSMW problem are given.
It is offered a new algorithm which has a ratio
bound in polynomial time for solving the DSMW
problem and also the algorithm is used to solve
the subproblem.
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1. Formulation of the Subproblem
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Introduce the function h(x)
, where I( )i gt0.
min
)
(
k
I
i
l
Î
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The Subproblem
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3. Transformation of the Subproblem To an
Equivalent Problem
Example. Matrix L(24)
Matrix U(22)
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So the Subproblem (1)-(2) is transformed into the
following Boolean (0 1) programming problem
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In the paper 4, the following theorem was
proved Theorem 1. The Subproblem (1)-(2) and
problem (3)-(9) are equivalent.
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Clearly, the problem is generalized of the
Assignment problem 8. Although its known that
the assignment problem can be solved by Hungarian
method at a complexity of O(r3) (r maxp, n),
it is proved that DSMW problem is NP-Hard
6-8.
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5. Some Notations and Definitions
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5. Some Notations and Definitions

• weight.
• weight of the subset

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Here, Optimal solution corresponds to Dominating
Subset with Minimal Weight.
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D4. Critical element, Dominant critical
element of column i.
2 3 5
Critical elements
Dom. Cri. El.
2 -- 5
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6. The Properties of the DSMW Problem 9
Proposition 1. The set of the dominant critical
elements of the problem gives better feasible
solution than the set of critical elements.
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2 3 5
.
10 12 9
?
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7. A Heuristic Algorithm for the DSMW Problem
I propose a heuristic algorithm by using above
properties for near solving the problem. The
complexity of the proposed algorithm is O(n. p2
).
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8. Computational Experiments
.
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Table 1. Results for input matrices which ensure
condition (a)
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Table 2. Results for input matrices which ensure
condition (b).
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Table 3. Results for matrices which have 25
column, 40 row and
ensure condition (a).
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Table 4. Results for input matrices which ensure
condition (a).
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Table 5. Results for input matrices which ensure
condition (b).
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9. Conclusion In this paper, it is proposed a
new heuristic algorithm which has a ratio bound
in polynomial time, by using the properties
of a combinatorial optimization problem, the
DSMW problem. The computational experiments
show the effectiveness of the proposed
algorithm.
It is proposed a new method for solving the
Global Optimization Problem by using this
algorithm with C.A.M.
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1 M.Yu.Andramonov, A.M.Rubinov and
B.M.Glover, Cutting Angle methods in Global
Optimization, Applied Mathematics
Letters, Vol.12, (1999), 95-100. 2
Dj.A.Babayev, An Exact Method for Solving the
Subproblem of the Cutting Angle Method of Global
Optimization In book  "Optimization and
Related Topics", in Kluwer Academic Publishers,
series "Applied Optimization", Vol 47,
December, Dordrecht/ Boston/ London, (2000),
472-482. 3 V. Chvatal, A Greedy Heuristic
for the Set Covering Problem, Mathematics of
Operations Research, Vol.4, No3,
(1979), 233-235. 4 U.G. Nuriyev , On
Transformation of Global Optimization
Subproblem., In proceeding of 3. th Joint
Seminar on Appied Mathematics, Baku State
University Zanjan University, Baku, (2002),
96. 5 U.G. Nuriyev , An approach to the
subproblem of the Cutting Angle Method of global
optimization, Journal of Global
Optimization (To appear), 2004. 6 U.G.
Nuriyev , On Complexity of a Global Optimization
Problem, Mathematical Computational
Applications, Vol.8, No.1, pp. , (2003),
27-34. 7 U.G. Nuriyev , B. Ordin , Dominating
Subset with Minimal Weight Problem and the survey
of its complexity, Proceeding of XXIII
National Meeting onOperational Research and
Industrial Engineering, Istanbul,
(2002), 33. 8 U.G. Nuriyev , B. Ordin,
Analysis of the Complexity of the Dominating
Subset with Minimal Weight Problem,
EURO/INFORMS Conferences, Istanbul,
Turkey,(2003). 9 B. Ordin, On Some Properties
of the Solution of the Dominating Subset with
Minimal Weight Problem, A Euro Conference
for young OR researchers and practitioners ORP3,
September 21-26, Kaiserslautern, Germany,
(2003), 351-360. 10 A.M. Rubinov, Abstract
Convexity and Global Optimization, Dordrecht,
Kluwer Academic Publishers, (2000).
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