A Parametric Linearization Approach for Solving Zero-One Nonlinear Programming Problems

In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objec-tive functions of this sequence of linear programming problems defined by (P.L.A).


Introduction
Integer programming is one of the most interesting and difficult research areas in mathematical programming and operations research.During the past years, many works has been devoted to linear integer programming, linear 0-1 programming and nonlinear 0-1programming problems [1][2][3][4].
Nnonlinear constrained integer programming problem have many applications in sciences and engineering.A number of research paper dealing with reliability optimization problems are reported in the literature.These are integer programming problems with nonlinear separable objective function and nonlinear multi choice constrained [5,6].
A developed optimization method for solving integer nonlinear programming problem (INLP) with 0-1 variable could be found in [7].This method is closely related to the lexicographic method of Gilmore and Gomory [8], for the knapsack problem and additive algorithm of Balas [9].
One of the conventional methods for solving zero-one nonlinear programming problem is to transform it to a linear programming problem.The main difficulty of this method is the very large number of variables and constraints which increases the problem-size.
A linearization involving a linear number of variables and constraints was first proposed by Glover [10] and improved by Oral and Kettani [11,12].The resulting linearization involving only n -1 additional variable and 2(n -1) linear constraint.
Hanssen and Meyer in [13] compare different ways for linearization the unconstraint quadratic zero-one minimization problem.This approaches involves to increase the number of variables and constraints.
Consider the zero-one nonlinear programming problem as follow: where   , In this paper we present a new approach call it parametric linearization approach for finding the approxi-ond section contains a description of parametric lin .The Parametric Linearization Approach et mated global optimum of zero-one NP problem by solving a sequence of linear programming problems over sub-regions.The solution of the sequence of LP problems tends to the optimal solution of the original nonlinear problem.The reminder of the paper is organized as follow: The sec We call as a parametric piecewise linear approximation o earization approximation and the convergence results.The third section contain a description of using the parametric linearization approximation for solving zero-one nonlinear programming problem.In the forth section the approach will be followed to decrease the number of sub-problems which must be solved in each iteration.Numerical examples used to illustrate the efficiency of the proposed approach and they are given in fifth section.The last section draws overall conclusions.x a b  as follows: where x 0 be an arbitrary point in (a,b).In ual Taylor us expansion x 0 is a fixed point but in our definition this is an arbitrary point.Thus we may call it moving linear Taylor expansion of   we define is an arbitrary point.Now   r x as the following G form: where  ollowin is the characteristic function and defined as the f g: following theorems it is shown that w 1: In th hen the norm of partition tends to zero it means In the other word we may shown that: Proof: The proof is a simple conclusion of the definition.
 in a metric space un uicontinuous on A if for every 0 be an equicontinuous sequ t ence of func ions on a compact set A and   here We know th is pointwise ce sequence the ing a natural num convergen n there e ber M such that for each we have: Therefore we have the following inequalities: Then according to the theorem 7.8 in [6] the sequence is uniformly continuous on A and the proof is leted.Theorem 2.2: ] as the following form: , Hen partitioned to M cells where 1 2 , , ( , , shown the cell of this by 1, , k M   and as a pricewise linear apximation

Description of the Approach
A ach we consider the nonlinear programming problems which re shown in (2).
In our approach we introduce the parametric linearization approximation for nonlinear functions.For solving the optimization problem (2) the nonlinear objective function and constraints are transformed approximately to the piecewise linear functions.
We consider

 
M P A of the cell [ , , , Therefore in each sub-region of the above form we have the following linear programming problem: , For solving the above linear programming problem we divided the interval [0,1] to r equal sub-interval of the form in the solution of the optimization problem (3) all variables must be equal zero or one.Therefore after dividing the interval [0,1] we only should consider the first and the last sub-intervals 1 0, In these sub-intervals let 1 2 be shown in ( 4).Le ms be solved the optimum solut we have 2 n linear programming problems of the form which t this linear programming proble ion and the optimal value of the objective function of this linear programming problems are shown with i x  and i z  respectively then the optimal solution of the original optimization problem of the form (2) may be calculated as follows: ms and lemmas which are ٍ According to the theore proved in Section 2, we know that if in any partitions of [0,1] the norm of partitions tends to zero we have Therefore if an arbitrary partition of [0, 1] be refined then each linear approximation of nonlinear functions   According to the described approach nonlinear programming problems this problems must be converted to a sequence of linear programming problems.
In this situation if n the number of variable in the main optimization problem is a large with 2 n sub-problems which must be solved.
According to the following manner this number may be decreased.At first the sub-problems are solved sequentially until the first feasible solution has been determined.For the reminder sub-regions by substituting x j with nds to values x j = 0 or x j = 1 which is the hole feasible solution of the original zero-one non-linear programming problem (2).

Numerical Examples
Now we give the numerical examples to sho ency of our proposed algorithm.Example 5.1: Consider the following zero-one nonlinear programming problem: In the first stage we divided the region [0, 1] 3 to r 3 equal sub-regions.According to our notation in (4) we consider the following sub-regions: In each sub-region of the above form we choose the element , ,   equal to the middle point of each intervals.In the other word we have So in each sub-region nonlinear 0-1 programming problem converted approximately to the following linear programming problem: , This linear programming problem he optimum value of the original 0-1 nonlinear programming problem is calculated according (5).In the following tableau we show the approximate optimal solution for different values of r.For solving this problem at first we transform it to the following convex nonlinear programming: 3 Now using the parametric linearization technique we transform the above nonlinear programming problem to the following sequence of linear programming problems: smooth function on [a,b].We know the linear Taylor expansion of   f x at the point 0 [ , ]

Definition 2 . 2 :
Let   f x is a nonlinear function on [a,b] and