一非線性規劃問題的全域最佳解之充分條件

1

行政院國家科學委員會專題研究計畫成果報告

一非線性整數規劃問題的全域最佳化之充分條件

Sufficient Conditions for Global Optimum of A Class of

Nonlinear Integer Pr ogr ams

計畫編號：NSC 89-2213-E-009-044

執行期限：88 年 8 月 1 日至 89 年 7 月 31 日

主持人：黎漢林 國立交通大學資訊管理研究所

計畫參與人員：胡念祖、張李志平、陳孟敏 國立交通大學資訊管理研究所

一、中英文摘要 傳統非線性整數規劃方法甚少探討全 域最佳化的條件，本研究目的在於提出一 種特殊形式整數規劃全域最佳化的必要與 充分條件。我們首先引介橋變數（Bridge variable）及其所組成的新函數，接下來我 們證明一個全域最佳解就是這新函數的最 小解。本文所提出的全域最佳解條件對於 特定之非線性整數規劃的求解非常有幫 助。 關鍵詞：橋變數，非線性整數規劃，非線 性整數規劃的必要與充分條件 Abstr act

Classical nonlinear integer programming methods do not recognize conditions for global optimality. This study proposes a necessary and sufficient condition for global optimality of a special structured integer programs. By denoting a bridge variable as one appears in more than one cross product term in the program, we define a function composed of all bridge variables. Following that we prove that a global optimum is one which has minimal value in this function. The proposed condition is very helpful in finding the global optimum of a nonlinear integer program where many variables are non-bridge variables.

Keywor ds: Bridge variable ,

Box-constrained nonlinear integer program , Necessary and sufficient condition for nonlinear integer program.

二、緣由與目的

The conventional optimization tools such as gradients, subgradients, and second order constructions as Hessiane, cannot be expected to yield conditions of optimality for a nonlinear integer program. An exhaustive enumeration method (Vizvari and Yilmaz,1994) requires to evaluate most of feasible points for finding the global optimum of a nonlinear integer program. Several special types of nonlinear integer programs were studied under different assumptions: Various Lagrangean decomposition methods (Floudas,1995) solve a class of nonlinear integer programming problems based on nonlinear duality theory. Some outer approximation algorithms (Duran and Grossmann,1986,Horst and Tuy,1990) solve specific nonlinear integer programs with linear constraints.

This paper proposes a necessary and sufficient condition of global optimality for a special structured Nonlinear Integer Program (NIP). This study classifies all variables as bridge variables and non-bridge variables. A bridge variable is one which appears in more than one cross product terms in a NIP. The special structured program we are interested in is one where many of variables are non-bridge variables. By defining a new function based on bridge variables, our necessary condition of global optimality states that if a point has minimal value of this new function, then this point is a global minimum of a NIP. Our sufficient condition of global optimality states that a global minimum of a NIP should have minimal value of the new function.

2

nonlinear integer program discussed in this paper is formulated as following box-constrained program. It could be straightforwardly extended into constrained nonlinear programs: NIP

Min f(X) = k i k i ikx a α

∑

, + ik jk j k j i i ijkx x c β γ

∑

, , subject to x_i ≤ x_i ≤ x_i for i = 1,2,...,n, x_i

are integers, x and _i x are respectively the_i

lower and upper bounds, a_ik,c_ijk,α_k,β_ik,γ _jk are real, where X = (x₁,x₂,...,x_n).

Follows is an example of a NIP : Min f(X) = 2x -12 4 1 x + 6 1 x + 2.5 2 x + 2 3 x -2x1x3 +x₁2x₃-2x₂x₄+x₃x₄ (1) subject to 0≤ x1 ≤4 , 0≤ x2 ≤3 , 0≤ x3 ≤2 , 0≤ x₄ ≤1, x ,₁ x ,2 x ,3 x are integers.4

If solving this problem by an exhaustive enumeration method, the number of integer points required to check is 5x4x3x2 = 120 .

Observing NIP problem in (1) we know that there are four cross product terms composed by following three pairs of variables

(

x₁,x₃

)

,

(

x₂,x₄

)

,

(

x₃,x₄

)

where

3

x and x appear twice, and 4 x and 1 x2

appear once. Here x and ₃ x are called4

“bridge variables” since they serve as bridges to link related variables in the cross product terms. For instance, x links 3

(

x1,x3

)

with

(

x3,x4

)

and x links 4

(

x2,x4

)

with

(

x3,x4

)

.

This paper is interested in a NIP problem where many of variables are non-bridge variables. We propose a necessary and sufficient condition of global optimality for this special structured NIP problems. The proposed condition is quite useful in finding a global solution of such a problem.

三、結果與討論 　　

Denote Y as a non-bridge set composed of all non-bridge variables, and Z as a bridge set composed by all bridge variables, the

decision vector can be expressed as X = {Y,

Z}, where Y = (y1,y2,...,y_m) and Z =

(z₁,z₂,...,z_q).

Then a NIP problem can be rewritten as Min f(X) = f(Y,Z) =

( )

_i i i y f

∑

+

( )

_j j j z f

∑

+

(

_{i j}

)

j i ij y z f , ,

∑

+

(

_{i k}

)

k i ik z z f , ,

∑

(2) subject to y_i ≤ y_i ≤ y_i , z_j ≤z_j ≤z_j

where f is composed by a non-bridge_i

variable, f is composed by a bridge_j

variable, f is composed by one non-bridge_ij

and one bridge variables, f is_jk

composed by two bridge variables.

A condition of global optimality for a BIP problem is proposed as follows.

Theorem 1： ( Necessary condition of global

optimality)

A global optimum

(

Y ,∗ Z∗

)

of (2) should satisfy

(

_Y∗ _Z∗

)

f , = Min

{ (

g z₁0,z,₂0,...,z_q0

)

for all z_k ≤z_k0 ≤z_k,k=1,2,...,q

}

Theorem 2 ( Sufficient condition of global

optimality for NIP problem )

If there is a point

(

Y ,∗ Z∗

)

satisfying following conditions

(

_Y∗ _Z∗

)

f , = Min

{ (

02 0

)

0 1,z, ,...,zq z g for all zk zk zk,k 1,2,...,q 0 ≤ = ≤

}

,then

(

_{Y ,}∗ _Z∗

)

is a global minimum for NIP problem of (2)

We use some numerical examples to illustrate that the condition is useful in solving a NIP problem .

Example 1 Consider following problem which does not have cross product term : Minimize f

( )

x = x₁3 −x₂2 +x₃

3

i =1,2,3.

In this example Q = Φ and BS = Φ, The optimal solution

(

x₁*,x*₂,x₃*

)

should satisfy g

(

x₁*,x₂*,x₃*

)

= Min

{

3 , 2 ₁ 2

}

1 − ≤ x ≤ x + Min

{

−x₂2 , −2≤ x₂ ≤2

}

+ Min

{

x₃ , −2≤x₃ ≤2

}

= -8-4-2 = -14, with x = -2, ₁* x = 2 or –2 , and*2 * 3 x = -2.

Example 2 Consider following three camel

NIP problem Minimize f

( )

x = 3 2 6 3 4 3 2 3 2 2 2 1 6 1 4 1 2 1 2 3 1 3 1 . 6 1 19 . 1 2x − x + x −xx + x + x −x + x − xx subject to −2≤ x₁ ≤1, −2≤ x₂ ≤2, 2 2≤ ₃ ≤ − x ; x ,₁ x ,2 x are integers.3 Here Q=

{

(

x₁,x₂

) (

, x₂,x₃

)

}

and BS =

{ }

x . Following Definition 2, 2

( )

0 2 x g can be specified as g

( )

x20 = Min

( )

{

(

,

)

, 2 1 2

}

0 2 1 12 1 1 x + f x x − ≤x ≤ f + Min

( )

{

(

)

}

( )

0 2 2 3 3 0 2 23 3 3 x f x ,x , 2 x 2 f x f + − ≤ ≤ + By specifying x as –2, 20

( )

0 2 x g becomes g

(

x20 =−2

)

=Min

{

2 2 2

}

6 1 19 . 1 2 1 1 6 1 4 1 2 1 − x + x + x − ≤ x ≤ x + Min

{

4 2 2

}

3 1 3x₃2 −x₃4 + x₃6 + x₃− ≤x₃ ≤ +.4 = -3.974

The best solution is

(

x₁ =−2,x₂ =−2,x₃ =0

)

for given x = -2.₂

Similarly, we have

(

x2 =−1

)

=

g -2.274 with best known x = -1

2 , x =0,3

(

x₂ =0

)

=

g -.374 with best known x =1 or₁

-2 , x =0,₃

(

x₂ =1

)

=

g -2.274 with best known x = 2 ,₁

3

x =0,

(

x2 =2

)

=

g -1.974 with best known x = 1 ,1 3

x =0,

the global optimum is then

(

x₁*,x*₂,x₃*

)

=

(

−2,−2,0

)

with objective value –3.974 .

To find a global minimum of this example by an exhaustive enumeration way , the total number of points required to be checked is 4x5x5=100 . By utilizing the proposed condition of global optimality, the number of required check points becomes 4x5 + 5x5 = 45.

Example 3 Consider a NIP problem , where Q=

{

(

y1,z1

) (

, y2,z2

) (

, z1,z2

)

}

and BS=

{

z₁,z₂

}

. The total number of points corresponding to

(

z₁,z₂

)

is 3x2 = 6.

A g

(

z₁ =1,z₂ =1

)

value is computed. Similarly all other 24 g

(

z₁0,z0₂

)

can be obtained. The global solution of this example is

(

y₁*,y₂*,z₁*,z₂*

)

=

(

1,1,1,1

)

. Here the total number of points required to be checked is 5x3 + 3x4 + 4x2 = 35.

As described before, if solving this example by an exhaustive enumeration way, the required number of check points is 5x4x3x2=120.

The proposed global optimality condition can also be extended to solve a NIP problem where a cross term contains more than two variables. Consider following example:

Example 4 Solving following integer problem Minimize 4 3 2 3 1 3 3 2 2 3 1 ) (x x x x xx x x x f = + − + − subject to −2≤ x₁ ≤4 , −2≤ x₂ ≤3 ,

4 2 2≤ 3 ≤ − x , −2≤ x4 ≤1 where Q=

{

(

x1,x3

) (

, x2,x3,x4

)

}

and BS =

{ }

x .3 A g

( )

x₃0 is expressed as

( )

0 3 x g =Min

{

x₁3 +x₁x₃0 −2≤ x₁ ≤4

}

+ Min

{

x₂2 −x₂x₃0x₄ −2≤x₂ ≤3,−2≤x₄ ≤1

}

-( )

₀ 3 3 x

where x₂x₃0x₄ is a cross term..

To compute a g

( )

x30 requires to check

7+6x4= 31 integer points. The total number required to examine for finding a global minimum is 31x6= 186. It is less than 7x6x5x4= 840 which is the required check number for using an exhaustive way to solve the problem, the obtained global minimum is

) , , ,

(x₁* x*₂ x₃* x₄* = ( -2 , 1 , 2 , 2 ).

This research proposes a necessary and sufficient condition for global optimum of a specific structured nonlinear integer programs. This condition is quite useful for finding the global minimum of a NIP problem where most of variables are non-bridge variables. 四、計畫成果自評 1.本文找到一非線性整數規劃最佳解的充 分條件，此條件陳述如定理一及定理 二。此條件對求解非線性整數規劃問題 甚有助益。 2.但此條件目前只適用於特殊題型，尚無 法擴充至一般性整數問題此仍待後續研 究。 3. 本研究結果已撰寫成兩篇論文投稿至 Journal of Global Optimization 及 Journal of Mathematical Analysis and

Applications。

五、參考文獻

[1] Duran , M. and I.E.Grossmann (1986),

An Outer-approximation algorithm for a class of mixed integer nonlinear programs , Mathematical Programming , 36 , 307-339. [2]Floudas , C.A. et al. (1999) ,Handbook of Test Problems in Local and Global

Optimization , kluwer Academic Publishers Netherlands.

[3]Floud as, C.A. (1995) , Nonlinear and Mixed Integer Optimization : Fundamentals and Applications. Oxford University Press , New York.

[4]Vizvari , B. and F. Yilmaz (1994) , An Order (Enumerative ) Algorithm for Nonlinear 0-1 Programming , Journal of Global Optimization 3 , 277-290 .

[5] Horst , R. and Hoang Tuy (1990) , Global optimization , Springer – Verlag , Berlin.