Treatment of Continuous Function in GGP

= The above conditions can be represented by a set of linear inequalities below:

⎪⎪

Expression (3.11) implies:

(i) If λ_i = 0∀i then f(x)=0,

(ii) If λ_i =1 ,θ_i =1 ,andα_i is odd for all i∈{1,2,...,n} then f(x)=−f⁰(x).

PROPOSITION 3.4 Consider a signomial continuous function

∏

= expressed with the following inequalities:

λ

J i M x x M

x_i⁰ − θ_i ≤ _i ≤ _i⁰ + θ_i,∀ ∈ , J i M

x x M

x_i − − _i ≤ _i ≤− _i + − _i ∀ ∈

− ⁰ (1 θ ) ⁰ (1 θ ), ,

J i x x_i ≤ _i ∉

< ,

0 ⁰ ,

J i x

x_i = ,_i⁰ ∉ , where

(i)

∏

=

= ⁿ

i

i ⁱ

x c f

1 0

0(x) ( )^α ,

(ii) },λ,θ,λ_i,θ_i∈{0 ,1 M =2max{−x_i,x_i},

(iii) t is an integer variable,

(iv) J is a set containing all i where the lower bound of x areis negative, and I _i denotes a set of all i where i∈J and α_i is odd, J and I are expressed as

J={i |x_i < 0∀i} and

I={i |x_i <0 and α_i is odd∀i}.

PROOF Referring to Li and Tsai (2005). □

In order to globally solve a GGP problem, f⁰(x) in PROPOSITION 3.4 should be approximately linearized. The most convenient approach to linearizing f⁰(x) is to take logarithms to reformulate f⁰(x) as the differences of some convex functions in (3.1) as proposed by Floudas’s method. Such an approach, however, requires that we add numerous new binary variables. For instance, to treat a constraint

10z=x₁^−.5x₂⁻¹x₃² ≤ where 1≤x₁,x₂,x₃ ≤5. (3.12) Floudas’s method first denotes z₁ =logx₁ ,z₂ =logx₂ ,andz₃ =logx₃, then to reformulates

≤10

z as the following linear inequalities:

10−0.5z₁ −z₂ +2z₃ ≤log ,

i i

i z x

x

L(log )≤ ≤log , i=1,2,3.

Since log are concave functions, linearizing x_i L(logx_i) with m break points requires the addition of 3(m−1) binary variables for piecewise linearization. Here we propose some converxification and concavification strategies to reduce the number of binary variables required to linearize L(logx_i). Consider the following propositions:

PROPOSITION 3.5 A piecewise linearized function L(f(x)) of a concave function f(x) with m break points b₁ <b₂ <...<b_m can be expressed as

(C1) b_k −M⋅A_k ≤ x≤b_k+1+M ⋅A_k, k =1,...,m−1,

(C2) f(b_k)+s_k(x−b_k)−M ⋅A_k ≤ L(f(x))≤ f(b_k)+s_k(x−b_k)+M⋅A_k, k =1,...,m−1 where

(i)

k k

k k k

b b

b f b s f

−

= −

+ +

1

1) ( )

( ,

(ii) A are specified as the similar way in (2.7), _k

(iii) M is a large enough positive value.

Obviously, only

⎡

^log2⁽m−¹⁾

⎤

binary variables are required to create the piecewise linear function L(f(x)).

PROOF Referring to Tsai et al. (2002). □

PROPOSITION 3.6 A twice-differentiable function f(x)=cx₁^α1x₂^α2...x_n^αn,x_i ≥0∀i=1,...,n, is a convex function in one of the following conditions:

(C1) c≥0, iα_i <0∀ ,

(C2) c<0, iα >0∀ ,

∑

^{n α} ≤1^.

PROOF Referring to Tsai et al. (2002). □

To simplify the expression, here we use a signomial term with three variables as instances to illustrate the convexification and concaveification techniques.

CONVEXIFICATION AND CONCAVIFICATION STRATEGIES 1

Consider a constraint z =x₁^α1x₂^α2x₃^α3 ≤constant with x_i ≥ i0( =1 ,2 ,3) , α₁,α₂ <0,

3 >0

α . This constraint can be converted into the following inequalities:

constantx₁^α1x₂^α2w₃^−α3 ≤ , ) ( ₃¹

3 1 3

−

− ≤w ≤ L x

x

where x₁^α1x₂^α2w₃^−α3 ,x₃⁻¹ are convex (following (C1) of PROPOSITION 3.6, and L(x₃⁻¹) is a piecewise linear function by PROPOSITION 3.5.

Take the example in (3.12) as an introduction. The constraint z≤10 is converted into the following inequalities base on CONVEXIFICATION AND CONCAVIFICATION STRATEGIES 1:

10x₁^−0.5x₂⁻¹w₃⁻² ≤ , ) ( ₃¹

3 1 3

−

− ≤w ≤L x

x ,

where only

⎡

^log2⁽m−¹⁾

⎤

binary variables are used instead of the 3(m−1) binary variables required by conventional methods.

CONVEXIFICATION AND CONCAVIFICATION STRATEGIES 2

A constraint z=−x₁^α1x₂^α2x₃^α3 ≤constant with x_i ≥ i0( =1 ,2 ,3) , 0α₁,α₂,α₃ < . This constraint can be converted into following inequalities:

constant

3 1 3 3 1 3 2 1

1 ≤

−w w w ,

) ( ³

3 i i

i i

i w L x

x ^α ≤ ≤ ^α for i=1 ,2 ,3,

where w w w³ x_i^3αi

1 3 3 1 3 2 1

1 ,

− are convex (following (C2) of PROPOSITION 3.6), and L(x_i^3αi) is a piecewise linear function by PROPOSITION 3.5.

3.5 Numerical Examples

EXAMPLE 3.1 GGP with Linear Continuous Function

Consider following program with one continuous free-sign variable x and two discrete variables y₁andy₂:

PROGRAM 1

Min xy₁³y₂ +xy₁y₂²

s.t. (C1) 500xy₁² + yy_{1 2} ≤− , (C2) −xy₁+ y₁²y₂ ≤500, (C3) −5≤x≤5,

(C4) y₁∈{−1, 0, 1, 4,5,6,7.5,8,9,10},

(C5) y₂∈{−27, −18, −9, −7 ,−4 ,−1, 1,3,4,5}.

Program 1 can not be solved by Floudas’s methods (1997, 1999, 2000) or Li and Tasi’s (2005) approach, since there are zero and negative values for continuous and discrete variables. Our solution proceeds as follows:

(i) Denote y₁∈{−1 ,0 ,1 ,4,5,6,7.5,8,9,10}={d₁,...,d₁₀}. Since

⎡

^log2¹⁰

⎤

= , four binary ⁴

variables u₁,u₂,u₃ ,andu₄ are used to express y . Specifying ₁ k as:

10k =1+u₁ +2u₂ +2²u₃ +2³u₄ ≤ . (3.13)

by . Specifying A_1,k as

x are introduced to form following inequalities based on PROPOSITION 3.4:

⎪⎪

(iii) Other nonconvex terms are treated similarly as in (3.18) and (3.19). These terms are denoted as z₂ =x₁y₁y₂², z₃ =x₁y₁², z₄ = y₁y₂, z₅ =x₁y₁, and z₆ = y₁²y₂.

The original problem is then converted into a mixed 0-1 linear program below, which contains 162 linear constraints and 8 binary variables.

PROGRAM 2 with the objective value 4851.

Here we use a signomial continuous function x₁³x₂ to replace x in Program 1. The example is formulated below:

PROGRAM 3

Min (The same objective function in Program 1)

s.t. (C1) The same constraints in Program 1 but erase the constraint:−5≤ x≤5, (C2) x=x₁³x₂, −5≤x₁ ≤5, −5≤ x₂ ≤5.

Compared with the solution process of Program 2, the extra work is to treat the equality constraint x= x₁³x₂ as follows:

(i) Expressing free-sign variables x₁andx₂ by PROPOSITION 3.4 as inequalities below:

2 inequalities based on Proposition 5:

θ

θ x x M

M

x⁰− ′ ≤ ≤ ⁰+ ′ ,

)

(iii) Using Convexification and Concavification Strategies to convert the inequalities in (ii) as follows:

The whole program is reformulated as the following mixed 0-1 convex program:

PROGRAM 4

Min z₁+z₂

s.t. (C1) The same constraints in Program 2, (C2) Constraints in (3.20)－(3.23).

This is a mixed binary convex program which can be solved to reach finite ε -convergence to the global optimal solution as

)

EXAMPLE 3.3 Comparisons of Computational Efficiency for Signomial Function

This example is used to compare the computational efficiency between the proposed method and Floudas’s methods. Consider following GGP problem:

PROGRAM 5 For n=3 , the objective function of above problem becomes

3 method to have the following convex integer program:

PROGRAM 6

Resolving this problem with n=3 by the proposed method to have the following linear mixed binary program:

PROGRAM 7

Min z₁₂₃−z₁₂−z₂₃−z₁₃

s.t. (C1) b≤z₁₂ +z₂₃+z₁₃ ≤2b,

(C2) , 0, 1, _, 0, 123

1 , 1

, , 1

,

, p p A p p i , ,

d

z ^m _{i j}

j j i m

j

j i j i m

j

j i j i

i =

∑

ⁱ

∑

=

∑

= ≥ =

=

=

=

α ,

(C3) z₁⋅d₂^α_,²_j−M⋅A_2,j ≤z₁₂ ≤z₁⋅d₂^α_,²_j+M⋅A_2,j ∀j, (C4) z₂⋅d₃^α_,³_j −M⋅A_3,j ≤z₂₃ ≤z₂⋅d₃^α_,³_j+M⋅A_3,j ∀j, (C5) z₁⋅d₃^α_,³_j −M⋅A_3,j ≤z₁₃ ≤z₁⋅d₃^α_,³_j+M⋅A_3,j ∀j, (C6) z₁₂⋅d₃^α_,³_j −M⋅A_3,j ≤z₁₂₃≤z₁₂⋅d₃^α_,³_j+M⋅A_3,j ∀j,

(C7) ^{⎡ m⎤} u m i u_ik i,k

k ik

k- ⋅ ≤ ∀ ∈ ∀

∑

=

} 1 , 0 { , )

2

( _,

log

1 ,

2 1

.

All M ′ come from (3.7). By specifying various values for _x m and b, we solve both programs using LINGO (2004). The related CPU time, number of binary variables, and number of constraints are reported in Table 3.2.

Table 3.2 indicates that both methods reach the same ε global optimum for − ε =10⁻⁸. The number of binary variables used in Floudas’s method is larger than the proposed method, while the proposed method uses more constraints than Floudas’s method. The CPU time demonstrates that the proposed method is much more computationally efficient than Floudas’s methods.

Table 3.2 Experiment result of Example 3.3

Experiment Conditions Proposed method

Objective Value －5.389143

CPU Time 3sec 1min 44sec

Objective Value －7.452201

CPU Time 11sec 6min 28sec

Number of binary variables 12 48

Number of constraints 223 9

Objective Value －19.73737

CPU Time 2min 12sec 32min 45sec

Number of binary variables 15 96

Number of constraints 447 9

Objective Value －61.78579

*(α₁ ,α₂ ,α₃)=(−2, 0.5, 1.2)

EXAMPLE 3.4 Comparison of Computational Efficiency for Discrete Variables with Large

Sizes

Consider following program with three discrete variables y₁ ,y₂ and y , where ₃ y₁ contains 8 possible values and y₂ ,y₃ both contain 128 possible values.

PROGRAM 8

Floudas's method ( or Li and Tsai's method) converts this program into following form:

Our proposed method (i.e., THEOREM 3.1 and PROPOSITION 3.3) transforms this program into following form:

PROGRAM 10

5

The number of binary variables, constraints, continuous variables, as well as the number of iterations and CPU time are listed in Table 3.3. Table 3.3 illustractes that both programs obtain the same global optimum with y₁ =16.8 ,y₂ =3.1 ,y₃ =14, and objective value is 685.0155 as solved by LINGO (2004). However, the proposed method is more computational effecient than Floudas's method and Li&Tsai's method.

Table 3.3 Comparisons of existing methods with proposed method No. of binary

variables

No. of constraints

No. of continuous variables

No. of iterations

CPU time (second) Floudas's method and

Li&Tsai's method 264 13 3 53588 72

Proposed method 17 30 269 11139 3

Chapter 4 Extension 2－Solving Task Allocation

在文檔中 整數規劃之高效率求解方法及其運用 (頁 53-68)