a branch and reduce approach for solving generalized linear multiplicative programming

Volume 2011, Article ID 409491, 12 pagesdoi:10.1155/2011/409491 Research Article A Branch-and-Reduce Approach for Solving Generalized Linear Multiplicative Programming 1 Department of Ma

Volume 2011, Article ID 409491, 12 pages

doi:10.1155/2011/409491

Research Article

A Branch-and-Reduce Approach for Solving

Generalized Linear Multiplicative Programming

1 Department of Mathematical Sciences, Xidian University, Xi’an 710071, China

2 Department of Mathematics, Henan Normal University, Xinxiang 453007, China

3 School of Computer Science and Technology, Xidian University, Xi’an 710071, China

Correspondence should be addressed to Chun-Feng Wang,wangchunfeng09@126.com

Received 15 March 2011; Accepted 12 May 2011

Academic Editor: Victoria Vampa

Copyrightq 2011 Chun-Feng Wang et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We consider a branch-and-reduce approach for solving generalized linear multiplicative pro-gramming First, a new lower approximate linearization method is proposed; then, by using this linearization method, the initial nonconvex problem is reduced to a sequence of linear programming problems Some techniques at improving the overall performance of this algorithm are presented The proposed algorithm is proved to be convergent, and some experiments are provided to show the feasibility and efficiency of this algorithm

1 Introduction

In this paper, the following generalized linear multiplicative programming is considered:

min

p0



i1



c T

0i x  d0iγ 0i

s.t.

p j



i1



c T ji x  d ji

γ ji

≤ β j , j  1, , m,

x ∈ X0 l, u ⊂ R n ,

P

where c ji  c ji1 , c ji2 , , c jinT ∈ R n , d ji ∈ R, and β j ∈ R, γ ji ∈ R, β j > 0 and for all x ∈ X0,

c T ji x  d ji > 0, j  0, , m, i  1, , p j

Since a large number of practical applications in various fields can be put into problem

P, including VLSI chip design 1, decision tree optimization 2, multicriteria optimization

problem 3, robust optimization 4, and so on, this problem has attracted considerable attention in the past years

It is well known that the product of affine functions need not be quasi convex, thus the problem can have multiple locally optimal solutions, many of which fail to be globally optimal, that is, problemP is multiextremal 5

In the last decade, many solution algorithms have been proposed for globally solving special forms of P They can be generally classified as outer-approximation method 6, decomposition method 7, finite branch and bound algorithms 8, 9, and cutting plane method 10 However, the global optimization algorithms based on the general form P have been little studied Recently, several algorithms were presented for solving problemP

11–15

The aim of this paper is to provide a new branch-and-reduce algorithm for globally solving problem P Firstly, by using the property of logarithmic function, we derive an equivalent problem Q of the initial problem P, which has the same optimal solution

as the problem P Secondly, by utilizing the special structure of Q, we present a new linear relaxation technique, which can be used to construct the linear relaxation programming problem forQ Finally, the initial nonconvex problem P is systematically converted into a series of linear programming problems The solutions of these converted problems can be as close as possible to the globally optimal solution ofQ by successive refinement process The main features of this algorithm: 1 the problem investigated in this paper has

a more general form than those in 6 10; 2 a new linearization method for solving the problemQ is proposed; 3 these generated linear relaxation programming problems are embedded within a branch and bound algorithm without increasing the number of variables and constraints;4 some techniques are proposed to improve the convergence speed of our algorithm

This paper is organized as follows In Section 2, an equivalent transformation and a new linear relaxation technique are presented for generating the linear relaxation programming problemLRP for Q, which can provide a lower bound for the optimal value

ofQ InSection 3, in order to improve the convergence speed of our algorithm, we present

a reducing technique InSection 4, the global optimization algorithm is described in which the linear relaxation problem and reducing technique are embedded, and the convergence

of this algorithm is established Numerical results are reported to show the feasibility of our algorithm inSection 5

2 Linear Relaxation Problem

Without loss of generality, assume that, for 0≤ i ≤ T j , γ ji > 0, T j 1 ≤ i ≤ p j , γ ji < 0, j  0, , m,

i  1, , p j

By using the property of logarithmic function, the equivalent problemQ of P can

be derived, which has the same optimal solution asP,

min φ0 x T0

i1

γ0iln

c T 0i x  d0i



p0



iT0 1

γ0iln

c T 0i x  d0i

s.t φ j x 

T j



i1

γ jiln

c T

ji x  d ji





p j



iT j1

γ jiln

c T

ji x  d ji



≤ ln β j ,

x ∈ X0 l, u ⊂ R n , j  1, , m.

Q

Thus, for solving problem P, we may solve its equivalent problem Q instead Toward this end, we present a branch-and-reduce algorithm In this algorithm, the principal aim is to construct linear relaxation programming problemLRP for Q, which can provide

a lower bound for the optimal value ofQ

Suppose that X  x, x represents either the initial rectangle of problem Q, or modified rectangle as defined for some partitioned subproblem in a branch and bound scheme The problemLRP can be realized through underestimating every function φ j x with a linear relaxation function φ l j x j  0, , m All the details of this linearization

method for generating relaxations will be given below

Consider the function φ j x j  0, , m Let φ j1 x  T j

i1 γ jilncT

ji x  d ji, and

φ j2 x  p j

iT j1γ jilncT

ji x  d ji , then, φ j1 x and φ j2 x are concave function and convex

function, respectively

First, we consider the function φ j1 x For convenience in expression, we introduce the

following notations:

X ji  c T

ji x  d jin

t1

c jit x t  d ji ,

X jin

t1

min

c jit x t , c jit x t



 d ji ,

X jin

t1

max

c jit x t , c jit x t



 d ji ,

K ji ln



X ji



− lnX ji

X ji − X ji ,

f ji x  lnc T ji x  d ji



 ln X ji ,

h ji x  lnX ji

 K ji



X ji − X ji lnX ji

 K ji

n

t1

c jit x t  d ji − X ji

.

2.1

By Theorem 1 in11, we can derive the lower bound function φ l

j1 x of φ j1 x as

follows:

φ l j1 x 

T j



i1

γ ji h ji x ≤

T j



i1

Second, we consider function φ j2 x j  0, , m Since φ j2 x is a convex function,

by the property of the convex function, we have

φ j2 x ≥ φ j2 xmid  ∇φ j2 xmidT x − xmid  φ l

where xmid 1/2x  x,

∇φ j2 x 

⎛

⎜

⎜

⎜

⎜

γ j,T j1cj,T j 1,1

c T j,T j1x  d j,T j1  γ j,T j2cj,T j 2,1

c T j,T j2x  d j,T j2 · · ·  γ j,p j c j,p j ,1

c T j,p j x  d j,p j

γ j,T j1cj,T j 1,n

c T j,T j1x  d j,T j1  γ j,T j2cj,T j 2,n

c T j,T j2x  d j,T j2 · · ·  γ j,p j c j,p j ,n

c T jp j x  d jp j

⎞

⎟

⎟

⎟

Finally, from2.2 and 2.3, for all x ∈ X, we have

φ j l x  φ l

j1 x  φ l

j x, j  0, , m Then, the difference between φ l

j x and φ j x satisfies

φ j x − φ l

where x − x  max{x i − x i | i  1, , n}.

Proof LetΔ1  φ j1 x − φ l

j1 x, Δ2  φ j2 x − φ l

j2 x Since φ j x − φ l

j x  φ j1 x − φ l

j1 x 

φ j2 x − φ l

j2 x  Δ1 Δ2, we only need to proveΔ1 → 0, Δ2 → 0 as x − x → 0.

First, considerΔ1 By the definition ofΔ1, we have

Δ1 φ j1 x − φ l

j1 x 

T j



i1

γ ji

Furthermore, by Theorem 1 in11, we know that f ji x − h ji x → 0 as x − x → 0 Thus,

we haveΔ1 → 0 as x − x → 0.

Second, considerΔ2 From the definition ofΔ2, it follows that

Δ2 φ j2 x − φ l

j2 x

 φ j2 x − φ j2 xmid − ∇φ j2 xmidT x − xmid

 ∇φ j2 ξ T x − xmid − ∇φ j2 xmidT x − xmid

≤∇2φ j2

η ξ − xmidx − xmid,

2.8

where ξ, η are constant vectors, which satisfy φ j2 x − φ j2 xmid  ∇φ j2 ξ T x − xmid and

∇φ j2 ξ − ∇φ j2 xmid  ∇2φ j2 η T ξ − xmid, respectively Since ∇2φ j2 x is continuous, and X

is a compact set, there exists some M > 0 such that ∇2φ j2 x ≤ M From 2.8, it implies thatΔ2≤ Mx − x2 Furthermore, we haveΔ2 → 0 as x − x → 0.

Taken together above, it implies that φ j x − φ l

j x  Δ1 Δ2 → 0 as x − x → 0, and

the proof is complete

From Theorem 2.1, it follows that the function φ l j x can approximate enough the function φ j x as x − x → 0.

Based on the above discussion, the linear relaxation programming problemLRP of

Q over X can be obtained as follows:

min φ l

0x

s.t φ j l x ≤ ln β j , j  1, , m,

x ∈ X 

x, x

⊂ R n

LRP

Obviously, the feasible region for the problemQ is contained in the new feasible region for the problemLRP, thus, the minimum V LRP of LRP provides a lower bound

for the optimal value V Q of problem Q over the rectangle X, that is V LRP ≤ V Q.

3 Reducing Technique

In this section, we pay our attention on how to form the new reducing technique for eliminate the region in which the global minimum ofQ does not exist

Assume that UB is the current known upper bound of the optimal value φ∗0 of the problemQ Let

α tT0

i1

γ0iK0ic0it  ∇φ j2 xmidt , t  1, , n,

T 

T0



i1

γ0i

ln

X 0i  K 0id0i − K 0iX 0i

 φ02xmid − ∇φ02xmidT xmid,

ρ k UB − n

t1,t /  k

min

α t x t , α t x t



− T, k  1, , n.

3.1

The reducing technique is derived as in the following theorem

Theorem 3.1 For any subrectangle X  X tn×1 ⊆ X0with X t  x t , x t  If there exists some index

k ∈ {1, 2, , n} such that α k > 0 and ρ k < α k x k , then there is no globally optimal solution of Q

over X1; if α k < 0 and ρ k < α k x k , for some k, then there is no globally optimal solution of Q over X2,

X1X t1

n×1 ⊆ X, with X1

t 

⎧

⎪

⎪



ρ k

α k , x k

 

X t , t  k,

X2X t2

n×1 ⊆ X, with X2

t 

⎧

⎪

⎪

x k , ρ k

α k

X t , t  k.

3.2

Proof First, we show that for all x ∈ X1, φ0x > UB Consider the kth component x k of x Since x k ∈ ρ k /α k , x k, it follows that

ρ k

From α k > 0, we have ρ k < α k x k For all x ∈ X1, by the above inequality and the definition of

ρ k, it implies that

UB− n

t1,t /  k

min

α t x t , α t x t



that is

UB <

n



t1,t /  k

min

α t x t , α t x t



 α k x k  T

≤n

t1

α t x t  T  φ l

0x.

3.5

Thus, for all x ∈ X1, we have φ0x ≥ φ l

0x > UB ≥ φ∗

0, that is, for all x ∈ X1, φ0x is always greater than the optimal value φ∗0of the problemQ Therefore, there cannot exist globally optimal solution ofQ over X1

For all x ∈ X2, if there exists some k such that α k < 0 and ρ k < α k x k, from arguments similar to the above, it can be derived that there is no globally optimal solution ofQ over

X2

4 Algorithm and Its Convergence

In this section, based on the former results, we present a branch-and-reduce algorithm to solve the problemQ There are three fundamental processes in the algorithm procedure: a reducing process, a branching process, and an updating upper and lower bounds process Firstly, based onSection 3, when some conditions are satisfied, the reducing process can cut away a large part of the currently investigated feasible region in which the global optimal solution does not exist

The second fundamental process iteratively subdivides the rectangle X into two

subrectangles During each iteration of the algorithm, the branching process creates a more refined partition that cannot yet be excluded from further consideration in searching for a global optimal solution for problem Q In this paper we choose a simple and standard bisection rule This branching rule is sufficient to ensure convergence since it drives the intervals shrinking to a singleton for all the variables along any infinite branch of the branch

and bound tree Consider any node subproblem identified by rectangle X  {x ∈ R n | x i ≤

x i ≤ x i ,  1, , n} ⊆ X0 This branching rule is as follows

i Let p  arg max{x i − x i | i  1, , n}.

ii Let γ  x p  x p /2.

iii Let

X 

x ∈ R n | x i ≤ x i ≤ x i , i /  p, x p ≤ x p ≤ γ ,

X 

x ∈ R n | x i ≤ x i ≤ x i , i /  p, γ ≤ x p ≤ x p



.

4.1

By this branching rule, the rectangle X is partitioned into two subrectangles X and X.

The third process is to update the upper and lower bounds of the optimal value ofQ This process needs to solve a sequence of linear programming problems and to compute the objective function value ofQ at the midpoint of the subrectangle X for the problem Q In addition, some bound tightening strategies are applied to the proposed algorithm

The basic steps of the proposed algorithm are summarized as follows In this algorithm, let LBXk be the optimal value of LRP over the subrectangle X  X k , and x k 

xX k  be an element of corresponding arg min Since φ l

j x j  0, , m is a linear function, for convenience in expression, assume that it is expressed as follows φ j l x n

t1 a jt x t  b j,

where a jt , b j ∈ R Thus, we have min x∈X φ j l x n

t1min{ajt x t , a jt x t }  b j

4.1 Algorithm Statement

Step 1 initialization Let the set all active node Q0  {X0}, the upper bound UB  ∞, the

set of feasible points F  ∅, some accuracy tolerance > 0 and the iteration counter k  0.

Solve the problemLRP for X  X0 Let LB0  LBX0 and x0  xX0 If x0 is a feasible point ofQ, then let

UB φ0



x0

, F  F!

If UB < LB0 , then stop: x0is an -optimal solution of Q Otherwise, proceed

Step 2 updating the upper bound Select the midpoint x k

midof X k ; if x k

midis feasible toQ,

then F  F ∪ {x kmid} Let the upper bound UB  min{φ0x k

mid, UB} and the best known feasible point x∗ arg minx∈F φ0 x.

Step 3 branching and reducing Using the branching rule to partition X k into two new subrectangles, and denote the set of new partition rectangles as X k For each X ∈ X k, utilize the reducing technique of Theorem 3.1 to reduce box X, and compute the lower bound φ l

j x of φ j x over the rectangle X If for j  1, , m, there exists some j such that

minx∈X φ l j x > ln β j , or for j  0, min x∈X φ l0x > UB, then the corresponding subrectangle X will be removed from X k , that is, X k  X k \ X, and skip to the next element of X k

Step 4 bounding If X k / ∅, solve LRP to obtain LBX and xX for each X ∈ X k If LBX > UB, set Xk  X k \ X; otherwise, update the best available solution UB, F and x∗

if possible, as in theStep 2 The partition set remaining is now Q k  Q k \ X k"X k, and a new lower bound is LBk infX∈Q kLBX

Step 5convergence checking Set

If Q k1  ∅, then stop: UB is the -optimal value of Q, and x∗ is an -optimal solution Otherwise, select an active node X k1 such that X k1  arg minX∈Q k1LBX, xk1  xX k1

Set k  k  1, and return toStep 2

4.2 Convergence Analysis

In this subsection, we give the global convergence properties of the above algorithm

Theorem 4.1 convergence The above algorithm either terminates finitely with a globally

-optimal solution, or generates an infinite sequence {x k } which any accumulation point is a globally optimal solution of Q.

Proof When the algorithm is finite, by the algorithm, it terminates at some step k ≥ 0 Upon

termination, it follows that

FromStep 1 andStep 5in the algorithm, a feasible solution x∗for the problemQ can be found, and the following relation holds

Let v denote the optimal value of problem Q BySection 2, we have

Since x∗is a feasible solution of problemQ, φ0x∗ ≥ v Taken together above, it implies

that

v ≤ φ0 x∗ ≤ LBk  ≤ v  , 4.7

and so x∗is a global -optimal solution to the problem Q in the sense that

When the algorithm is infinite, by5, a sufficient condition for a global optimization

to be convergent to the global minimum, requires that the bounding operation must be consistent and the selection operation is bound improving

A bounding operation is called consistent if at every step any unfathomed partition can be further refined, and if any infinitely decreasing sequence of successively refined partition elements satisfies

lim

k → ∞UB − LBk   0, 4.9

where LBk is a computed lower bound in stage k and UB is the best upper bound at iteration

k not necessarily occurring inside the same subrectangle with LB k Now, we show that4.9 holds

Since the employed subdivision process is rectangle bisection, the process is exhaustive Consequently, from Theorem 2.1 and the relationship V LRP ≤ V Q, the

formulation4.9 holds, this implies that the employed bounding operation is consistent

A selection operation is called bound improving if at least one partition element where the actual lower bound is attained is selected for further partition after a finite number

of refinements Clearly, the employed selection operation is bound improving because the partition element where the actual lower bound is attained is selected for further partition in the immediately following iteration

From the above discussion, and Theorem IV.3 in5, the branch-and-reduce algorithm presented in this paper is convergent to the global minimum ofQ

5 Numerical Experiments

In this section, some numerical experiments are reported to verify the performance of the proposed algorithm The algorithm is coded in Matlab 7.1 The simplex method is applied to solve the linear relaxation programming problems The test problems are implemented on a Pentium IV3.06 GHZ microcomputer, and the convergence tolerance is set at  1.0e − 4 in

our experiments

Example 5.1see 12,15.

min x1 x2 12.5 2x1 x2 11.1 x1 2x2 11.9 s.t x1 2x2 11.1 2x1 2x2 21.3 ≤ 50,

1≤ x1≤ 3, 1 ≤ x2≤ 3.

5.1

Example 5.2see 15

min 2x1 x2− x3 1−0.2 2x1− x2 x3 1x1 2x2 10.5 s.t 3x1− x2 10.3 2x1− x2 x3 2−0.1 ≤ 10,

1.2x1 x2 1−12x1 2x2 10.5 ≤ 12,

x1 x2 20.2 1.5x1 x2 1−2≤ 15,

1≤ x1≤ 2, 1 ≤ x2≤ 2, 1 ≤ x3≤ 2.

5.2

Example 5.3see 12,15

min x1 x2 x32x1 x2 x3x1 2x2 2x3

s.t x1 2x2 x31.1 2x1 2x2 x31.3 ≤ 100,

1≤ x1≤ 3, 1 ≤ x2≤ 3, 1 ≤ x3≤ 3.

5.3

Example 5.4see 13,16

min −x1 2x2 24x1− 3x2 43x1− 4x2 5−1−2x1 x2 3−1

s.t x1  x2≤ 1.5,

x1 − x2≤ 0,

0≤ x1≤ 1, 0 ≤ x2≤ 1.

5.4

Example 5.5see 11,15

min 2x1 x2 11.5 2x1 x2 12.1 0.5x1 2x2 10.5 s.t x1 2x2 11.2 2x1 2x2 20.1 ≤ 18,

1.5x1 2x2 12x1 2x2 10.5 ≤ 25,

1≤ x1≤ 3, 1 ≤ x2≤ 3.

5.5

Step branching and reducing Using the branching rule to partition X k into two new subrectangles,... choose a simple and standard bisection rule This branching rule is sufficient to ensure convergence since it drives the intervals shrinking to a singleton for all the variables along any infinite branch. ..

0x > UB ≥ φ∗

0, that is, for all x ∈ X1, φ0x is always greater than the optimal value φ∗0of