A Branch-and-Reduce Approach for Solving Generalized Linear Multiplicative Programming

We consider a branch-and-reduce approach for solving generalized linear multiplicative programming. 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.


Introduction
In this paper, the following generalized linear multiplicative programming is considered: x ∈ X 0 l, u ⊂ R n , P where c ji c ji1 , c ji2 , . . ., c jin T ∈ R n , d ji ∈ R, and β j ∈ R, γ ji ∈ R, β j > 0 and for all x ∈ X 0 , 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, problem P 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 problem P 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 for Q .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 of Q 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 problem Q 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 problem LRP for Q , which can provide a lower bound for the optimal value of Q .In Section 3, in order to improve the convergence speed of our algorithm, we present a reducing technique.In Section 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 in Section 5.
By using the property of logarithmic function, the equivalent problem Q of P can be derived, which has the same optimal solution as P , 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 problem LRP for Q , which can provide a lower bound for the optimal value of Q .
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 problem LRP 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 i 1 γ ji ln c T ji x d ji , and φ j2 x p j i T j 1 γ ji ln c T 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:

2.1
By Theorem 1 in 11 , we can derive the lower bound function φ l j1 x of φ j1 x as follows: 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 where Finally, from 2.2 and 2.3 , for all x ∈ X, we have
Taken together above, it implies that φ j x − φ l j x Δ 1 Δ 2 → 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 problem LRP of Q over X can be obtained as follows:

LRP
Obviously, the feasible region for the problem Q is contained in the new feasible region for the problem LRP , 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 .

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 of Q does not exist.
Assume that UB is the current known upper bound of the optimal value φ * 0 of the problem Q .Let

3.1
The reducing technique is derived as in the following theorem.
Theorem 3.1.For any subrectangle X X t n×1 ⊆ X 0 with 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 X 1 ; if α k < 0 and ρ k < α k x k , for some k, then there is no globally optimal solution of Q over X 2 , where

3.2
Proof.First, we show that for all x ∈ X 1 , φ 0 x > UB.Consider the kth component From α k > 0, we have ρ k < α k x k .For all x ∈ X 1 , by the above inequality and the definition of α t x t T φ l 0 x .

3.5
Thus, for all x ∈ X 1 , we have φ 0 x ≥ φ l 0 x > UB ≥ φ * 0 , that is, for all x ∈ X 1 , φ 0 x is always greater than the optimal value φ * 0 of the problem Q .Therefore, there cannot exist globally optimal solution of Q over X 1 .
For all x ∈ X 2 , 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 of Q over X 2 .

Algorithm and Its Convergence
In this section, based on the former results, we present a branch-and-reduce algorithm to solve the problem Q .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 on Section 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} ⊆ X 0 .This branching rule is as follows.
iii Let

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 of Q .This process needs to solve a sequence of linear programming problems and to compute the objective function value of Q 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 LB X k be the optimal value of LRP over the subrectangle X X k , and x k x X 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 l j x n t 1 a jt x t b j , where a jt , b j ∈ R. Thus, we have min x∈X φ l j x n t 1 min{a jt x t , a jt x t } b j .

Algorithm Statement
Step 1 initialization .Let the set all active node Q 0 {X 0 }, the upper bound UB ∞, the set of feasible points F ∅, some accuracy tolerance > 0 and the iteration counter k 0.
Solve the problem LRP for X X 0 .Let LB 0 LB X 0 and If UB < LB 0 , then stop: x 0 is an -optimal solution of Q .Otherwise, proceed.
Step 2 updating the upper bound .Select the midpoint Let the upper bound UB min{φ 0 x k mid , UB} and the best known feasible point x * arg min x∈F φ 0 x .

Mathematical Problems in Engineering
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 min x∈X φ l j x > ln β j , or for j 0, min x∈X φ l 0 x > 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 Step 5 convergence checking .Set ∅, then stop: UB is the -optimal value of Q , and x * is an -optimal solution.Otherwise, select an active node X k 1 such that Set k k 1, and return to Step 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 globallyoptimal 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 UB − LB k ≤ . 4.4 From Step 1 and Step 5 in the algorithm, a feasible solution x * for the problem Q can be found, and the following relation holds Let v denote the optimal value of problem Q .By Section 2, we have Since x * is a feasible solution of problem Q , φ 0 x * ≥ v. Taken together above, it implies that and so x * is a global -optimal solution to the problem Q in the sense that When the algorithm is infinite, by 5 , 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 where LB k 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 that 4.9 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 formulation 4.9 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 in 5 , the branch-and-reduce algorithm presented in this paper is convergent to the global minimum of Q .

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 IV 3.06 GHZ microcomputer, and the convergence tolerance is set at 1.0e − 4 in our experiments.

5.5
The results in Table 1 show that our algorithm is both feasible and efficient.