A Simplicial Branch and Bound Duality-Bounds Algorithm to Linear Multiplicative Programming

A simplicial branch and bound duality-bounds algorithm is presented to globally solving the linear multiplicative programming (LMP).We firstly convert the problem (LMP) into an equivalent programming one by introducing p auxiliary variables. During the branch and bound search, the required lower bounds are computed by solving ordinary linear programming problems derived by using a Lagrangian duality theory. The proposed algorithm proves that it is convergent to a global minimum through the solutions to a series of linear programming problems. Some examples are given to illustrate the feasibility of the present algorithm.

Problem (LMP) has many important applications.Since it subsumes quadratic programming, bilinear programming and linear zero-one programming as special cases, the applications appear quite numerous.Readers may refer to Benson [2] for the following analysis.Now the quadratic programming problem is given as follows: min 1 2    +    + , where  is an  ×  symmetric matrix of rank ,  ∈   and  ∈ .From Tuy [3], there exist linearly independent sets of -dimensional vectors, {V 1 , V 2 , . . ., V  } and { 1 ,  2 , . . .,   }, such that, for all  ∈   , Thus, problem (LMP) encompasses the general quadratic programming problem as a special case, and the applications of problem (LMP) include all of the applications of general quadratic programming.Among the latter, for example, quadratic assignment problems [4], problems in economies of scale [5], the constrained linear regression problem [6], VLSI chip design problems [7], the linear complementarity problem [5], and portfolio analysis problems [6].
The bilinear programming problem can be converted into the LMP and it may be written by where  ∈   ,  ∈   ,  ∈   ,  ∈   ,  is an  ×  matrix of rank ,  and  are  ×  matrix and  ×  matrix, respectively.From Konno and Yajima [8], by using a constructive procedure, it can be written in the form where V 1 , V 2 , . . ., V  ∈   and  1 ,  2 , . . .,   ∈   .The latter is a special case of the LMP with . ., , and  = {(, ) ∈  + |  ⩽ ,  ⩽ , ,  ⩾ 0}.Therefore, among the applications problem (P) are all of the applications of bilinear programming, including, for example, location-allocation problems [9], constrained bimatrix games [10], the three-dimensional assignment problem [11], certain linear max-min problems [12], and many problems in engineering design, economic management and operations research.
A linear zero-one programming problem may be written as where  ∈   ,  ∈   ,  is  ×  matrix.From Raghavachari [13], for  > 0 sufficiently large,  * is an optimal solution to problem (Q) if and only if  * is an optimal solution to the problem Since problem (QC) is a special case of the LMP, it follows that all of the numerous applications of linear zero-one programming are embodied among the applications of the LMP.For an overview of some of these applications, see Nemhauser and Wolsey [14].
In this paper, a simplicial branch and bound dualitybounds algorithm is presented to the problem (LMP) by solving a sequence of linear programming one over partitioned subsets.The algorithm implements a simplicial branch and bound search, finding a global optimal solution to the problem, equivalent to the problem (LMP).Branching takes place in a space of only dimension  in the algorithm, where  is the number of terms in the objective function of problem (LMP).During the search, the required lower bounds are computed by solving ordinary linear programming problems.When the algorithm is infinite, any accumulation point of this sequence of feasible solutions is guaranteed to globally solve the problem.The proposed branch and bound algorithm is summarized as follows.Firstly, the branch and bound search takes place in a space of only dimension , where  is the number of terms in the objective function of problem (LMP), rather than in the decision space R  .Secondly, the subproblems that must be solved during the search are all linear programming problems that can be solved very efficiently, for example, by a simplex method.The algorithms in this article are motivated by the seminal works of [34], the sum of linear ratios problem, and Horst and Tuy [35] by using branch and bound for global optimization.
The organization and content of this article can be summarized as follows.In Section 2, some preliminary results and operations are presented to implement the simplicial branch and bound duality-bounds algorithm.The simplicial branch and bound duality-bounds algorithm is given in Section 3. In Section 4, the convergence of the algorithm is established.In Section 5 some examples are solved to demonstrate that the proposed algorithm is effective.Some concluding remarks are given in Section 6.

Preliminaries
In this section, we firstly show how to convert the LMP into an equivalent nonconvex programming (LMP( 0 )) by introducing a -dimension vector  for finding a simplex.Then, for each -dimensional simplex  created by the branching process, the lower bound LB() can be found by solving an ordinary linear program by using the Lagrangian weak duality theorem of nonlinear programming.
is a single point set and  * is a global optimal solution to the LMP, where  * is any optimal solution to the linear program obtained by setting  equal to  * in problem (LMP( 0 )).Therefore, we will assume in the remainder of this article that  0 is a dimensional simplex.

Equivalent Problem.
For any simplex  ⊂ R  , define the problem In order to solve the LMP, the branch and bound algorithm is used to solves problem (LMP( 0 )) instead.The validity of solving problem (LMP( 0 )), in order to solve the LMP, follows from the next result.Theorem 3. If ( * ,  * ) is a global optimal solution for problem (LMP( 0 )), then  * is a global optimal solution for problem (LMP).If  * is a global optimal solution for problem (LMP), then ( * ,  * ) is a global optimal solution for problem (LMP( 0 )), where  *  =     * +  0 ,  = 1, 2, . . ., .The global optimal values V and V( 0 ) of problems (LMP) and (LMP( 0 )), respectively, are equal.Proof.By using the fact that  0 ⊇ , the proof of this theorem follows easily from the defintions of problem (LMP( 0 )).

Duality Bound.
For each -dimensional simplex  created by the branching process, the algorithm computes a lower bound LB() for the optimal value V() of problem (LMP()).The next theorem shows that, by using the Lagrangian weak duality theorem of nonlinear programming, the lower bound LB() can be found by solving an ordinary linear programming.Theorem 4. Let  ⊆ R  be a -dimensional simplex with vertices  0 ,  1 , . . .,   , and let  = {0, 1, 2, . . ., }.Then LB() ⩽ V(), where LB() is the optimal value of the linear programming problem Proof.By the definition of V() and the weak duality theorem of Lagrangian duality, V() ⩾ LB(), where Since it follows that, Since  is a compact polyhedron with extreme points   ,  = 0, 1, 2, . . ., , for each  ∈ R  and  ⩾ 0, ∑  =1 (     +      ) +    ⩾ 0 holds for all  ∈  if and only if it holds for all  ∈ { 0 ,  1 , . . .,   }.So, for all  ∈ , we can get Notice that for all  ∈ , the left-hand-side of ( 15 That is, For any  ∈ R  ,   ( 0 −   ) is a linear function.Because simplex  is a compact polyhedron with extreme points  0 ,  1 , . . .,   , this implies for any  ∈ R  , ∑  =1   ( 0 −   ) ⩾  holds if and only if The proof is complete.
Proof.The proof is similar to [34, Proposition 3], it is omitted here.
Remark 6.From part (ii) of Proposition 5, for any dimensional simplex  created by the algorithm during the branch and bound search, the duality bounds-based lower bound LB() for the optimal value V() of problem (LMP()) is either finite or equal to +∞.When LB() = +∞, problem (LMP()) is infeasible and, as we shall see,  will be eliminated from further consideration by the deletion by bounding process of the algorithm.The monotonicity property in part (i) of Proposition 5 will be used to help to show the convergence of the algorithm.Now, we show how to determine an upper bound of the global optimal value for (LMP()).For each -dimensional simplex  generated by the algorithm such that LB() is finite, the algorithm generates a feasible solution  to problem (LMP).As the algorithm finds more and more feasible solutions to it an upper bound for the optimal value V of it improves iteratively.These feasible solutions are found from dual optimal ones to the lower bounding problems (LP()) that are solved by the algorithm, as given in the following result.
Proposition 7. Let  ⊆ R  be a -dimensional simplex with vertices  0 ,  1 , . . .,   , and suppose that LB() ̸ = + ∞.Let  ∈ R  be optimal dual variables corresponding to the first  constraints of linear program LP().Then  is a feasible solution for problem (LMP).

Global Optimizing Algorithm
To globally solve problem (LMP()), the algorithm to be presented uses a branch and bound approach.There are three fundamental processes in the algorithm, a branching process, a lower bounding one, and an upper bounding one.

Branching Rule.
The branch and bound approach is based on partitioning the -dimensional simplex  0 into smaller subsimplices that are also of dimension , each concerned with a node of the branch and bound tree, and each node is associated with a linear subproblem on each subsimplicie.These subsimplices are obtained by the branching process, which helps the branch and bound procedure identify a location in the feasible region of problem (LMP( 0 )) that contains a global optimal solution to the problem.
During each iteration of the algorithm, the branching process creates a more refined partition of a portion in  =  0 that cannot yet be excluded from consideration in the search for a global optimal solution for problem (LMP()).The initial partition  1 consists simply of , since at the beginning of the branch and bound procedure, no portion of  can as yet be excluded from consideration.
During iteration  of the algorithm,  ⩾ 1, the branching process is used to help create a new partition  +1 .First, a screening procedure is used to remove any rectangle from   that can, at this point of the search, be excluded from further consideration, and  +1 is temporarily set equal to the set of simplices that remain.Later in iteration , a rectangle   in  +1 is identified for further examination.The branching process is then evoked to subdivide   into two subsimplices   1 ,   2 .This subdivision is accomplished by a process called simplicial bisection.

Lower Bound and Upper
Bound.The second fundamental process of the algorithm is the lower bounding one.For each simplex  ⊆  0 created by the branching process, this process gives a lower bound LB() for the optimal value V() of the following problem LMP(), For each simplex  created by the branching process, LB() is found by solving a single linear programming LP() as follows, where  0 ,  1 , . . .,   denote the vertices of the -dimensional simplex .
During each iteration  ⩾ 0, the lower bounding process computes a lower bound LB  for the optimal value V( 0 ) of problem (LMP( 0 )).For each  ⩾ 0, this lower process bound LB  is given by The upper bounding process is the third fundamental one of the branch and bound algorithm.For each -dimensional simplex  created by the branching process, this process finds an upper bound for (LMP()).Let  ∈ R  be optimal dual variables corresponding to the first  constraints of linear program LP(S), and set  * =  * .Then, from definition of problem (DLP()), we have that  * ⩽ ,  * ⩾ 0. This implies that  * is a feasible solution to (LMP()).Therefore, an upper bound UB() of (LMP()) is ℎ( * ).In each iteration of the algorithm, this process finds an upper bound for V.For each  ⩾ 0, let  ∈ R  be optimal dual variables corresponding to the first  constraints of linear program LP(), then this upper bound UB  is given by where  is the incumbent feasible solution to the problem.

Deleting Technique.
As the branch and bound search proceeds, certain -dimensional simplices created by the algorithm are eliminated from further consideration.There are two ways occuring, either by deletion by bounding or by deletion by infeasibility.During any iteration ,  ⩾ 1, let UB  be the smallest objective function value achieved in problem (LMP) by the feasible solutions to problem (LMP(S)) thus far generated by the algorithm.A simplex  ⊆  0 is deleted by bounding when LB () ⩾ UB  (24) holds.When (30) holds, searching simplex  further will not improve upon the best feasible solution found thus far for problem (LMP).
As soon as each -dimensional simplex  is created by simplicial bisection in the algorithm, it is subjected to the deletion by infeasibility test.Let  0 ,  1 , . . .,   denote the vertices of such a simplex .If for some  ∈ {1, 2, . . ., }, either or then simplex  is said to pass the deletion by infeasibility test and it is eliminated by the algorithm from further consideration.If for each  ∈ {1, 2, . . ., }, both ( 25) and ( 26) fail to hold, then simplex  fails the deletion by infeasibility test and it is retained for further scrutiny by the algorithm.
The validity of the deletion by infeasibility test follows from the fact that if (25) or (26) holds for some , then for each  ∈ , there is no  ∈  such that This implies problem (LMP(S)) infeasible.

Branch and Bound Algorithm.
Based on the results and algorithmic process discussed in this section, the branch and bound algorithm for globally solving the LMP can be stated as follows.
Step 4 (Infeasiblity Test).Delete from R each simplex that passes the deletion by infeasiblity test.Let  represent the subset of R thereby obtained.
Step 9 (Convergence).If   = 0, then stop.UB  is an optimal value of the LMP, and   is a global -optimal solution for problem (LMP).Otherwise, set  + 1 and go to Step 2.

Convergence of the Algorithm
In this section we give a global convergence of algorithm above.By the construction of the algorithm, when the algorithm is finite, then   = 0, so that V( 0 ) ⩾ UB  + = ℎ(  )+.Since, by Proposition 5, V = V( 0 ) and since   ∈ , this implies that V ⩾ ℎ(  )+ and   is a global -optimal solution to the LMP.Thus, when the algorithm is finite, it globally solves the LMP as desired.
If the algorithm does not terminate after finitely many iterations, then it is easy to show that it generates at least one infinite nested subsequence {  } of simplices, that is, where  +1 ⊆   for all .In this case, the following result is a key to convergence in the algorithm.Theorem 9. Suppose that the Branch and Bound Algorithm is infinite, and that {  } is an infinite nested subsequence of simplices generated by the algorithm.Let  * denote any accumulation point of {  } ∞ =0 where, for each ,   ∈ R  denotes any optimal dual variables corresponding to the first  constraints of linear program (LP(  )).Then  * is a global optimal solution for problem (LMP).
Proof.Suppose that the algorithm is infinite, and let {  } be chosen as in the theorem.Then, from Horst and Tuy [35], ⋂    = { * } for some point  * ∈ R  .
Let {  } ∈ denote any such subsequence, and let  * = lim ∈   .Then, since  is closed,  * ∈ .Now, we show that  * is a global optimal solution for problem (LMP).
With Theorem 9, we can easily show two fundamental convergence properties of the algorithm as follows.
Corollary 10.Suppose that the Branch and Bound Algorithm is infinite.Then each accumulation point of {  } ∞ =0 is a global optimal solution for problem.
Proof.The proof is similar to in [34, Corollary 1], it is omitted here.
Corollary 11.Suppose that the Branch and Bound Algorithm is infinite.Then lim  → ∞ LB  = lim  → ∞ LB  = V.
Proof.The proof is similar to [34,Corollary 2], it is omitted here.

Numerical Examples
Now we give numerical experiments for the proposed global optimization algorithm to illustrate its efficiency.and bound search, finding a global optimal solution to the problem that is equivalent to the LMP.We believe that the new algorithm has advantage in several potentially practical and computational cases.Besides, numerical examples show that the proposed algorithm is feasible.