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

Abstract

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.

1. Introduction

1.1. Problem and Applications

In this paper, linear multiplicative programming problems are given by: (LMP)𝑣=minℎ(𝑥)=^𝑝_𝑖=1𝑐^𝑇_𝑖𝑥+𝑐_𝑖0𝑑^𝑇_𝑖𝑥+𝑑_𝑖0,s.t.𝑥∈𝑋=𝑥∈𝑅^𝑛∣𝐴𝑥⩽𝑏,𝑥⩾0, (1) where 𝑝⩾2, 𝑐^𝑇_𝑖=(𝑐_𝑖1,𝑐_𝑖2,…,𝑐_𝑖𝑛), 𝑑^𝑇_𝑖=(𝑑_𝑖1,𝑑_𝑖2,…,𝑑_𝑖𝑛)∈ℝ^𝑛,𝑖=1,2,…,𝑝, 𝑏^𝑇=(𝑏₁,𝑏₂,…,𝑏_𝑚)∈ℝ^𝑚, 𝑐_𝑖0,𝑑_𝑖0∈ℝ,𝑖=1,2,…,𝑝, 𝐴 is an 𝑚×𝑛 matrix, 𝑋⊆𝑅^𝑛 is a nonempty, compact convex set. We assume also that for each 𝑖=1,2,…,𝑝, 𝑑^𝑇_𝑖𝑥+𝑑_𝑖0>0. Generally, the problem (LMP) is a special case of non-convex programming problem, known to be an NP-hard even at 𝑝=1 [1].

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: 1min2𝑦^𝑇𝑄𝑦+𝑑^𝑇𝑦+𝑐,s.t.𝑦∈𝑌=𝑦∈𝑅^𝑛∣𝐴𝑦⩽𝑏,𝑦⩾0, (2) where 𝑄 is an 𝑛×𝑛 symmetric matrix of rank 𝑝,𝑑∈𝑅^𝑛 and 𝑐∈𝑅. From Tuy [3], there exist linearly independent sets of 𝑛-dimensional vectors, {𝑣₁,𝑣₂,…,𝑣_𝑝} and {𝑤₁,𝑤₂,…,𝑤_𝑝}, such that, for all 𝑦∈𝑅^𝑛, 12𝑦^𝑇1𝑄𝑦=2^𝑝_𝑖=1𝑣^𝑇_𝑖𝑦𝑤^𝑇_𝑖𝑦. (3) 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 1min2𝑏^𝑇𝑦+𝑑^𝑇𝑧+𝑦^𝑇𝑄𝑧,s.t.𝑦∈𝑌=𝑦∈𝑅^𝑚∣𝐶𝑦⩽𝑐,𝑦⩾0,𝑧∈𝑍=𝑧∈𝑅^𝑛∣𝐸𝑧⩽𝑒,𝑧⩾0, (4) where 𝑏∈𝑅^𝑚, 𝑑∈𝑅^𝑛, 𝑐∈𝑅^𝑞, 𝑒∈𝑅^𝑟, 𝑄 is an 𝑚×𝑛 matrix of rank 𝑝, 𝐶 and 𝐸 are 𝑞×𝑚 matrix and 𝑟×𝑛 matrix, respectively. From Konno and Yajima [8], by using a constructive procedure, it can be written in the form 1min2𝑏^𝑇𝑦+𝑑^𝑇𝑧+^𝑝_𝑖=1𝑣^𝑇_𝑖𝑦𝑤^𝑇_𝑖𝑧,s.t.𝑦∈𝑌,𝑧∈𝑍, (5) where 𝑣₁,𝑣₂,…,𝑣_𝑝∈𝑅^𝑚 and 𝑤₁,𝑤₂,…,𝑤_𝑝∈𝑅^𝑛. The latter is a special case of the LMP with 𝑥^𝑇=[𝑦^𝑇,𝑧^𝑇], ℎ(𝑥)=𝑏^𝑇𝑦+𝑑^𝑇𝑧+∑^𝑝_𝑖=1(𝛼^𝑇_𝑖𝑥)(𝛽^𝑇_𝑖𝑥), 𝛼^𝑇_𝑖=[𝑣^𝑇_𝑖,0^𝑇] and 𝛽^𝑇_𝑖=[0^𝑇,𝑤^𝑇_𝑖], 𝛼_𝑖,𝛽_𝑖∈𝑅^𝑚+𝑛,𝑖=1,2,…,𝑝, and 𝑋={(𝑦,𝑧)∈𝑅^𝑚+𝑛∣𝐶𝑦⩽𝑐,𝐸𝑧⩽𝑒,𝑦,𝑧⩾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 (Q)1min2𝑏^𝑇𝑦,s.t.𝑦∈𝑌=𝑦∈𝑅^𝑚,𝑦∣𝐶𝑦⩽𝑐,𝑦⩾0_𝑖∈{0,1},𝑖=1,2,…,𝑛, (6) where 𝑏∈𝑅^𝑚, 𝑐∈𝑅^𝑝,  𝐶 is 𝑝×𝑚 matrix. From Raghavachari [13], for 𝑀>0 sufficiently large, 𝑦^∗ is an optimal solution to problem (Q) if and only if 𝑦^∗ is an optimal solution to the problem ()1QCmin2𝑏^𝑇𝑦+𝑀^𝑛_𝑖=1𝑦_𝑖1−𝑦_𝑖,s.t.𝑦∈𝑌=𝑦∈𝑅^𝑚∣𝐶𝑦⩽𝑐,𝑦⩾0,0⩽𝑦_𝑖⩽1,𝑖=1,2,…,𝑛. (7) 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].

1.2. Purpose and Content

The LMP is a global optimization problem. In the past 20 years, many algorithms have been proposed globally to a multiplicative programming one. The methods can be classified as parameterization based methods [2, 15–17], outer-approximation and branch-and-bound methods [18–25], vertex enumeration methods [26, 27], a method based on image space analysis [28], a primal and dual simplex method [29], an outcome-space cutting plane method [30], heuristic methods [31, 32], and decomposition method [33].

In this paper, a simplicial branch and bound duality-bounds 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 𝑝 in the algorithm, where 𝑝 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 𝑝, where 𝑝 is the number of terms in the objective function of problem (LMP), rather than in the decision space ℝ^𝑛. 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.

2. Preliminaries

In this section, we firstly show how to convert the LMP into an equivalent nonconvex programming (LMP(𝑆⁰)) by introducing a 𝑝-dimension vector 𝑦 for finding a simplex. Then, for each 𝑝-dimensional simplex 𝑆 created by the branching process, the lower bound LB(𝑆) can be found by solving an ordinary linear program by using the Lagrangian weak duality theorem of nonlinear programming.

2.1. Initial Simplex

To globally solve the LMP, the branch and bound algorithm to be developed will be used for searching values of 𝑑^𝑇_𝑖𝑥+𝑑_𝑖0,𝑖=1,2,…,𝑝, at optimality. For each 𝑖=1,2,…,𝑝, let 𝑙_𝑖=min_𝑥∈𝑋{𝑑^𝑇_𝑖𝑥+𝑑_𝑖0}, and 𝛾_𝑖=max_𝑥∈𝑋{𝑑^𝑇_𝑖𝑥+𝑑_𝑖0}. By introducing additional variable vector 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑝)∈ℝ^𝑝, we construct a set 𝑌 as follows: 𝑌=𝑦∈ℝ^𝑝∣𝑑^𝑇_𝑖𝑥+𝑑_𝑖0−𝑦_𝑖,⩽0,𝑖=1,2,…,𝑝,𝑦∈𝐷,forsome𝑥∈𝑋 (8) where 𝐷≜{𝑦∈ℝ^𝑝∣𝑙_𝑖⩽𝑦_𝑖⩽𝛾_𝑖,𝑖=1,2,…,𝑝}.

In order to construct simplex 𝑆⁰, denoting 𝛾=min_𝑥∈𝑋∑^𝑝_𝑖=1(𝑑^𝑇_𝑖𝑥+𝑑_𝑖0). Let 𝑆⁰∈𝑅^𝑝 be the convex hull of 𝑣⁰,𝑣¹,…,𝑣^𝑝, where 𝑣⁰_𝑗=𝛾_𝑗,𝑗=1,2,…,𝑛, and, for each 𝑗=1,2,…,𝑛, 𝑣^𝑖_𝑗=𝛾_𝑗if𝑗≠𝑖,𝛾−∑_𝑗≠𝑖𝛾_𝑗if𝑗=𝑖. (9) Notice that either 𝛾=∑^𝑝_𝑖=1𝛾_𝑖 or 𝛾<∑^𝑝_𝑖=1𝛾_𝑖, and for any 𝑗=1,…,𝑝,(𝑣^𝑗−𝑣⁰)^𝑇=(0,0,…,𝛾−∑^𝑝_𝑖=1𝛾_𝑖,…,0), where 𝛾−∑^𝑝_𝑖=1𝛾_𝑖 is the 𝑗th component of 𝑣^𝑗−𝑣⁰. We can easily show the following theorem (see Benson [34]).

Theorem 1. 𝑆⁰ is either a single point or 𝑆⁰ is a 𝑝-dimensional simplex. In either case, 𝑆⁰⊇𝑌.

Remark 2. From Theorem 1, 𝑆⁰ is either a single point or 𝑆⁰ is a 𝑝-dimensional simplex. Notice that if 𝑆⁰ is a single one, that is, 𝑆⁰={𝑦^∗} is a single one, then 𝑋={𝑥^∗} is a single point set and 𝑥^∗ is a global optimal solution to the LMP, where 𝑥^∗ is any optimal solution to the linear program obtained by setting 𝑥 equal to 𝑥^∗ in problem (LMP(𝑆⁰)). Therefore, we will assume in the remainder of this article that 𝑆⁰ is a 𝑝-dimensional simplex.

2.2. Equivalent Problem

For any simplex 𝑆⊂ℝ^𝑝, define the problem (LMP(𝑆))𝑣(𝑆)=minℎ𝑥,𝑦=^𝑝_𝑖=1𝑐^𝑇_𝑖𝑥+𝑐_𝑖0𝑦_𝑖,s.t.𝑑^𝑇_𝑖𝑥+𝑑_𝑖0−𝑦_𝑖⩽0,𝑖=1,2,…,𝑝,𝐴𝑥−𝑏⩽0,𝑥⩾0,𝑦∈𝑆. (10)

In order to solve the LMP, the branch and bound algorithm is used to solves problem (LMP(𝑆⁰)) instead. The validity of solving problem (LMP(𝑆⁰)), in order to solve the LMP, follows from the next result.

Theorem 3. If (𝑥^∗,𝑦^∗) is a global optimal solution for problem (LMP(𝑆⁰)), then 𝑥^∗ is a global optimal solution for problem (LMP). If 𝑥^∗ is a global optimal solution for problem (LMP), then (𝑥^∗,𝑦^∗) is a global optimal solution for problem (LMP(𝑆⁰)), where 𝑦^∗_𝑖=𝑑^𝑇_𝑖𝑥^∗+𝑑_𝑖0,𝑖=1,2,…,𝑝. The global optimal values 𝑣 and 𝑣(𝑆⁰) of problems (LMP) and (LMP(𝑆⁰)), respectively, are equal.

Proof. By using the fact that 𝑆⁰⊇𝑌, the proof of this theorem follows easily from the defintions of problem (LMP(𝑆⁰)).

2.3. Duality Bound

For each 𝑝-dimensional simplex 𝑆 created by the branching process, the algorithm computes a lower bound LB(𝑆) for the optimal value 𝑣(𝑆) of problem (LMP(𝑆)). The next theorem shows that, by using the Lagrangian weak duality theorem of nonlinear programming, the lower bound LB(𝑆) can be found by solving an ordinary linear programming.

Theorem 4. Let 𝑆⊆ℝ^𝑝 be a 𝑝-dimensional simplex with vertices 𝑦⁰,𝑦¹,…,𝑦^𝑝, and let 𝐽={0,1,2,…,𝑝}. Then LB(𝑆)⩽𝑣(𝑆), where LB(𝑆) is the optimal value of the linear programming problem (𝐿𝑃(𝑆))𝐿𝐵(𝑆)=max^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+𝑡𝑠.𝑡.−^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖−𝐴^𝑇𝜆⩽𝐶,_𝑝𝑖=1𝑦^𝑗_𝑖𝜃_𝑖+𝑡⩽^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖0,𝑗=0,1,2,…,𝑝,𝜃⩾0,𝜆⩾0,𝑡𝑓𝑟𝑒𝑒, (11) where 𝐶∈ℝ^𝑛 and 𝐶=(min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖1), min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖2),…,min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖𝑛))^𝑇.

Proof. By the definition of 𝑣(𝑆) and the weak duality theorem of Lagrangian duality, 𝑣(𝑆)⩾LB(𝑆), where LB(𝑆)=max_{𝜃⩾0𝜆⩾0}min_{𝑦∈𝑆𝑥⩾0}^𝑝_𝑖=1𝑐^𝑇_𝑖𝑥+𝑐_𝑖0𝑦_𝑖+^𝑝_𝑖=1𝜃_𝑖𝑑^𝑇_𝑖𝑥+𝑑_𝑖0−𝑦_𝑖+𝜆^𝑇𝐴𝑥−𝜆^𝑇𝑏=max_{𝜃⩾0𝜆⩾0}^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+min_{𝑦∈𝑆𝑥⩾0}^𝑝_𝑖=1𝑦_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖+𝜆^𝑇+𝐴,𝑥_𝑝𝑖=1𝑦_𝑖𝑐_𝑖0−𝜃_𝑖. (12)
Since min_𝑦∈𝑆^𝑝_𝑖=1𝑦_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖+𝜆^𝑇=𝐴,𝑥0,∑^𝑝_𝑖=1𝑦_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖+𝜆^𝑇𝐴⩾0,∀𝑥⩾0,−∞,otherwise, (13) it follows that, LB(𝑆)=max^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+min_𝑦∈𝑆^𝑝_𝑖=1𝑦_𝑖𝑐_𝑖0−𝜃_𝑖s.t.^𝑝_𝑖=1𝑦_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖+𝜆^𝑇𝐴⩾0,∀𝑦∈𝑆,𝜆⩾0,𝜃⩾0. (14) Since 𝑆 is a compact polyhedron with extreme points 𝑦^𝑗,𝑗=0,1,2,…,𝑝, for each 𝜃∈ℝ^𝑝 and 𝜆⩾0, ∑^𝑝_𝑖=1(𝑦_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖)+𝜆^𝑇𝐴⩾0 holds for all 𝑦∈𝑆 if and only if it holds for all 𝑦∈{𝑦⁰,𝑦¹,…,𝑦^𝑝}. So, for all 𝑗∈𝐽, we can get ∑^𝑝_𝑖=1(𝑦^𝑗_𝑖𝑐^𝑇_𝑖+𝜃_𝑖𝑑^𝑇_𝑖)+𝜆^𝑇𝐴⩾0, that is, −^𝑝_𝑖=1𝜃_𝑖𝑑^𝑇_𝑖−𝜆^𝑇𝐴⩽^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐^𝑇_𝑖,∀𝑗∈𝐽. (15) Notice that for all 𝑗∈𝐽, the left-hand-side of (15) is the same linear function of 𝜃 and 𝜆, then (15) is equivalent to −∑^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖−𝐴^𝑇𝜆⩽𝐶, where 𝐶∈ℝ^𝑛 and 𝐶=(min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖1),…,min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖𝑛))^𝑇.
Therefore, (LB(𝑆))=max^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+min_𝑦∈𝑆^𝑝_𝑖=1𝑦_𝑖𝑐_𝑖0−𝜃_𝑖s.t.−^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖−𝐴^𝑇𝜆⩽𝐶,𝜃⩾0,𝜆⩾0. (16) That is, (LB(𝑆))=max^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+𝑡s.t.−^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖−𝐴^𝑇𝜆⩽𝐶,_𝑝𝑖=1𝑦_𝑖𝑐_𝑖0−𝜃_𝑖⩾𝑡,𝜃⩾0,𝜆⩾0. (17) For any 𝜃∈ℝ^𝑝, 𝑦_𝑖(𝑐_𝑖0−𝜃_𝑖) is a linear function. Because simplex 𝑆 is a compact polyhedron with extreme points 𝑦⁰,𝑦¹,…,𝑦^𝑝, this implies for any 𝜃∈ℝ^𝑝, ∑^𝑝_𝑖=1𝑦_𝑖(𝑐_𝑖0−𝜃_𝑖)⩾𝑡 holds if and only if ^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖0−𝜃_𝑖−𝑡⩾0,𝑗=0,1,2,…,𝑝. (18) The proof is complete.

Proposition 5. Let 𝑆¹,𝑆²⊆ℝ^𝑝 be a 𝑝-dimensional subsimplices of 𝑆 formed by the branching process such that 𝑆¹⊆𝑆²⊆𝑆⁰. Then(i)𝐿𝐵(𝑆¹)⩾𝐿𝐵(𝑆²).(ii)Let 𝑆⊆ℝ^𝑝be a 𝑝-dimensional simplex with vertices 𝑦⁰,𝑦¹,…,𝑦^𝑝. Then 𝐿𝐵>−∞.

Proof. The proof is similar to [34, Proposition 3], it is omitted here.

Remark 6. From part (ii) of Proposition 5, for any 𝑝-dimensional simplex 𝑆 created by the algorithm during the branch and bound search, the duality bounds-based lower bound LB(𝑆) for the optimal value 𝑣(𝑆) of problem (LMP(𝑆)) is either finite or equal to +∞. When LB(𝑆)=+∞, problem (LMP(𝑆)) is infeasible and, as we shall see, 𝑆 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(𝑆)). For each 𝑝-dimensional simplex 𝑆 generated by the algorithm such that LB(𝑆) is finite, the algorithm generates a feasible solution 𝑤 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(𝑆)) that are solved by the algorithm, as given in the following result.

Proposition 7. Let 𝑆⊆ℝ^𝑝 be a 𝑝-dimensional simplex with vertices 𝑦⁰,𝑦¹,…,𝑦^𝑝, and suppose that LB(𝑆)≠+∞. Let 𝑤∈ℝ^𝑛 be optimal dual variables corresponding to the first 𝑛 constraints of linear program LP(𝑆). Then 𝑤 is a feasible solution for problem (LMP).

Proof. The dual linear program to problem (LP(S)) is DLP(𝑆)LB(𝑆)=min𝐶^𝑇𝑤+^𝑝_𝑗=0𝑞_𝑗^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖0s.t.−𝑑^𝑇_𝑖𝑤+^𝑝_𝑗=0𝑦^𝑗_𝑖𝑞_𝑗⩾𝑑_𝑖0,𝑖=1,2,…,𝑝,−𝐴𝑤⩾−𝑏,^𝑝_𝑗=0𝑞_𝑗=1,𝑤⩾0. (19) The constraints of problem (DLP(S)) imply that 𝐴𝑤⩽𝑏, 𝑤⩾0.

3. Global Optimizing Algorithm

To globally solve problem (LMP(𝑆)), 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.

3.1. Branching Rule

The branch and bound approach is based on partitioning the 𝑝-dimensional simplex 𝑆⁰ into smaller subsimplices that are also of dimension 𝑝, 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(𝑆⁰)) 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 𝑆=𝑆⁰ that cannot yet be excluded from consideration in the search for a global optimal solution for problem (LMP(𝑆)). The initial partition 𝑄₁ consists simply of 𝑆, since at the beginning of the branch and bound procedure, no portion of 𝑆 can as yet be excluded from consideration.

During iteration 𝑘 of the algorithm, 𝑘⩾1, the branching process is used to help create a new partition 𝑄_𝑘+1. First, a screening procedure is used to remove any rectangle from 𝑄_𝑘 that can, at this point of the search, be excluded from further consideration, and 𝑄_𝑘+1 is temporarily set equal to the set of simplices that remain. Later in iteration 𝑘, a rectangle 𝑆^𝑘 in 𝑄_𝑘+1 is identified for further examination. The branching process is then evoked to subdivide 𝑆^𝑘 into two subsimplices 𝑆^𝑘₁,𝑆^𝑘₂. This subdivision is accomplished by a process called simplicial bisection.

Definition 8 (see [35]). Let 𝑆 be a 𝑝-dimensional simplex with vertex set {𝑣⁰,𝑣¹,…,𝑣^𝑝}. Let 𝑤 be the midpoint of any of the longest edges [𝑣^𝑟,𝑣^𝑡] of 𝑆. Then {𝑆¹,𝑆²} is called a 𝑠𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑎𝑙𝑏𝑖𝑠𝑒𝑐𝑡𝑖𝑜𝑛 of 𝑆, where the vertex set of 𝑆¹ is {𝑣⁰,𝑣¹,…,𝑣^𝑟−1,𝑤,𝑣^𝑟+1,…,𝑣^𝑝} and the vertex set of 𝑆² is {𝑣⁰,𝑣¹,…,𝑣^𝑡−1,𝑤,𝑣^𝑡+1,…,𝑣^𝑝}.

3.2. Lower Bound and Upper Bound

The second fundamental process of the algorithm is the lower bounding one. For each simplex 𝑆⊆𝑆⁰ created by the branching process, this process gives a lower bound LB(𝑆) for the optimal value 𝑣(𝑆) of the following problem LMP(𝑆), 𝑣(𝑆)=minℎ(𝑥)=^𝑝_𝑖=1𝑐^𝑇_𝑖𝑥+𝑐_𝑖0𝑦_𝑖,s.t.𝑑^𝑇_𝑖𝑥+𝑑_𝑖0−𝑦_𝑖⩽0𝐴𝑥−𝑏⩽0,𝑥⩾0,𝑦∈𝑆. (20) For each simplex 𝑆 created by the branching process, LB(𝑆) is found by solving a single linear programming LP(𝑆) as follows, (LB(𝑆))=max^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖0−𝜆^𝑇𝑏+𝑡^∗s.t.−^𝑝_𝑖=1𝜃_𝑖𝑑_𝑖−𝐴^𝑇𝜆⩽𝐶,_𝑝𝑖=1𝑦^𝑗_𝑖𝜃_𝑖+𝑡⩽^𝑝_𝑖=1𝑦^𝑗_𝑖𝑐_𝑖0,𝑗=0,1,2,…,𝑝,𝜃⩾0,𝜆⩾0,𝑡free, (21) where 𝑦⁰,𝑦¹,…,𝑦^𝑝 denote the vertices of the 𝑝-dimensional simplex 𝑆.

During each iteration 𝑘⩾0, the lower bounding process computes a lower bound LB_𝑘 for the optimal value 𝑣(𝑆⁰) of problem (LMP(𝑆⁰)). For each 𝑘⩾0, this lower process bound LB_𝑘 is given by LB_𝑘=minLB(𝑆)∣𝑆∈𝑄_𝑘. (22)

The upper bounding process is the third fundamental one of the branch and bound algorithm. For each 𝑝-dimensional simplex 𝑆 created by the branching process, this process finds an upper bound for (LMP(𝑆)). Let 𝑤∈ℝ^𝑛 be optimal dual variables corresponding to the first 𝑛 constraints of linear program LP(S), and set 𝑥^∗=𝑤^∗. Then, from definition of problem (DLP(𝑆)), we have that 𝐴𝑥^∗⩽𝑏,𝑥^∗⩾0. This implies that 𝑥^∗ is a feasible solution to (LMP(𝑆)). Therefore, an upper bound UB(𝑆) of (LMP(𝑆)) is ℎ(𝑥^∗). In each iteration of the algorithm, this process finds an upper bound for 𝑣. For each 𝑘⩾0, let 𝑤∈ℝ^𝑛 be optimal dual variables corresponding to the first 𝑛 constraints of linear program LP(𝑆), then this upper bound UB_𝑘 is given by UB_𝑘=ℎ(𝑥), (23) where 𝑥 is the incumbent feasible solution to the problem.

3.3. Deleting Technique

As the branch and bound search proceeds, certain 𝑝-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 𝑘, 𝑘⩾1, let UB_𝑘 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 𝑆⊆𝑆⁰ is deleted by bounding when LB(𝑆)⩾UB_𝑘 (24) holds. When (30) holds, searching simplex 𝑆 further will not improve upon the best feasible solution found thus far for problem (LMP).

As soon as each 𝑝-dimensional simplex 𝑆 is created by simplicial bisection in the algorithm, it is subjected to the deletion by infeasibility test. Let 𝑦⁰,𝑦¹,…,𝑦^𝑝 denote the vertices of such a simplex 𝑆. If for some 𝑖∈{1,2,…,𝑝}, either min𝑦⁰_𝑖,𝑦¹_𝑖,…,𝑦^𝑝_𝑖>𝐿_𝑖, (25) or max𝑦⁰_𝑖,𝑦¹_𝑖,…,𝑦^𝑝_𝑖<𝑙_𝑖, (26) then simplex 𝑆 is said to pass the deletion by infeasibility test and it is eliminated by the algorithm from further consideration. If for each 𝑖∈{1,2,…,𝑝}, both (25) and (26) fail to hold, then simplex 𝑆 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 𝑖, then for each 𝑦∈𝑆, there is no 𝑥∈𝑋 such that 𝑑^𝑇_𝑖𝑥+𝑑_𝑖0−𝑦_𝑖=0. (27) This implies problem (LMP(S)) infeasible.

3.4. 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 1 (Initialization). (0.1) Initialize the iteration counter 𝑘∶=0; the set of all active nodes 𝑄₀∶={𝑆⁰}; the upper bound UB₀=+∞.(0.2) Solve linear program (LP(𝑆⁰)) for its finite optimal value LB(𝑆⁰). Let 𝑤∈ℝ^𝑛 be optimal dual variables corresponding to the first 𝑛 constraints of linear program LP(𝑆⁰). Set 𝑥⁰=𝑤,UB₀=ℎ(𝑥⁰),LB₀=LB(𝑆⁰). Set 𝑘=1, and go to iteration 𝑘.
Main Step (at iteration 𝑘)

Step 2 (Termination). If UB_𝑘−1−LB_𝑘−1⩽𝜀, where 𝜀>0 is some accuracy tolerance, then stop, and 𝑥^𝑘−1 is a global 𝜀-optimal solution for problem (LMP) and 𝑣=LB_𝑘−1. Otherwise, set 𝑥^𝑘=𝑥^𝑘−1, LB_𝑘=LB_𝑘−1, UB_𝑘=UB_𝑘−1. Go to Step 3.

Step 3 (Branching). Let 𝑆^𝑘∈𝑄_𝑘−1 satisfy 𝑆^𝑘∈argmin{LB(𝑆)∣𝑆∈𝑄_𝑘−1}. Use simplicial bisection to divide 𝑆^𝑘 into 𝑆^𝑘₁ and 𝑆^𝑘₂. Let 𝑅={𝑆^𝑘₁,𝑆^𝑘₂}.

Step 4 (Infeasiblity Test). Delete from 𝑅 each simplex that passes the deletion by infeasiblity test. Let 𝑅 represent the subset of 𝑅 thereby obtained.

Step 5 (Fathoming). For each new sub-simplex 𝑆∈𝑅, compute optimal value LB(𝑆) of linear program (LP(𝑆⁰)). If LB(S) is finite, let 𝑤∈ℝ^𝑛 be optimal dual variables corresponding to the first (𝑝+1)𝑛 constraints of linear program (LP(S)).

Step 6 (Updating upper Bound). If ℎ(𝑤)<ℎ(𝑥^𝑘), set 𝑥^𝑘=𝑤, UB_𝑘=ℎ(𝑥^𝑘).

Step 7 (New Partition). Let 𝑄_𝑘={𝑄_𝑘−1⧵{𝑆^𝑘}}⋃𝑅.

Step 8 (Deletion). 𝑄_𝑘=𝑄_𝑘⧵{𝑆∶LB(𝑆)−UB_𝑘⩾𝜀}.

Step 9 (Convergence). If 𝑄_𝑘=∅, then stop. UB_𝑘 is an optimal value of the LMP, and 𝑥^𝑘 is a global 𝜀-optimal solution for problem (LMP). Otherwise, set 𝑘+1 and go to Step 2.

4. 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 𝑄_𝑘=∅, so that 𝑣(𝑆⁰)⩾UB_𝑘+𝜀=ℎ(𝑥^𝑘)+𝜀. Since, by Proposition 5, 𝑣=𝑣(𝑆⁰) and since 𝑥^𝑘∈𝑋, this implies that 𝑣⩾ℎ(𝑥^𝑘)+𝜀 and 𝑥^𝑘 is a global 𝜀-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 {𝑆^𝑟} of simplices, that is, where 𝑆^𝑟+1⊆𝑆^𝑟 for all 𝑟. 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 {𝑆^𝑟} is an infinite nested subsequence of simplices generated by the algorithm. Let 𝑤^∗ denote any accumulation point of {𝑤^𝑟}^∞_𝑟=0 where, for each 𝑟,𝑤^𝑟∈ℝ^𝑛 denotes any optimal dual variables corresponding to the first 𝑛 constraints of linear program (LP(𝑆^𝑟)). Then 𝑤^∗ is a global optimal solution for problem (LMP).

Proof. Suppose that the algorithm is infinite, and let {𝑆^𝑟} be chosen as in the theorem. Then, from Horst and Tuy [35], ⋂_𝑟𝑆^𝑟={𝑦^∗} for some point 𝑦^∗∈ℝ^𝑝.
For each simplex 𝑆^𝑟, denote its vertices by 𝑦^𝑟,𝑗,𝑗=0,1,…,𝑝, denote (𝑤^𝑟,𝑞^𝑟₀,𝑞^𝑟₁,…,𝑞^𝑟_𝑝) as an optimal dual solution to linear program (LP(𝑆^𝑟)). Set 𝑈={𝑞∈ℝ^𝑝+1∣∑^𝑝_𝑗=0𝑞_𝑗=1,𝑞_𝑗⩾0,𝑗=0,1,…,𝑝}, then 𝑈 is compact and for each 𝑟, 𝑞^𝑟≜(𝑞^𝑟₀,𝑞^𝑟₁,…,𝑞^𝑟_𝑝)∈𝑈.
By Proposition 7, for each 𝑟, 𝑤^𝑟∈𝑋. Since 𝑋 is bounded, this implies that {𝑤^𝑟} has at least one convergent subsequence. Let {𝑤^𝑟}_𝑟∈𝑅 denote any such subsequence, and let 𝑤^∗=lim_𝑟∈𝑅𝑤^𝑟. Then, since 𝑋 is closed, 𝑤^∗∈𝑋. Now, we show that 𝑤^∗ is a global optimal solution for problem (LMP).
Since 𝑈 is bounded, then there exists an infinite subsequence 𝑅^′ of 𝑅 such that for each 𝑗=0,1,2,…,𝑝,lim_{𝑟∈𝑅′}𝑞^𝑟=𝑞^∗. We have 𝑞^∗∈𝑈, since 𝑈 is close. Notice that since lim_𝑟∈𝑅𝑤^𝑟=𝑤^∗, lim_{𝑟∈𝑅′}𝑤^𝑟=𝑤^∗. Also, since ⋂_𝑟𝑆^𝑟={𝑦^∗},lim_{𝑟∈𝑅′}𝑦^𝑟,𝑗=𝑦^∗ for each 𝑗=0,1,2,…,𝑝. Set (𝐶^𝑟)^𝑇=(min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦_𝑖^𝑟,𝑗𝑐_𝑖1), min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦_𝑖^𝑟,𝑗𝑐_𝑖2),…,min_𝑗∈𝐽(∑^𝑝_𝑖=1𝑦_𝑖^𝑟,𝑗𝑐_𝑖𝑛)). Then, we have lim_{𝑟∈𝑅′}𝐶^𝑟𝑇=𝐶^∗𝑇=min_𝑗∈𝐽^𝑝_𝑖=1𝑦^∗_𝑖𝑐_𝑖1,min_𝑗∈𝐽^𝑝_𝑖=1𝑦^∗_𝑖𝑐_𝑖2,…,min_𝑗∈𝐽^𝑝_𝑖=1𝑦^∗_𝑖𝑐_𝑖𝑛=𝑦^∗_𝑖𝑐_𝑖1,𝑦^∗_𝑖𝑐_𝑖2,…,𝑦^∗_𝑖𝑐_𝑖𝑛. (28)
From Theorems 3–4, Proposition 5, and the algorithm, for each 𝑟,𝑟^′∈𝑅^′ such that 𝑟^′>𝑟, LB(𝑆^𝑟)⩽LB(𝑆^𝑟′)⩽𝑣. Therefore, for some infinite subsequence 𝑅^′′ of 𝑅^′, lim_{𝑟∈𝑅′′}LB(𝑆^𝑟) exists and satisfies lim_{𝑟∈𝑅′′}LB(𝑆^𝑟)⩽𝑣.
Form an objective function of problem (DLP(𝑆^𝑟)), we obtain the equation LB𝑆^𝑟=𝐶^𝑟𝑇𝑤^𝑟+^𝑝_𝑗=0𝑞^𝑟_𝑗^𝑝_𝑖=1𝑦_𝑖^𝑟,𝑗𝑐_𝑖0,∀𝑟∈𝑅^′′. (29) Taking limits over 𝑟∈𝑅^′′ in this equation yields lim_{𝑟∈𝑅′′}LB𝑆^𝑟=𝐶^∗𝑇𝑤^∗+^𝑝_𝑗=0𝑞^∗_𝑗^𝑝_𝑖=1𝑦^∗_𝑖𝑐_𝑖0=𝐶^∗𝑇𝑤^∗+^𝑝_𝑖=1𝑦^∗_𝑖𝑐_𝑖0=^𝑝_𝑖=1𝑐^𝑇_𝑖𝑤^∗+𝑐_𝑖0𝑦^∗_𝑖⩽𝑣, (30) where the first equation follows from lim_{𝑟∈𝑅′′}𝑤^𝑟=𝑤^∗, lim_{𝑟∈𝑅′}(𝐶^𝑟)^𝑇=𝐶^∗𝑇,lim_{𝑟∈𝑅′′}𝑦^𝑟,𝑗=𝑦^∗,𝑗=0,1,…,𝑝, and lim_{𝑟∈𝑅′′}𝑞^𝑟=𝑞^∗, the second equation holds because 𝑞^∗∈𝑈, the third equation follows form 𝐶^∗𝑇=(𝑦^∗_𝑖𝑐_𝑖1,𝑦^∗_𝑖𝑐_𝑖2,…,𝑦^∗_𝑖𝑐_𝑖𝑛), and the inequality holds because lim_{𝑟∈𝑅′′}LB(𝑆^𝑟)⩽𝑣.
We will now show that (𝑥,𝑦)=(𝑤^∗,𝑦^∗) is a feasible solution to problem LMP(𝑆⁰). First, notice that since 𝑆⁰ is closed, 𝑦^∗∈𝑆⁰, and, since 𝑤^∗∈𝑋,𝐴𝑤^∗⩽𝑏 and 𝑤^∗⩾0. Let 𝑖∈{1,2,…,𝑝}, For each 𝑟∈𝑅^′′, from constraint 𝑖 of problem (DLP(𝑆^𝑟)), −𝑑^𝑇_𝑖𝑤^𝑟+∑^𝑝_𝑗=0𝑦₁^𝑟,𝑗𝑞^𝑟_𝑗=𝑑_𝑖0. Taking limits over 𝑟∈𝑅^′′ in the above equation, we obtain 𝑑_𝑖0=−𝑑^𝑇_𝑖𝑤^∗+^𝑝_𝑗=0𝑦^∗_𝑖𝑞^∗_𝑗=−𝑑^𝑇_𝑖𝑤^∗+𝑦^∗_𝑖^𝑝_𝑗=0𝑞^∗_𝑗=−𝑑^𝑇_𝑖𝑤^∗+𝑦^∗_𝑖, (31) where the first equation follows from lim_{𝑟∈𝑅′′}𝑤^𝑟=𝑤^∗,lim_{𝑟∈𝑅′′}𝑦^𝑟,𝑗=𝑦^∗,𝑗=0,1,…,𝑝, and lim_{𝑟∈𝑅′′}𝑞^𝑟=𝑞^∗, the last equation is due to the fact that 𝑞^∗∈𝑈. By the choice of 𝑖, since 𝑦^∗∈𝑆⁰,𝐴𝑤^∗⩽𝑏 and 𝑤^∗⩾0, (31) implies that (𝑥,𝑦)=(𝑤^∗,𝑦^∗) is a feasible solution for problem LMP(𝑆⁰). Therefore, by Theorem 3, ^𝑝_𝑖=1𝑐^𝑇_𝑖𝑤^∗+𝑐_𝑖0𝑦^∗_𝑖⩾𝑣. (32) From (30) and Theorem 3, (32) implies that the feasible solution (𝑤^∗,𝑦^∗) to problem LMP(𝑆⁰) is a global optimal solution to problem LMP(𝑆⁰). Therefore, by Theorem 3, 𝑤^∗ is a global optimal solution to the 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 {𝑥^𝑘}^∞_𝑘=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_𝑘→∞LB_𝑘=lim_𝑘→∞LB_𝑘=𝑣.

Proof. The proof is similar to [34, Corollary 2], it is omitted here.

5. Numerical Examples

Now we give numerical experiments for the proposed global optimization algorithm to illustrate its efficiency.

Example 12. min𝐺(𝑥)=𝑥₁+2𝑥₂−2−2𝑥₁−𝑥₂++33𝑥₁−2𝑥₂+3𝑥₁−𝑥2−1s.t.−2𝑥₁+3𝑥₂⩽6,4𝑥₁−5𝑥₂⩽8,5𝑥₁+3𝑥₂⩽15,−4𝑥₁−3𝑥₂𝑥⩽−12,₁⩾0,𝑥₂⩾0. (33)
Prior to initiating the algorithm, we use Theorem 1 to find a simplex 𝑆⁰ containing 𝑋. By solving three linear programs, we determine that the vertices 𝑆⁰ are given by 𝑣⁰=𝑦⁰=(−1.6667,1.1351),𝑣¹=𝑦¹=(3.8649,1.1351),𝑣²=𝑦²=(−1.6667,6.6667). Also, for each 𝑖=1,2, we compute the minimum 𝑙_𝑖 and the maximum 𝐿_𝑖 of ⟨𝑑_𝑖,𝑥⟩+𝑑_𝑖0 over 𝑋 for use in the deletion by infeasiblity test (25) and (26). This entails solving four linear programs and get 𝑙₁=−2.8919,𝑙₂=−2.6667,𝐿₁=−1.6667,𝐿₂=1.1351.
Initialization. By solving the following linear programming (LP(𝑆⁰)), we get LB(𝑆⁰)=−49.7086 and the dual variables 𝑤⁰=(1.2857,2.8571), max3𝜃₁−𝜃₂−6𝜆₁−8𝜆₂−15𝜆₃+12𝜆₄+𝑡s.t.−2𝜃₁+𝜃₂−2𝜆₁+4𝜆₂+5𝜆₃−4𝜆₄⩾−1.7386,−𝜃₁−𝜃₂+3𝜆₁−5𝜆₂+3𝜆₃−3𝜆₄⩾16.6668,−1.6667𝜃₁+1.1351𝜃₂+𝑡⩽10.71153.8649𝜃₁+1.1351𝜃₂+𝑡⩽−0.35165,−1.6667𝜃₁+6.6667𝜃₂+𝑡⩽46.6669,𝜃⩾0,𝜆⩾0,𝑡free. (34) We set 𝑥⁰=𝑤^∗,UB₀=−16.224,LB₀=−49.7086,𝐺⁰={𝑆⁰}, and 𝑘=1. Select the convergence tolerance to be equal to 𝜀=10⁻².
Iteration 1. Since UB₀−LB₀<𝜀, 𝑆⁰ is split by simplicial bisection into 𝑆¹₁ and 𝑆¹₂ where the vertices of 𝑆¹₁ are (−1.6667,1.1351),(1.0991,3.9009),(−1.6667,6.6667) and the vertices of 𝑆¹₂ are (−1.6667,1.1351),(3.8649,1.1351),(1.0991,3.9009). Neither 𝑆¹₁ nor 𝑆¹₂ is deleted by the deletion by infeasibility test. By solving problem (LP(𝑆¹₁)), we obtain the lower bound LB(𝑆¹₁)=−38.6454 and the dual variable 𝑤=(1.2857,2.8571). Since ℎ(𝑤)=−16.224=ℎ(𝑥¹), where 𝑥¹=𝑥⁰, we do not update 𝑦¹ and UB₁=UB₀. By solving problem (LP(𝑆¹₂)), we obtain the lower bound LB(𝑆¹₂)=−18.09944 and the dual variable 𝑤=(1.2857,2.8571). With problem (LP(𝑆¹₁)), the dual to problem (LP(𝑆¹₂)) does not lead to an update of 𝑦¹ and UB₁. We have 𝑄¹={𝑆¹₁,𝑆¹₂}. and neither 𝑆¹₁ nor 𝑆¹₂ is deleted by Step 8 from 𝑄¹. At the end of Iteration 1, 𝑥¹=(1.2857,2.8571),UB₁=−16.224,LB₁=−38.6454,𝐺¹={𝑆¹₁,𝑆¹₂}.
The algorithm finds a global 𝜀-optimal value −16.2837 after 7 iterations at the global 𝜀-optimal solution 𝑥^∗=(1.547,2.421).

Example 13. min𝐺(𝑥)=−𝑥₁+2𝑥₂−0.5−2𝑥₁+𝑥₂++63𝑥₁−2𝑥₂+6.5𝑥₁+𝑥₂−1s.t.−5𝑥₁+8𝑥₂⩽24,5𝑥₁+8𝑥₂⩽44,6𝑥₁−3𝑥₂⩽15,−4𝑥₁−5𝑥₂𝑥⩽−10,₁⩾0,𝑥₂⩾0. (35)
Prior to initiating the algorithm, we use Theorem 1 to find a simplex 𝑆⁰ containing 𝑋. By solving three linear programs, we determine that the vertices 𝑆⁰ are given by 𝑣⁰=𝑦⁰=(9,6),𝑣¹=𝑦¹=(−6.5,6),𝑣²=𝑦²=(9,3.5). Also, for each 𝑖=1,2, we compute the minimum 𝑙_𝑖 and the maximum 𝐿_𝑖 of ⟨𝑑_𝑖,𝑥⟩+𝑑_𝑖0 over 𝑋 for use in the deletion by infeasiblity test (25) and (26). This entails solving four linear programs and get 𝑙₁=1,𝑙₂=1,𝐿₁=9,𝐿₂=6.
The algorithm terminates at the beginning of Iteration 29 with the global 𝜀-optimal solution 𝑥^∗=(1.5549,0.7561) and 𝜀-optimal value 10.6756 for problem (35).

Example 14. In this example, we solve 10 different random instances for various sizes and objective function structures. These test problems are generated by fixing 𝜀=10⁻⁵. And then all elements of 𝐴, 𝑐_𝑖,𝑑_𝑖,𝑐_𝑖0 and 𝑑_𝑖0 are randomly generated, whose ranges are [1,10]. Since the test problems are coded in C++ and the experiments are conducted on a Pentium IV (3.06 GHZ) microcomputer, the computational results of five problems are summarized in Table 1. The following indices characterize the performance in algorithm: Iter: the average number of iterations; Time: the average execution time in seconds.

Table 1 

Computational results of test problems.

6. Conclusion

We have presented and validated a new simplicial branch and bound algorithm globally to the linear multiplicative programming. The algorithm implements a simplicial branch 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.

Acknowledgment

Supported by National Natural Science Foundation of China (no. 70771030, no. 70271047 and no. 11261010) and Project Science Foundation of Guangdong University of Finance (no. 2012RCYJ005).