Generating Efficient Outcome Points for Convex Multiobjective Programming Problems and Its Application to Convex Multiplicative Programming

Abstract

Convex multiobjective programming problems and multiplicative programming problems have important applications in areas such as finance, economics, bond portfolio optimization, engineering, and other fields. This paper presents a quite easy algorithm for generating a number of efficient outcome solutions for convex multiobjective programming problems. As an application, we propose an outer approximation algorithm in the outcome space for solving the multiplicative convex program. The computational results are provided on several test problems.

1. Introduction

The convex multiobjective programming problem involves the simultaneously minimize 𝑝≥2 noncomparable convex objective functions 𝑓_𝑗∶ℝ^𝑛→ℝ, 𝑗=1,…,𝑝, over nonempty convex feasible region 𝑋 in ℝ^𝑛 and may be written byMin𝑓(𝑥),s.t.𝑥∈𝑋, (𝑉𝑃_𝑋) where 𝑓(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑝(𝑥))^𝑇. When 𝑋⊂ℝ^𝑛 is a polyhedral convex set and 𝑓_𝑗, 𝑗=1,…,𝑝 are linear functions, problem (𝑉𝑃_𝑋) is said to be a linear multiobjective programming problem (𝐿𝑃_𝑋).

For a nonempty set 𝑄⊂ℝ^𝑝, we denote by 𝑄_𝐸 and 𝑄_WE the sets of all efficient points and weakly efficient points of 𝑄, respectively, that are𝑄_𝐸∶=𝑞^∗∈𝑄∣∄𝑞∈𝑄suchthat𝑞^∗≥𝑞and𝑞^∗=𝑞≠𝑞_∗∈𝑄∣𝑞^∗−ℝ^𝑝₊∩𝑄=𝑞^∗,𝑄_WE∶=𝑞^∗∈𝑄∣∄𝑞∈𝑄suchthat𝑞^∗=𝑞≫𝑞_∗∈𝑄∣𝑞^∗−intℝ^𝑝₊∩𝑄=∅. (1.1)

Here for any two vectors 𝑎,𝑏∈ℝ^𝑝, the notations 𝑎≥𝑏 and 𝑎≫𝑏 mean 𝑎−𝑏∈ℝ^𝑝₊ and 𝑎−𝑏∈intℝ^𝑝₊, respectively, where ℝ^𝑝₊ is the nonnegative orthant of ℝ^𝑝, and intℝ^𝑝₊ is the interior of ℝ^𝑝₊. By definition, 𝑄_𝐸⊆𝑄_WE.

The set𝑌^image=𝑓(𝑋)=𝑓(𝑥)∣𝑥∈𝑋 (1.2) is called the outcome set (or image) of 𝑋 under 𝑓. A point 𝑥⁰∈𝑋 is said to be an efficient solution for problem (𝑉𝑃_𝑋) if 𝑓(𝑥⁰)∈𝑌^image_𝐸. For simplicity of notation, let 𝑋_𝐸 denote the set of all efficient solutions for problem (𝑉𝑃_𝑋). When 𝑓(𝑥⁰)∈𝑌_WE, 𝑥⁰ is called weakly efficient solution for problem (𝑉𝑃_𝑋) and the set of all weakly efficient solutions is denoted by 𝑋_WE, it is clear that 𝑋_𝐸 and 𝑋_WE are the preimage of 𝑌^image_𝐸 and 𝑌^image_WE under 𝑓, respectively. We will refer to 𝑌^image_𝐸 and 𝑌^image_WE as the efficient outcome set and weakly efficient outcome set for problem (𝑉𝑃_𝑋), respectively.

The goal of problem (𝑉𝑃_𝑋) is to generate the sets 𝑋_𝐸 and 𝑋_WE or at least their subsets. However, it has been shown that, in practice, the decision maker prefers basing his or her choice of the most preferred solution primarily on 𝑌^image_𝐸 and 𝑌^image_WE rather than 𝑋_𝐸 and 𝑋_WE. Arguments to this effect are given in [1].

It is well known that the task of generating 𝑋_𝐸, 𝑋_WE, 𝑌^image_𝐸, 𝑌^image_WE, or significant portions of these sets for problem (𝑉𝑃_𝑋) is a difficult problem. This is because they are, in general, nonconvex sets, even in the case of linear multiobjective programming problem (𝐿𝑃_𝑋).

Problem (𝑉𝑃_𝑋) arises in a wide variety of applications in engineering, economics, network planning, production planning, operations research, especially in multicriteria design and in multicriteria decision making (see, for instance, [2, 3]). Many of the approaches for analyzing convex multiobjective programming problems involve generating either the sets 𝑋_𝐸, 𝑋_WE, 𝑌^image_𝐸, and 𝑌^image_WE or a subset thereof, without any input from decision maker (see, e.g., [1, 2, 4–13] and references therein). For a survey of recent developments see [6].

This paper has two purposes:(i)the first is to propose an algorithm for generating a number of efficient outcome points for convex multiobjective programming problem (𝑉𝑃_𝑋) depending on the requirements of decision makers (Algorithm 1 in Section 2). Computational experiments show that this algorithm is quite efficient;(ii)as an application, we present an outer approximation algorithm for solving the convex multiplicative programming problem (𝐶𝑃_𝑋) associated with the problem (𝑉𝑃_𝑋) in outcome space ℝ^𝑝 (Algorithm 2 in Section 3), where the problem (𝐶𝑃_𝑋) can be formulated as min^𝑝_𝑗=1𝑓_𝑗(𝑥),s.t.𝑥∈𝑋. (𝐶𝑃_𝑋)

It is well known that problem (𝐶𝑃_𝑋) is a global optimization problem and is known to be NP-hard, even special cases such as when 𝑝=2, 𝑋 is a polyhedron, and 𝑓_𝑗 is linear for each 𝑗=1,2 (see [14]). Because of the wide range of its applications, this problem attracts a lot of attention from both researchers and practitioners. Many algorithms have been proposed for globally solving the problem (𝐶𝑃_𝑋), (see, e.g., [10, 14–20] and references therein).

The paper is organized as follows. In Section 2, we present Algorithm 1 for generating efficient outcome points for convex multiobjective programming problem (𝑉𝑃_𝑋) and its theoretical basis. To illustrate the performance of Algorithm 1, we use it to generate efficient points for a sample problem. The Algorithm 2 for solving the convex multiplicative programming (𝐶𝑃_𝑋) associated with the problem (𝑉𝑃_𝑋), and numerical examples are described in Section 3.

2. Generating Efficient Outcome Points for Problem (𝑉𝑃_𝑋)

2.1. Theoretical Basis

Assume henceforth that 𝑋⊂ℝ^𝑛 is a nonempty, compact convex set given by𝑋∶=𝑥∈ℝ^𝑛∣𝑔_𝑖(𝑥)≤0,𝑖=1,…,𝑚, (2.1) where all the 𝑔₁,𝑔₂,…,𝑔_𝑚 are convex functions on ℝ^𝑛. Then the set𝐺⁰=𝑌^image+ℝ^𝑝₊ (2.2) is nonempty full-dimensional convex set in ℝ^𝑝. Furthermore, from Yu [21, page 22, Theorem  3.2], we know that 𝑌^image_𝐸=𝐺⁰_𝐸.

For each 𝑘=1,…,𝑝, we consider the following problem:min𝑦_𝑘,s.t.𝑦∈𝐺⁰.(𝑃⁰ (𝑘)) We denote the optimal value for the problem (𝑃⁰(𝑘)) by 𝑦^opt_𝑘 and the optimal solution for this problem by 𝑦^𝑘=(𝑦^𝑘₁,…,𝑦^𝑘_𝑝)∈ℝ^𝑝.

Let 𝑦^𝑚=(𝑦^𝑚₁,𝑦^𝑚₂,…,𝑦^𝑚_𝑝), where𝑦^𝑚_𝑘=𝑦^opt_𝑘∀𝑘=1,2,…,𝑝. (2.3) As usual, the point 𝑦^𝑚 is said to be an ideal efficient point of 𝐺⁰. It is clear that if 𝑦^𝑚∈𝐺⁰ then 𝑌^image_𝐸=𝐺⁰_𝐸={y^𝑚}. Therefore, we suppose that 𝑦^𝑚∉𝐺⁰. Obviously, by definition, if (𝑥^∗,𝑦^∗)∈ℝ^𝑛+𝑝 is an optimal solution for the problem (𝑃(𝑘)) given by min𝑦_𝑘s.t.𝑓_𝑗(𝑥)−𝑦_𝑗𝑔≤0,𝑗=1,…,𝑝,_𝑖(𝑥)≤0,𝑖=1,…,𝑚,(𝑃 (𝑘)) then 𝑦^𝑘=𝑦^∗ is an optimal solution for the problem (𝑃⁰(𝑘)), and the optimal values of these two problems are equal.

To generate various efficient outcome points in 𝑌^image_𝐸, the algorithm will rely upon the point 𝑦^𝑀=(𝑦^𝑀₁,𝑦^𝑀₂,…,𝑦^𝑀_𝑝), where for each 𝑗=1,2,…,𝑝,𝑦^𝑀_𝑗=𝛼, and 𝛼 is any real number satisfying𝛼>max𝑦^𝑘_𝑗,𝑘=1,…,𝑝,𝑗=1,…,𝑝. (2.4)

We consider the set 𝐺 given by𝐺=𝐺⁰∩𝑦^𝑀−ℝ^𝑝₊. (2.5) It is obvious that 𝐺 is a nonempty, full-dimensional compact convex set in ℝ^𝑝. The set 𝐺 is instrumental in Algorithm 1 to be presented in Section 2.2 for generating efficient outcome points for problem (𝑉𝑃_𝑋).

Remark 2.1. In 1998, Benson [1] presented the outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiobjective linear programming problem. Here, the 𝐺 seems to be analogous with the set 𝑌 considered by Benson [1]. However, note that 𝑌⊃𝑌^image and 𝑌^image is not necessarily a subset of 𝐺. The Figure 1 illustrates the set 𝐺 in the case of 𝑝=2.

Figure 1 

Proposition 2.2. 𝑌^image_𝐸=𝐺_𝐸.

Proof. This result is easy to show by using Yu [21, page 22, Theorem  3.2] and the definition of the point 𝑦^𝑀. Therefore, the proof is omitted.

Let𝐵⁰=𝑦^𝑚+ℝ^𝑝₊∩𝑦^𝑀−ℝ^𝑝₊. (2.6) It is clear that 𝐺⊂𝐵⁰. The following fact will play an important role in establishing the validity of our algorithm.

Proposition 2.3. For each point 𝑦∈𝐵⁰⧵𝐺, let 𝑦^𝑤 denote the unique point on the boundary of 𝐺 that belongs to the line segment connecting 𝑦 and 𝑦^𝑀. Then 𝑦^𝑤∈𝐺_𝐸.

Proof. Let 𝐷=𝐺−𝑦^𝑤. Since 𝐺 is the compact convex set and 𝑦^𝑤 belongs to the boundary of 𝐺, the set 𝐷 is also the compact convex set containing the origin 0 of the outcome space ℝ^𝑝 and 0 belongs to the boundary of 𝐷. According to the separation theorem [22], there is a nonzero vector 𝑞∈ℝ^𝑝 such that 𝑞,𝑢≥0∀𝑢∈𝐷. (2.7) Let 𝑆⊂ℝ^𝑝 be a 𝑝-simplex with vertices 0, 𝑒¹,𝑒²,…,𝑒^𝑝, where 𝑒¹,𝑒²,…,𝑒^𝑝 are the unit vectors of ℝ^𝑝. By definition, we can choose the point 𝑦^𝑀 such that (𝑦^𝑤+𝑆)⊂𝐺. This implies that 𝑆⊂𝐷. From (2.7), by taking 𝑢 to be 𝑒¹,𝑒²,…,𝑒^𝑝 to see that 𝑞≥0. (2.8) Furthermore, the expression (2.7) can be written by 𝑞,𝑦−𝑦^𝑤≥0∀𝑦∈𝐺, (2.9) that is, 𝑞,𝑦≥𝑞,𝑦^𝑤∀𝑦∈𝐺. (2.10)
According to [23, Chapter 4, Theorem  2.10], a point 𝑦^∗∈𝐺 is a weakly efficient point of 𝐺 if and only if there is a nonzero vector 𝑣∈ℝ^𝑝 and 𝑣≥0 such that 𝑦^∗ is an optimal solution to the convex programming problemmin𝑣,𝑦∣𝑦∈𝐺. (2.11) Combining this fact, (2.8) and (2.10) give 𝑦^𝑤∈𝐺_𝑊𝐸. To complete the proof, it remains to show that 𝑦^𝑤∈𝐺_𝐸. Assume the contrary that 𝑦^𝑤∈𝐺_𝑊𝐸⧵𝐺_𝐸. (2.12) By the definition, we have 𝐺_𝑊𝐸=𝐺_𝐸∪^𝑝_𝑘=1𝐹_𝑘, (2.13) where for each 𝑘=1,…,𝑝, 𝐹_𝑘 is the optimal solution set for the following problem: min𝑦_𝑘,s.t.𝑦∈𝐺.(𝑃^𝐺 (𝑘)) It is easy to see that the optimal values of two problems (𝑃^𝐺(𝑘)) and (𝑃⁰(𝑘)) are the same. From this fact and the definition of the point 𝑦^𝑚, it follows that 𝐹_𝑘=𝑦∈𝐺∣𝑦_𝑘=𝑦^𝑚_𝑘. (2.14) Therefore, if 𝑦^𝑤∈𝐺_𝑊𝐸⧵𝐺_𝐸 then there is 𝑖₀∈{1,2,…,𝑝} such that 𝑦^𝑤_𝑖0=𝑦^𝑚_𝑖0. Since 𝑦∈𝐵⁰⧵𝐺, we always have 𝑦≥𝑦^𝑚 and 𝑦^𝑤=𝜆𝑦^𝑀+(1−𝜆)𝑦 with 0<𝜆<1. By the choice of the point 𝑦^𝑀, we have 𝑦^𝑀≫𝑦^𝑚. Hence, 𝑦^𝑚_𝑖0=𝑦^𝑤_𝑖0=𝜆𝑦^𝑀_𝑖0+(1−𝜆)𝑦_𝑖0>𝑦^𝑚_𝑖0. (2.15) This contradiction proves that 𝑦^𝑤 must belong to the efficient outcome set 𝐺_𝐸.

Remark 2.4. This Proposition 2.3 can be viewed as an Benson's extension in [1, Proposition  2.4]. Benson showed that the unique point on the boundary of 𝑌 (which is considered in [1]) and corresponding segment belongs to the weakly efficient outcome set 𝑌_𝑊𝐸, here we prove that 𝑦^𝑤 belongs to the efficient outcome set 𝐺_𝐸.

Remark 2.5. Let 𝑦 be a given point in 𝐵⁰⧵𝐺. From Propositions 2.3 and 2.2, the line segment connecting 𝑦 and 𝑦^𝑀 contains a unique point 𝑦^𝑤∈𝑌^image_𝐸 which lies on the boundary of 𝐺. To determine this efficient outcome point 𝑦^𝑤, we have to find the unique value 𝜆^∗ of 𝜆, 0<𝜆<1, such that 𝑦^𝑤=𝜆^∗𝑦^𝑀+1−𝜆^∗𝑦=𝑦+𝜆^∗𝑦^𝑀−𝑦 (2.16) belongs to the boundary of 𝐺 (see Figure 2). It means that 𝜆^∗ is the optimal value for the problem

Figure 2 

min𝜆s.t.𝑦+𝜆𝑦^𝑀−𝑦∈𝐺,0≤𝜆≤1. (2.17) By definition, it is easy to see that 𝜆^∗ is also the optimal value for the following convex programming problem with linear objective function(𝑥)𝑦min𝜆s.t.𝑓−𝜆^𝑀−𝑦−𝑔𝑦≤0,_𝑖(𝑥)≤0,𝑖=1,…,𝑚,0≤𝜆≤1. (𝑇𝑦) Note that 𝜆^∗ exists because the feasible region of problem (𝑇𝑦) is a nonempty, compact convex set. Furthermore, by the definition, it is easy to show that if (𝑥^∗,𝜆^∗)∈ℝ^𝑛+1 is an optimal solution for Problem (𝑇𝑦), then 𝑥^∗ is an efficient solution for the convex multiobjective programming problem (𝑉𝑃_𝑋), that is, 𝑥^∗∈𝑋_𝐸. For the sake of convenience, 𝑥^∗ is said to be an efficient solution associated with 𝑦^𝑤, and 𝑦^𝑤 is said to be an efficient outcome point generated by 𝑦. It is easily seen that by varying the choices of points 𝑦∈𝐵⁰⧵𝐺, the decision maker can generate multiple points in 𝑌^image_𝐸. In theory, if 𝑦 could vary over all of 𝐵⁰⧵𝐺, we could generate all of the properly efficient points of 𝑌 (see [24]) in this way.

The following Proposition 2.6 shows that for each efficient outcome point 𝑦^𝑤 generated by a given point 𝑦, we can determine 𝑝 new points which belong to 𝐵⁰⧵𝐺 and differ 𝑦. This work is accomplished by a technique called cutting reverse polyblock.

A set of the form 𝐵=⋃_𝑦∈𝑉[𝑦,𝑦^𝑀]⊂ℝ^𝑝, where [𝑦,𝑦^𝑀]∶={𝑦|𝑦≤𝑦≤𝑦^𝑀}, 𝑉⊂𝐵⁰=[𝑦^𝑚,𝑦^𝑀], and |𝑉|<+∞, is called a reverse polyblock in hyperrectangle 𝐵⁰ with vertex set 𝑉. A vertex 𝑦∈𝑉 is said to be proper if there is no 𝑦′∈𝑉⧵{𝑦} such that [𝑦,𝑦^𝑀]⊂[𝑦′,𝑦^𝑀]. It is clear that a reverse polyblock is completely determined by its proper vertices.

Proposition 2.6 (see, e.g., [20]). Let 𝐺=(𝑌^image+ℝ^𝑝₊)∩(𝑦^𝑀−ℝ^𝑝₊) be a nonempty compact convex set contained in a reverse polyblock 𝐵=⋃_𝑦∈𝑉[𝑦,𝑦^𝑀] with vertex set 𝑉⊂𝐵⁰. Let 𝑣∈𝑉⧵𝐺 and 𝑦^𝑤 is the unique point on the boundary of 𝐺 that belongs to the line segment connecting 𝑣 and 𝑦^𝑀. Then, 𝐵=𝐵⧵[𝑣,𝑦^𝑤] is a reverse polyblock with vertex set 𝑉=(𝑉⧵{𝑣})∪{𝑣¹,𝑣²,…,𝑣^𝑝}, where 𝑣^𝑖=𝑦^𝑤−𝑦^𝑤_𝑖−𝑣_𝑖𝑒^𝑖,𝑖=1,2,…,𝑝, (2.18) where, as usual, 𝑒^𝑖 denotes the 𝑖th unit vector of ℝ^𝑝, and 𝐺⊂𝐵⊂𝐵,𝑣∈𝐵⧵𝐵. (2.19)

Remark 2.7. By Proposition 2.3, the point 𝑦^𝑤 as described in Proposition 2.6 belongs to 𝐺_𝐸. From (2.18), it is easy to see that 𝑣^𝑖≠𝑣 for all 𝑖=1,…,𝑝 and for each 𝑖=1,…,𝑝, the vertex 𝑣^𝑖∈𝐵⁰⧵𝐺 because 𝑦^𝑤≥𝑣^𝑖 and 𝑦^𝑤≠𝑣^𝑖. The points 𝑣¹,…,𝑣^𝑝 are called new vertices corresponding to 𝑦^𝑤.

2.2. The Algorithm

After the initialization step, the algorithm can execute the iteration step many times to generate a number of efficient outcome points for problem (CMP) depending on the user's requirements. Note that the construction of the box 𝐵⁰ in Substep 1.1 of Algorithm 1 involves solving 𝑝 convex programming problems, each of which has a simple linear objective function and the same feasible region. Let 𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡 be a positive integer number. The algorithm for generating 𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡 efficient outcome points to the problem (𝑉𝑃_𝑋) and 𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡 efficient solutions associated with them are described as follows.

Algorithm 1. Step 1 (Initialization). (1.1)Construct 𝐵⁰=[𝑦^𝑚,𝑦^𝑀], where 𝑦^𝑚 and 𝑦^𝑀 are described in Section 2.1. Set 𝐵=𝐵⁰.(1.2) Set 𝑌^out_𝐸=∅ (the set of efficient outcome points),  𝑋^out_𝐸=∅ (the set of efficient solutions),  Nef∶=𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡,  𝑘∶=0 (the number of elements of the set 𝑌^out_𝐸),  𝑆={𝑦^𝑚}.

Step 2 (Iteration). See Steps 2.1 through 2.3 below.(2.1) Set 𝑆=∅.(2.2)For each 𝑦∈𝑆 dobegin𝑘∶=𝑘+1.Find an optimal solution (𝑥^∗,𝜆^∗)∈ℝ^𝑛+1 and the optimal value 𝜆^∗ to the problem (𝑇𝑦).Set 𝑤^𝑘=𝑦+𝜆^∗(𝑦^𝑀−𝑦), 𝑌^out_𝐸=𝑌^out_𝐸∪{𝑤^𝑘} and 𝑋^out_𝐸=𝑋^out_𝐸∪{𝑥^∗} (see Remark 2.5),𝐵=𝐵⧵[𝑣,𝑦^𝑤] with 𝑣=𝑦 and 𝑦^𝑤=𝑤^𝑘. Determine 𝑝 vertices 𝑣¹,…,𝑣^𝑝 corresponding to 𝑦^𝑤 via formula (2.18). Set 𝑆=𝑆∪{𝑣¹,…,𝑣^𝑝} and 𝐵=𝐵.end(2.3)If𝑘≥NefThen Terminate the algorithmElse Set 𝑆=𝑆 and return the Step 2.

2.3. Example

To illustrate the Algorithm 1, we consider the convex multiobjective programming problem (𝑉𝑃_𝑋) (see, Benson in [14]) with 𝑝=𝑛=2, where𝑓₁(𝑥)=𝑥₁−2²+1,𝑓₂(𝑥)=𝑥₂−4²+1, (2.20) and 𝑋⊂ℝ² that satisfies the constraints𝑔₁(𝑥)=25𝑥²₁+4𝑥²₂−100≤0,𝑔₂(𝑥)=𝑥₁+2𝑥₂−4≤0. (2.21)

What follows is a brief summary of the results of executing Algorithm 1 for determining 𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡=7 different efficient outcome points for this sample problem, and 7 efficient solutions associated with them.

Step 1. By solving the problem (𝑃(1)), we obtain the optimal solution (1.0,0.0,1.0,100.0) and the optimal value 𝑦^opt₁=1.0. Then, 𝑦¹=(1.0,100.0) is the optimal solution for the problem (𝑃⁰(1)).
By solving problem (𝑃(2)), we obtain the optimal value 𝑦^opt₂=2.380437 and the optimal solution (−1.6501599,2.8250799,100.0,2.380437). Hence, 𝑦²=(100.0,2.380437) is the optimal solution for the problem (𝑃⁰(2)). From (2.4), we choose 𝛼=110.0>max𝑦¹₁,𝑦¹₂,𝑦²₁,𝑦²₂=max{1.0,100.0,100.0,2.380437}. (2.22) Then 𝑦^𝑚=(𝑦^opt₁,𝑦^opt₂)=(1.0,2.380437),and𝑦^𝑀=(𝛼,𝛼)=(110.0,110.0). Set 𝐵=𝐵⁰=[𝑦^𝑚,𝑦^𝑀], 𝑌^out_𝐸∶=∅, 𝑋^out_𝐸∶=∅,Nef∶=𝑁𝑢𝑚𝐸𝑥𝑝𝑒𝑐𝑡=7; 𝑘∶=0 and 𝑆∶={𝑦^𝑚}.

Step 2 (The first time). Substep 2.1. Set 𝑆=∅.Substep 2.2 (The set 𝑆 has only one element (1.0, 2.380437)). Let 𝑦∶=(1.0,2.380437) and 𝑘=1. By solving the problem (𝑇𝑦), we obtain the optimal solution (𝑥^∗,𝜆^∗)=(0.230078,1.884961,0.028740) and the optimal value 𝜆^∗=0.028740. Then,𝑤¹=𝑦+𝜆^∗𝑦^𝑀−𝑦=(4.132625,5.473389) (2.23) (the first efficient outcome point),𝑥¹=𝑥^∗=(0.230078,1.884961) (2.24) (the first efficient solution).Set 𝑌^out_𝐸∶=𝑌^out_𝐸∪𝑤¹=𝑤¹,𝑋^out_𝐸∶=𝑋^out_𝐸∪𝑥¹=𝑥¹,𝐵=𝐵⧵𝑣,𝑦^𝑤with𝑣=𝑦,𝑦^𝑤=𝑤¹. (2.25)Using formula (2.18), we obtain 𝑝=2 new vertices 𝑣¹=(1.0,5.473389) and 𝑣²=(4.132625,2.380437) corresponding to 𝑦^𝑤.Set 𝑆=𝑆∪{𝑣¹,𝑣²}={(1.0,5.473389),(4.132625,2.380437)} and 𝐵=𝐵.Substep 2.3. Since 𝑘=1<Nef=7, set 𝑆=𝑆 and return to the Step 2.

Step 2 (The second time). Substep 2.1. 𝑆∶=∅.Substep 2.2. (The set 𝑆 has 2 elements:𝑠¹=(1.0,5.473389),𝑠²=(4.132625,2.380437)) (i)Let 𝑦=𝑠¹=(1.0,5.473389) and 𝑘=2. The problem (𝑇𝑦) has the optimal solution (𝑥^∗,𝜆^∗)=(0.820117,1.589942,0.012772), and the optimal value 𝜆^∗=0.012772. We have 𝑤²=(),,𝑥2.392125,6.808381thesecondefficientoutcomepoint₂=(0.820117,1.589942),(thesecondefficientsolution). (2.26) Set 𝑌^out_𝐸=𝑌^out_𝐸∪𝑤²=𝑤¹,𝑤²,𝑋^out_𝐸=𝑋^out_𝐸∪𝑥²=𝑥¹,𝑥²,𝐵=𝐵⧵𝑣,𝑦^𝑤with𝑣=𝑦,𝑦^𝑤=𝑤². (2.27)Two new vertices corresponding to 𝑦^𝑤 are 𝑣¹=(1.0,6.808381), and 𝑣²=(2.392125,5.473389).Set 𝑆=𝑆∪{𝑣¹,𝑣²}={(1.0,6.808381),(2.392125,5.473389) and 𝐵=𝐵.(ii)Let 𝑦=𝑠²=(4.132625,2.380437) and 𝑘=3. Then,(𝑥^∗,𝜆^∗)=(−0.276796,2.138398,0.019375), and 𝜆^∗=0.019375 are the optimal solution and the optimal value for the problem (𝑇𝑦), respectively. Then, 𝑤³=(),,𝑥6.183801,4.465562thethirdefficientoutcomepoint₃=(−0.276796,2.138398),(thethirdefficientsolution). (2.28)Set 𝑌^out_𝐸=𝑌^out_𝐸∪𝑤³=𝑤¹,𝑤²,𝑤³,𝑋^out_𝐸=𝑋^out_𝐸∪𝑥³=𝑥¹,𝑥²,𝑥³,𝐵=𝐵⧵𝑣,𝑦^𝑤with𝑣=𝑦,𝑦^𝑤=𝑤³. (2.29)Two new vertices corresponding to 𝑦^𝑤 are 𝑣¹=(4.132625,4.465562), and 𝑣²=(6.1838011,2.380437).Set 𝑆=𝑆∪{𝑣¹,𝑣²}={(1.0,6.808381),(2.392125,5.473389),(4.132625,4.465562),(6.1838011,2.380437)}, and 𝐵=𝐵.Substep 2.3. Since 𝑘=3<Nef=7, set 𝑆∶=𝑆 and return to the Step 2.

Step 2 (The third time). Substep 2.1. 𝑆∶=∅.

Substep 2.2. (The set 𝑆 has 4 elements: 𝑠¹=(1.0,6.808381), 𝑠²=(2.392125,5.473389), 𝑠³=(4.132625,4.465562), 𝑠⁴=(6.1838011,2.380437)).
 By a calculation analogous to above, we yield four next efficient outcome points 𝑤⁴, 𝑤⁵, 𝑤⁶, and 𝑤⁷ generated by 𝑦=𝑠¹, 𝑦=𝑠², 𝑦=𝑠³, 𝑦=𝑠⁴, respectively, and four next efficient solution 𝑥⁴, 𝑥⁵, 𝑥⁶, 𝑥⁷ associated with 𝑤⁴, 𝑤⁵, 𝑤⁶, 𝑤⁷, respectively. Namely, 𝑤⁴=(1.779439,7.546285);𝑥⁴=();𝑤1.117142,1.441430₅=(3.109453,6.170177);𝑥⁵=();𝑤0.547605,1.726198₆=(4.780944,5.111841);𝑥⁶=();w0.055535,1.972232₇=(7.668913,3.919957);𝑥⁷=(−0.582424,2.291212). (2.30) Since 𝑘=Nef=7, the algorithm is terminated. Thus, after 3 iterations, we obtain 𝑌^out_𝐸={𝑤¹,𝑤²,…,𝑤⁷}, and 𝑋^out_𝐸={𝑥¹,𝑥²,…,𝑥⁷}.

3. Application to Problem (𝐶𝑃_𝑋)

Consider convex multiplicative programming problem (𝐶𝑃_𝑋) associated with the convex multiobjective programming problem (𝑉𝑃_𝑋)min^𝑝_𝑗=1𝑓_𝑗(𝑥)s.t.𝑥∈𝑋, (𝐶𝑃_𝑋) where 𝑋⊂ℝ^𝑛 is a nonempty compact convex set defined by (2.1) and (𝑓_𝑗∶ℝ^𝑛→ℝ) is convex on ℝ^𝑛 and positive on 𝑋, 𝑗=1,2,…,𝑝.

One of the direct reformulations of the problem (𝐶𝑃_𝑋) as an outcome-space problem is given byminℎ𝑦=^𝑝_𝑗=1𝑦_𝑗s.t.𝑦∈𝑌^image, (𝑂𝑃_𝑌) where 𝑌^image={𝑓(𝑥)=(𝑓₁(𝑥),…,𝑓_𝑝(𝑥))^𝑇|𝑥∈𝑋} is the outcome set of 𝑋 under 𝑓. By assumption, we have 𝑌^image⊂intℝ^𝑝₊.

The following proposition tells us a link between the global solution to the problem (𝑂𝑃_𝑌) and the efficient outcome set 𝑌^image_𝐸 for the problem (𝑉𝑃_𝑋).

Proposition 3.1. Any global optimal solution to problem (𝑂𝑃_𝑌) must belong to the efficient outcome set 𝑌^image_𝐸.

Proof. The proposition follows directly from the definition.

Remark 3.2. We invoke Propositions 3.1 and 2.2 to deduce that problem (𝑂𝑃_𝑌) is equivalent to the following problem: minℎ𝑦=^𝑝_𝑗=1𝑦_𝑗s.t.𝑦∈𝐺_𝐸. (𝑂𝑃_𝐺𝐸) The relationship between two problems (𝐶𝑃_𝑋) and (𝑂𝑃_𝑌) is described by the following theorem and was given in [14, Theorem  2.2]. However, we give here a full proof for the reader's convenience.

Theorem 3.3. If 𝑦^∗ is a global optimal solution to problem (𝑂𝑃_𝑌), then any 𝑥^∗∈𝑋 such that 𝑓(𝑥^∗)≤𝑦^∗ is a global optimal solution to problem (𝐶𝑃_𝑋). Furthermore, the global optimal values of two problems (𝐶𝑃_𝑋) and (𝑂𝑃_𝑌) are equal, that is, ℎ𝑦^∗=^𝑝_𝑗=1𝑦^∗_𝑗=^𝑝_𝑗=1𝑓_𝑗𝑥^∗. (3.1)

Proof. Suppose that 𝑦^∗ is a global optimal solution to problem (𝑂𝑃_𝑌) and 𝑥^∗∈𝑋 satisfies 𝑓𝑥^∗≤𝑦^∗. (3.2) By Proposition 3.1, 𝑦^∗∈𝑌^image_𝐸. Since 𝑓(𝑥^∗)∈𝑌^image and 𝑦^∗∈𝑌^image_𝐸, from (3.2), we have 𝑓(𝑥^∗)=𝑦^∗. Thus, ℎ(𝑦^∗)=∏^𝑝_𝑗=1𝑦^∗_𝑗=∏^𝑝_𝑗=1𝑓_𝑗(𝑥^∗). Now, we show that 𝑥^∗ is a global optimal solution of the problem (𝐶𝑃_𝑋). Indeed, assume the contrary that there is a point 𝑥∈𝑋 such that ∏^𝑝_𝑗=1𝑓_𝑗(𝑥)<∏^𝑝_𝑗=1𝑓_𝑗(𝑥^∗). Combining this fact and (3.2) gives ^𝑝_𝑗=1𝑓_𝑗(𝑥)<^𝑝_𝑗=1𝑓_𝑗𝑥^∗≤^𝑝_𝑗=1𝑦^∗_𝑗. (3.3) Since 𝑥∈𝑋, we have 𝑦=𝑓(𝑥)∈𝑌^image. Therefore, ∏^𝑝_𝑗=1𝑦_𝑗<∏^𝑝_𝑗=1𝑦^∗_𝑗. This contradicts the hypothesis that 𝑦^∗ is a global optimal solution to problem (𝑂𝑃_𝑌) and proves that 𝑥^∗ is a global optimal solution to problem (𝐶𝑃_𝑋). This completes the proof.

By Theorem 3.3 and Proposition 3.1, solving the problem (𝐶𝑃_𝑋) can be carried out in two stages:(1)finding a global optimal solution 𝑦^∗ to the problem (𝑂𝑃_𝐺𝐸). Then 𝑦^∗ is also a global optimal solution to the problem (𝑂𝑃_𝑌), (2)finding a global optimal solution 𝑥^∗∈𝑋 to the problem (𝐶𝑃_𝑋) which satisfies 𝑓(𝑥^∗)≤𝑦^∗.

In the next section, the outer approximation algorithm is developed for solving the problem (𝑂𝑃_𝐺𝐸).

3.1. Outer Approximation Algorithm for Solving Problem (𝑂𝑃_𝐺𝐸)

Starting with the polyblock 𝐵⁰=[𝑦^𝑚,𝑦^𝑀] (see Section 2.1), the algorithm will iteratively generate a sequence of reverse polyblocks 𝐵^𝑘, 𝑘=1,2,…, such that𝐵⁰⊃𝐵¹⊃𝐵²⊃⋯⊃𝐺⊃𝐺_𝐸. (3.4) For each 𝑘=0,1,2,…, the new reverse polyblock 𝐵_𝑘+1 is constructed via the formula𝐵^𝑘+1∶=𝐵^𝑘⧵𝑣,𝑦^𝑤, (3.5) where 𝑣=𝑦^𝑘,𝑦^𝑘 is a global optimal solution to the problem min{ℎ(𝑦)∣𝑦∈𝐵^𝑘}, and 𝑦^𝑤 is the efficient outcome point generated by 𝑦=𝑦^𝑘.

For each 𝑘, let 𝑉^𝑘 denote the vertex set of the reverse polyblock 𝐵^𝑘. The following Proposition 3.4 shows that the function ℎ(𝑦) achieves a minimum over the reverse polyblock 𝐵^𝑘 at a proper vertex.

Proposition 3.4. Let ℎ(𝑦)=∏^𝑝_𝑗=1𝑦_𝑗, and let 𝐵^𝑘 be a reverse polyblock. Consider the problem to minimize ℎ(𝑦) subject to 𝑦∈𝐵^𝑘. An optimal solution 𝑦^𝑘 to the problem then exists, where 𝑦^𝑘 is a proper vertex of 𝐵^𝑘.

Proof. By definition, note that the objective function ℎ(𝑦) is a continuous function on ℝ^𝑝, and 𝐵^𝑘 is a compact, and the problem min{ℎ(𝑦)∣𝑦∈𝐵^𝑘} has an optimal solution 𝑦^𝑘∈𝐵^𝑘. For each 𝑦∈𝐵^𝑘, there is a proper vertex 𝑣 of 𝐵^𝑘 such that 𝑦∈[𝑣,𝑦^𝑀]. That means 𝑣≤𝑦. By definition of the function ℎ(𝑦), we have ℎ(𝑣)≤ℎ(𝑦). This shows that ℎ(𝑦^𝑘)=min{ℎ(𝑦)|𝑦∈𝑉^𝑘}, where 𝑉^𝑘 is the vertex set of 𝐵^𝑘, and the proof is completed.

Remark 3.5. By Proposition 3.4, instead of solving problem min{ℎ(𝑦)|𝑦∈𝐵^𝑘}, we solve the simple problem min{ℎ(𝑦)|𝑦∈𝑉^𝑘}. From (3.4), it is clear that for each 𝑘=0,1,2…, the optimal value 𝛽_𝑘=minℎ𝑦=^𝑝_𝑗=1𝑦_𝑗𝑦∈𝐵^𝑘 (3.6) is a lower bound for the problem (𝑂𝑃_𝐺𝐸), and {𝛽_𝑘} is an increasing sequence, that is, 𝛽_𝑘+1>𝛽_𝑘 for all 𝑘=0,1,2,….

Let 𝜀 be a given sufficient small real number. Let 𝑦^∗∈𝐺_𝐸. Then ℎ(𝑦^∗) is an upper bound for the problem (𝑂𝑃_𝐺𝐸). A point 𝑦^∗ is said to be an 𝜀-optimal solution to problem (𝑂𝑃_𝐺𝐸) if there is a lower bound 𝛽_∗ for this problem such that ℎ(𝑦^∗)−𝛽_∗<𝜀.

Below, we will present an algorithm for finding 𝜀-optimal solution to problem (𝑂𝑃_𝐺𝐸).

At the beginning of a typical iteration 𝑘≥0 of the algorithm, we have from the previous iteration an available nonempty reverse polyblock 𝐵^𝑘⊂ℝ^𝑝 that contains 𝐺 and an upper bound 𝜃_𝑘 for the problem (𝑂𝑃_𝐺𝐸). In iteration 𝑘, firstly problem min{ℎ(𝑦)|𝑦∈𝑉_𝑘} is solved to obtain the optimal solution set 𝑇^opt_𝑘. By the construction, 𝑇^opt_𝑘⊂(𝐵⁰⧵𝐺). Take any 𝑦^𝑘∈𝑇^opt_𝑘. The optimal value 𝛽_𝑘=ℎ(𝑦^𝑘) is the best currently lower bound. Then, we solve the convex programming problem (𝑇𝑦) with 𝑦∶=𝑦^𝑘 to receive the optimal value 𝜆^∗. By Proposition 2.3, the feasible solution 𝜔^𝑘=𝑦+𝜆^∗(𝑦^𝑀−𝑦)∈𝐺_𝐸 is an outcome efficient point generated by 𝑦=𝑦^𝑘 (see Remark 2.5). Now, the best currently upper bound is 𝜃_𝑘=min{𝜃_𝑘,ℎ(𝜔^𝑘)} and the feasible solution 𝑦^best satisfying ℎ(𝑦^best)=𝜃_𝑘 is said to be a currently best feasible solution. If 𝜃_𝑘−𝛽_𝑘<𝜀, then the algorithm terminates, and 𝑦^best is an 𝜀-optimal solution for the problem (𝑂𝑃_𝐺𝐸). Otherwise, set 𝐵^𝑘+1∶=𝐵^𝑘⧵[𝑣,𝑦^𝑤], where 𝑣∶=𝑦^𝑘 and 𝑦^𝜔=𝜔^𝑘. According to Proposition 2.6, the vertex set 𝑉^𝑘+1 of the reverse polyblock 𝐵^𝑘+1 is𝑉^𝑘+1=𝑉^𝑘⧵{𝑣}∪𝑣¹,𝑣²,…,𝑣^𝑝, (3.7) where 𝑣¹,…,𝑣^𝑝 are determined by formula (2.18). Figure 3 illustrates two beginning steps of the algorithm in the case of 𝑝=2.

Figure 3 

By the construction, it is easy to see that {𝜃_𝑘} (the sequence of upper bounds for the problem (𝑂𝑃_𝐺𝐸) is the nonincreasing sequence, that is, 𝜃_𝑘+1≤𝜃_𝑘 for all 𝑘=0,1,2,…

Now, the outer approximation algorithm for solving (𝑂𝑃_𝐺𝐸) is stated as follows.

Algorithm 2 (Outer approximation algorithm). Initialization Step
Construct 𝐵⁰=[𝑦^𝑚,𝑦^𝑀], where 𝑦^𝑚 and 𝑦^𝑀 are described in Section 2.1. Choose 𝜀>0 (𝜀>0 is a sufficient small number). Set 𝑉⁰={𝑦^𝑚} and 𝜃₀∶=𝑁𝑢𝑚𝑙𝑎𝑟𝑔𝑒 (𝑁𝑢𝑚𝑙𝑎𝑟𝑔𝑒 is a sufficient large number. This number can be viewed as an initialization the upper bound). Set 𝑘=0 and go to Iteration step k.
Iteration Step 𝑘 [𝑘=0,1,2,…]. See Steps k.1 through k.5 below.𝑘.1Determine the optimal solution set 𝑇^opt_𝑘∶=Argmin{ℎ(𝑦)|𝑦∈𝑉^𝑘}. Choose an arbitrary 𝑦^𝑘∈𝑇^opt_𝑘 and set 𝛽_𝑘∶=ℎ(𝑦^𝑘) ( currently best lower bound).𝑘.2Let 𝑦=𝑦^𝑘. Find the optimal value 𝜆^∗ to the problem (𝑇𝑦). And set 𝜔^𝑘=𝑦+𝜆^∗(𝑦^𝑀−𝑦)∈𝐺_𝐸.𝑘.3 (Update the upper bound)If ℎ(𝜔^𝑘)<𝜃_𝑘 Then 𝜃_𝑘=ℎ(𝜔^𝑘) (currently best upper bound)          and 𝑦^best=𝜔^𝑘 (currently best feasible point).𝑘.4If  𝜃_𝑘−𝛽_𝑘≤𝜀Then Terminate the algorithm: 𝑦^best is an 𝜀-optimal solutionElse Set 𝐵^𝑘+1∶=𝐵^𝑘⧵[𝑣,𝑦^𝑤] where 𝑣∶=𝑦^𝑘 and 𝑦^𝜔=𝜔^𝑘  and determine the set 𝑉^𝑘+1 by formula (2.18).𝑘.5Set 𝜃_𝑘+1∶=𝜃_𝑘; 𝑘∶=𝑘+1 and go to iteration step 𝑘.

Theorem 3.6. The algorithm terminates after finitely many steps and yields an 𝜀-optimal solution to problem (𝑂𝑃_𝐺𝐸).

Proof. Let 𝜀 be a given positive number. Since the function ℎ(𝑦) is uniformly continuous on the compact set 𝐵⁰, we can choose an enough small number 𝛿>0 such that if 𝑦,𝑦^′∈𝐵₀ and ‖𝑦−𝑦^′‖<𝛿, then |ℎ(𝑦)−ℎ(𝑦^′)|<𝜀. Then, to prove the termination of the algorithm, we need to show only that lim_𝑘→∞𝑤^𝑘−𝑦^𝑘=0. (3.8)
Observe first that the positive series ∑^∞_𝑘=1Vol[𝑦^𝑘,𝑤^𝑘] is convergent, since the open boxes int[𝑦^𝑘,𝑤^𝑘] are disjoint, and all of them are contained in the closure of 𝐵⁰⧵𝐺. It follows thatlim_𝑘→∞Vol𝑦^𝑘,𝑤^𝑘=0. (3.9) Now, by the construction of 𝑤^𝑘 and 𝑦^𝑘, we have 𝑤^𝑘−𝑦^𝑘=𝜆_𝑘𝑦^𝑀−𝑦^𝑘𝑦,Vol^𝑘,𝑤^𝑘=𝜆^𝑝_𝑘Vol𝑦^𝑘,𝑦^𝑀, (3.10)(3.11) for 𝜆_𝑘∈(0,1). Note that all of the points 𝑦^𝑘 are contained in the set 𝐵⁰⧵𝐺. Furthermore, by the choice of 𝑦^𝑀 (see (2.4)), the closure of 𝐵⁰⧵𝐺 is a compact subset of the interior of the cone 𝑦^𝑀−ℝ^𝑝₊. This observation implies that Vol[𝑦^𝑘,𝑦^𝑀] has a lower bound far from zero. Combining this fact, (3.9) and (3.11) imply that lim_𝑘→∞𝜆_𝑘=0. Also, the observation implies that ‖𝑦^𝑀−𝑦^𝑘‖ is bounded. Finally, by (3.10), we have lim_𝑘→∞‖𝑤^𝑘−𝑦^𝑘‖=0.

3.2. Computational Results

First, the Algorithm 2 has been applied for solving the test example given by Benson in [14] (see the example in Section 2.3)min𝑓₁(𝑥)𝑓₂(𝑥)∣𝑔₁(𝑥)≤0,𝑔₂(𝑥)≤0, (3.12) where 𝑓₁(𝑥)=(𝑥₁−2)²+1, 𝑓₂(𝑥)=(𝑥₂−4)²+1, and 𝑔₁(𝑥)=25𝑥²₁+4𝑥²₂−100, 𝑔₂(𝑥)=𝑥₁+2𝑥₂−4.

The calculation process for solving this example is described as follows.

Initialization.
Similarly, the example in Section 2.3, we have 𝐵⁰=[𝑦^𝑚,𝑦^𝑀] with 𝑦^𝑚=(1.00000,2.380437), and 𝑦^𝑀=(110.000000,110.000000). We choose 𝜀=0.025 and set 𝑉⁰={𝑦^𝑚=(1.00000,2.380437)}, 𝜃₀=10000 (the initialization upper bound), 𝑘=0 and go to iteration step 𝑘=0.

Iteration (𝑘=0). Step 1. We have 𝑇^opt₀∶=argmin{ℎ(𝑦)|𝑦∈𝑉⁰}={𝑦^𝑚}. Choose 𝑦⁰∶=𝑦^𝑚∈𝑇^opt₀, and set 𝛽₀∶=ℎ(𝑦⁰)=2.380437 (currently best lower bound).Step 2. Let 𝑦=𝑦⁰. Solving the problem (𝑇𝑦), we obtain the optimal solution (𝑥^∗,𝜆^∗)=(0.230078,1.884961,0.028740)) and the optimal value 𝜆^∗=0.028740. Then, 𝑤⁰=𝑦+𝜆^∗(𝑦^𝑀−𝑦)=(4.132625,5.473389)∈𝐺_𝐸.Step 3. Since ℎ(𝑤⁰)=22.619464<𝜃₀, we have 𝜃₀=ℎ(𝑤⁰)=22.619464 (currently best upper bound), and 𝑦^best=𝑤⁰ (currently best feasible point).Step 4. Since 𝜃₀−𝛽₀=20.239027>𝜀, we set 𝐵¹∶=𝐵⁰⧵[𝑣,𝑦^𝑤], where 𝑣∶=𝑦⁰, 𝑦^𝑤=𝑤⁰ and determine the set 𝑉¹={(1.000000,5.473389),(4.132625,2.380437)}. (3.13)Step 5. Set 𝜃₁∶=𝜃₀=22.619464;𝑘∶=0+1=1, and go to iteration step 𝑘=1.

Iteration (𝑘=1). Step 1. We have 𝑇^opt₁∶=argmin{ℎ(𝑦)|𝑦∈𝑉¹}={𝑦¹=(1.000000,5.473389)}. Choose 𝑦¹∈𝑇^opt₁ and set 𝛽₁∶=ℎ(𝑦¹)=5.473389 (the currently lower bound).Step 2. Let 𝑦=𝑦¹. The convex programming (𝑇𝑦) has the optimal solution (𝑥^∗,𝜆^∗)=(0.82011,1.589942,0.012772)), and the optimal value 𝜆^∗=0.012772. Then, we have 𝑤¹=(2.392125,6.808381)∈𝐺_𝐸.Step 3. Since ℎ(𝑤¹)=16.286498<𝜃₁, we set 𝜃₁=ℎ(𝑤¹) (the currently best upper bound), and 𝑦^best=𝑤¹ (the currently best feasible point).Step 4. Now, 𝜃₁−𝛽₁=10.813109>𝜀. Therefore, 𝐵²∶=𝐵¹⧵[𝑣,𝑦^𝑤], where 𝑣∶=𝑦¹,𝑦^𝑤=𝑤¹, and 𝑉²={(4.132625,2.380437),(1.000000,6.808381),(2.392125,5.473389)}.Step 5. Set 𝜃₂=𝜃₁=16.286498;𝑘∶=1+1=2, and go to iteration step 𝑘=2.

Iteration (𝑘=2). Step 1. We have 𝑇^opt₂=argmin{ℎ(𝑦)|𝑦∈𝑉²}={𝑦²=(1.000000,6.808381)}, choose 𝑦²∶=(1.000000,6.808381)∈𝑇^opt₂, and set 𝛽₂∶=ℎ(𝑦²)=6.808381.Step 2. Let 𝑦=𝑦². The problem (𝑇𝑦) has the optimal solution (𝑥^∗,𝜆^∗)=(1.117142,1.441429,0.007151)), and the optimal value 𝜆^∗=0.007151. Then 𝑤²=(1.779439,7.546285)∈𝐺_𝐸.Step 3. Since ℎ(𝑤²)=13.428154<𝜃₂, the currently best upper bound 𝜃₂∶=ℎ(𝑤²) and the currently best feasible point 𝑦^best=𝑤²=(1.779439,7.546285).Step 4. Since 𝜃₂−𝛽₂=6.619773>𝜀, we set 𝐵³∶=𝐵²⧵[𝑣,𝑦^𝑤], where 𝑣∶=𝑦², 𝑦^𝑤=𝑤², and determine the set 𝑉³={(4.132625,2.380437),(2.392125,5.473389),(1.000000,7.546285),(1.779439,6.808381)}.Step 5. Set 𝜃₃∶=𝜃₂=13.428154,𝑘∶=2+1=3, and go to iteration step 𝑘=3.

After 42 iterations, the algorithm terminates with 𝜃₄₂=9.770252094 and 𝛽₄₂=9.745596873, where 𝜃₄₂=ℎ(𝑤²⁰). Then, the 𝜀-optimal solution for the problem (𝑂𝑃_𝑌) and for the problem (𝐶𝑃_𝑋) are given by 𝑦^∗=𝑦^best=𝑤²⁰=(1.023846379,9.542693410), (3.14) and 𝑥^∗=(1.845577271,1.077211364). The approximate optimal value of problem (3.12) is 9.770252094.

Below, we present the results of computational experiment for two types of problem. We take 𝜀=0.025 in the numerical experiments. We tested our algorithm for each problem with random data for several times. Then, we took results in average. Numerical results are summarized in Tables 1 and 2.

Table 1 

Computational results on the problem of Type 1.

Table 2 

Computational results on the problem of Type 2.

Type 1. min^𝑝_𝑗=1𝑓_𝑗(𝑥),s.t.𝐴𝑥≤𝑏,𝑥≥0, (3.15) where 𝑓_𝑗(𝑥)=⟨𝛼^𝑗,𝑥⟩+𝜇_𝑗, 𝛼^𝑗∈ℝ^𝑛, 𝜇_j∈ℝ,𝑗=1,…,𝑝, 𝐴 is an (𝑚×𝑛)-matrix, and 𝑏∈ℝ^𝑚.

Type 2. min^𝑝_𝑗=1𝑓_𝑗(𝑥),s.t.𝐴𝑥≤𝑏,𝑥≥0, (3.16) where 𝑓_𝑗(𝑥)=(𝛼^𝑗₁𝑥₁+𝛽^𝑗₁)²+⋯+(𝛼^𝑗_𝑛𝑥_𝑛+𝛽^𝑗_𝑛)²+𝜈_𝑗, with 𝛼^𝑗,𝛽^𝑗∈ℝ^𝑛, 𝜈_𝑗∈ℝ, 𝑗=1,…,𝑝, 𝐴 is an (𝑚×𝑛)-matrix, and 𝑏∈ℝ^𝑚.
Our program was coded in Matlab R2007a and was executed on our own PC with the configuration, CPU Intel Core 2 T5300 1.73 GHz, 1 G Ram. On the two above problems, the parameters are defined as follows:
(i)𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑚×𝑛 is a randomly generated matrix with elements in [−1,1];(ii)𝑏=(𝑏₁,…,𝑏_𝑚)^𝑇 is a random vector satisfying the formula 𝑏_𝑖=^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_0𝑖 (3.17)with 𝑏_0𝑖 being a random number in [0,2] for 𝑖=1,…,𝑚;(iii)𝛼^𝑗=(𝛼^𝑗₁,…,𝛼^𝑗_𝑛) and 𝛽^𝑗=(𝛽^𝑗₁,…,𝛽^𝑗_𝑛), 𝑗=1,…,𝑝 are vectors with elements randomly distributed on [0,1];(iv)the coefficients 𝜇_𝑗, 𝜈_𝑗, 𝑗=1,…,𝑝 are uniformly distributed on [0,1].

4. Conclusion

In this paper, we have presented Algorithm 1 for generating a finite set 𝑌^out_𝐸 of efficient outcome points for the convex multiobjective programming problem (𝑉𝑃_𝑋). The number of such points depends on the requirement of the decision makers. When the selected number is large enough, the convex set conv(𝑌^out_𝐸+ℝ^𝑝₊), where conv(𝑌^out_𝐸+ℝ^𝑝₊) is convex hull of the set (𝑌^out_𝐸+ℝ^𝑝₊), may be viewed as an inner approximation of 𝐺⁰=𝑌+ℝ^𝑝₊ and the efficient set (conv(𝑌^out_𝐸+ℝ^𝑝₊))_𝐸 may be viewed as an inner approximation of the efficient outcome set 𝑌^image_𝐸 for the problem (𝑉𝑃_𝑋). For each efficient outcome point 𝑦^𝑤∈𝑌^image_𝐸 to be generated by a given 𝑦∈𝐵⁰⧵𝐺, the algorithm calls for solving a convex programming problem with linear objective function (𝑇𝑦). Note that by solving problem (𝑇𝑦), we also obtain the efficient solution 𝑥^∗ associated with 𝑦^𝑤, where (𝑥^∗,𝜆^∗) is the optimal solution for the problem (𝑇𝑦).

In [7], Ehrgott et al. have proposed an outer approximation algorithm for representing an inner approximation of 𝐺₀. This algorithm is the combination of an extension of Benson's outer approximation algorithm [1] for multiobjective linear programming problems and linearization technique. In each iteration step of these algorithms, a polyhedron is obtained from the previous one by adding a new hyperplane to it. The vertices of the new polyhedron can be calculated from those of the previous polyhedron by some available global optimization methods. Unlike those algorithms, our algorithm constructs a reverse polyblock at each iteration step from the previous one by cutting out a box, and its vertices can be easily determined by the formula (2.18).

As an application, we have proposed the outer approximation algorithm (Algorithm 2) for solving the convex multiplicative programming problem (𝐶𝑃_𝑋) associated with the problem (𝑉𝑃_𝑋) in outcome space. Since the number of terms 𝑝 in the objective function of problem (𝑉𝑃_𝑋) is, in practice, often much smaller than the number of variables 𝑛, we hope that the algorithms help to reduce considerably the size of the problems.

Acknowledgments

The authors would like to express their sincere gratitude to the referees for their valuable comments and suggestions which enable them to improve substantially this paper. This paper is supported by NAFOSTED, Vietnam.