An Auxiliary Function Method for Global Minimization in Integer Programming

Abstract

An auxiliary function method is proposed for finding the global minimizer of integer programming problem. Firstly, we propose a method to transform the original problem into an integer programming with box constraint, which does not change the properties of the original problem. For the transformed problem, we propose an auxiliary function to escape from the current local minimizer and to get a better one. Then, based on the proposed auxiliary function, a new algorithm to find the global minimizer of integer programming is proposed. At last, numerical results are given to demonstrate the effectiveness and efficiency of the proposed method.

1. Introduction

Consider the following integer programming problem:min𝑓(𝑥),s.t.𝑔_𝑖(𝑥)ℎ⩽0,𝑖=1,2,…,𝑙,_𝑖(𝑥)=0,𝑖=𝑙+1,𝑙+2,…,𝑚,𝑥∈𝑍^𝑛, (𝑃) where 𝑍^𝑛 is the 𝑛-dimensional integer set, and the set 𝑆={𝑥∈𝑍^𝑛∣𝑔_𝑖(𝑥)⩽0,𝑖=1,2,…,𝑙,ℎ_𝑖(𝑥)=0,𝑖=𝑙+1,𝑙+2,…,𝑚} is bounded.

Many real-life applications can be modeled as problem (𝑃), such as production planning, scheduling, and operations research problem. The objective functions of most of the problems are nonlinear and have more than one local optimal solutions over feasible region 𝑆. This requires the global optimization techniques to find the best solution amongst multiple local optima.

Since integer programming problems are generally NP-hard, there are no efficient algorithms with polynomial-time complexity for solving them. Thus, many approximate algorithms have been rapidly developed in recent years, such as greedy search (see [1–4]), simulated annealing (see [5, 6]), genetic algorithm (see [7–9]), tabu search (see [10, 11]), and discrete filled function techniques (see [12–17]). The discrete filled function method is one of the more recently developed global optimization methods to solve integer programming problems. Once a local minimizer has been found by a local search method, the discrete filled function method introduces an auxiliary function to escape from the current local minimizer and to get a better one.

At present, the discrete filled function methods mainly focus on the unconstrained integer programming problems, and most of the existing filled functions contain parameters which are needed to adjust. Thus, solving general integer programming problems is difficult, and solving the unconstrained integer programming problems needs much computation. In this paper, we propose an auxiliary function method to solve problem (𝑃). The proposed auxiliary function has no parameters, and the computation of the proposed method is relatively small.

The remainder of this paper is organized as follows: Section 2 gives some useful notations and definitions. In Section 3, we propose a method to transform the original problem into an integer programming with box constraint. Based on the transformed problem, an auxiliary function is proposed in Section 4, and its properties are also analyzed. An algorithm for solving problem (𝑃) is proposed in Section 5 with several numerical experiments. Some concluding remarks are given in Section 6.

2. Preliminaries

In this section, some useful notations and definitions are listed firstly. By estimating the bound of 𝑆, we can find a box region Ω=∏^𝑛_𝑖=1[𝐿_𝑖,𝑈_𝑖] with 𝑆⊂Ω. In most practical problems, Ω can be easily estimated or is usually given.

Let 𝑀 be a constant which satisfies 𝑀⩾max_𝑧∈𝑆𝑓(𝑧)+1, (2.1) and let 𝐷=±𝑒_𝑖,𝑖=1,2,…,𝑛 (2.2) be a direction set, where 𝑒_𝑖 is an 𝑛-dimensional unit vector in which the 𝑖th component is 1, and the others are 0.

Definition 2.1. 𝑥 is said to be a vertex of 𝑍^𝑛⋂Ω, if and only if, for all 𝑑∈𝐷, 𝑥+𝑑∈𝑍^𝑛⋂Ω and 𝑥−𝑑∉𝑍^𝑛⋂Ω, or 𝑥+𝑑∉𝑍^𝑛⋂Ω and 𝑥−𝑑∈𝑍^𝑛⋂Ω hold.
The set of vertices of Ω⋂𝑍^𝑛 is denoted as 𝑉.

Definition 2.2 (see [18]). For all 𝑥∈𝑍^𝑛⋂Ω, the neighborhood of 𝑥 is denoted as 𝑁(𝑥)={𝑥}⋃{𝑤∈𝑍^𝑛⋂Ω∣𝑤=𝑥+𝑑,𝑑∈𝐷}.

Definition 2.3 (see [18]). A point 𝑥∈𝑍^𝑛⋂Ω is called a local minimizer (maximizer) of 𝑓(𝑥) over 𝑍^𝑛⋂Ω if for all 𝑦∈𝑁(𝑥), 𝑓(𝑦)⩾𝑓(𝑥)(𝑓(𝑦)⩽𝑓(𝑥)) holds; a point 𝑥∈𝑍^𝑛⋂Ω is called a global minimizer (maximizer) of 𝑓(𝑥) over 𝑍^𝑛⋂Ω if for all 𝑦∈𝑍^𝑛⋂Ω, 𝑓(𝑦)⩾𝑓(𝑥)(𝑓(𝑦)⩽𝑓(𝑥)) holds. In addition, if equality does not hold, 𝑥 is called a strictly local (global) minimizer (maximizer) of 𝑓(𝑥).

For finding a local minimizer of integer programming, the following local search algorithm is taken in most cases.

Local Search Algorithm (see [12])
Step 1. Choose an initial point 𝑥∈𝑍^𝑛⋂Ω.Step 2. If 𝑥 is a local minimizer of 𝑓(𝑥), then stop; otherwise, find another point 𝑦∈𝑁(𝑥) which minimizes 𝑓(𝑥) over 𝑁(𝑥).Step 3. Set 𝑥∶=𝑦, and go to Step 2.

3. Transformation of the Original Problem

The local search algorithm usually requires that the search region should be a connected one. However, the feasible region of the original problem may not be connected, so we need to transform the original problem into a simple problem with a connected search region. In order to transform the original problem, we need to define three unary functions firstly:𝑔(𝑡)=ℎ(𝑡)=𝑠(𝑡)=0,𝑡>0,1,𝑡⩽0,1,𝑡=0,0,𝑡≠0,0,𝑡=𝑚,1,𝑡<𝑚, (3.1) where 𝑚 is the number of constraints.

Consider the following function:𝐹(𝑥)=𝑓(𝑥)+𝑀−𝑓(𝑥)×𝑠^𝑙_𝑖=1𝑔𝑔_𝑖(𝑥)+^𝑚_𝑖=𝑙+1ℎℎ_𝑖(𝑥) (3.2) and the problem (𝑥),𝑍min𝐹s.t.𝑥∈Ω^𝑛, (𝑇𝑃) where 𝑀 is taken to satisfy (2.1). The following theorem will show that problem (𝑇𝑃) does not change the local minimizers of problem (𝑃).

Theorem 3.1. If 𝑥 is a local minimizer of problem (𝑃), then 𝑥 is a local minimizer of problem (𝑇𝑃); in turn, if 𝑥 is a local minimizer of the problem (𝑇𝑃), then 𝑥 is a local minimizer of the problem (𝑃), or 𝐹(𝑥)=𝑀.

Proof. If 𝑥 is a local minimizer of the problem (𝑃), then for all 𝑦∈𝑁(𝑥)⋂𝑆, one has 𝐹(𝑦)=𝑓(𝑦)⩾𝑓(𝑥)=𝐹(𝑥); for 𝑦=𝑁(𝑥)⧵𝑆 and by the definition of 𝑀, one has 𝐹(𝑦)=𝑀>max_𝑧∈𝑆𝑓(𝑧)⩾𝑓(𝑥). So 𝐹(𝑦)⩾𝐹(𝑥) for all 𝑦∈𝑁(𝑥), namely, 𝑥 is a local minimizer of the problem (𝑇𝑃).
If 𝑥 is a local minimizer of the problem (𝑇𝑃), then for all 𝑦∈𝑁(𝑥), one has 𝐹(𝑦)>𝐹(𝑥). If 𝐹(𝑥)<𝑀, namely, 𝑥∈𝑆, then one has 𝑓(𝑦)=𝐹(𝑦)>𝐹(𝑥)=𝑓(𝑥) for all 𝑦∈𝑁(𝑥)⋂𝑆. So 𝑥 is a local minimizer of the problem (𝑃).

Theorem 3.1 shows that problem (𝑃) and problem (𝑇𝑃) have the same minimizers except the infeasible points. If we can find a global minimizer of problem (𝑇𝑃), then problem (𝑃) is solved.

In most cases, using the local search algorithm to solve problem (𝑇𝑃) will trap in a local minimizer of problem (𝑇𝑃). We hope to find a method to escape from a local minimizer and to get a better one. Section 4 will give an auxiliary function for solving this problem.

4. Auxiliary Function and Its Properties Proof

Let 𝑥^∗₁ be the current best local minimizer of problem (𝑇𝑃), and define the following two integer sets:𝑆₁=𝑥∈Ω𝑍^𝑛∣𝐹(𝑥)⩾𝐹𝑥^∗₁,𝑆₂=𝑥∈Ω𝑍^𝑛∣𝐹(𝑥)<𝐹𝑥^∗₁. (4.1)

To begin with, a unary function is given as follows:𝑤(𝑡)=1,𝑡⩾0,𝑡,𝑡<0. (4.2)

Consider the following auxiliary function:𝐴𝐹𝑥,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖×𝑤𝐹(𝑥)−𝐹𝑥^∗₁1+‖𝑥−𝑥^∗₁‖, (4.3) and its properties are analyzed as follows.

Theorem 4.1. 𝐴𝐹(𝑥,𝑥^∗₁) has no local minimizer in set 𝑆₁⧵𝑉.

By definition of 𝑆₁, for all 𝑥∈𝑆₁⧵𝑉, one has 𝐹(𝑥)⩾𝐹(𝑥^∗₁), then 𝐴𝐹𝑥,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖>0. (4.4)

In order to prove that 𝑥 is not a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁), we only need to find a point 𝑦₀ in 𝑁(𝑥) such that 𝐴𝐹(𝑦₀,𝑥^∗₁)<𝐴𝐹(𝑥,𝑥^∗₁).

If there exists 𝑦₀∈𝑁(𝑥) satisfies 𝐹(𝑦₀)<𝐹(𝑥^∗₁), then 𝐴𝐹𝑦₀,𝑥^∗₁=𝐹𝑦₀−𝐹𝑥^∗₁<0<𝐴𝐹𝑥,𝑥^∗₁. (4.5) For all 𝑦∈𝑁(𝑥) with 𝐹(𝑦)⩾𝐹(𝑥^∗₁), since 𝑥∉𝑉, there exists 𝑦₀∈𝑁(𝑥) such that ‖𝑦₀−𝑥^∗₁‖>‖𝑥−𝑥^∗₁‖, then 𝐴𝐹𝑦₀,𝑥^∗₁=11+‖𝑦₀−𝑥^∗₁‖<11+‖𝑥−𝑥^∗₁‖=𝐴𝐹𝑥,𝑥^∗₁. (4.6)Therefore, 𝐴𝐹(𝑥,𝑥^∗₁) has no minimizer in the set 𝑆₁⧵𝑉.

Theorem 4.2. If there exists another local minimizer 𝑥^∗₂ of 𝐹(𝑥) such that 𝐹(𝑥^∗₂)<𝐹(𝑥^∗₁), then 𝑥^∗₂ is also a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁).

Proof. By 𝐹(𝑥^∗₂)<𝐹(𝑥^∗₁), one has 𝐴𝐹𝑥^∗₂,𝑥^∗₁=𝐹𝑥^∗₂−𝐹𝑥^∗₁<0. (4.7)

Since 𝑥^∗₂ is a local minimizer of 𝐹(𝑥), one has 𝐹(𝑦)⩾𝐹(𝑥^∗₂) for all 𝑦∈𝑁(𝑥^∗₂).

When 𝐹(𝑦)⩾𝐹(𝑥^∗₁), and by (4.3), one has 𝐴𝐹𝑦,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖>0>𝐴𝐹𝑥^∗₂,𝑥^∗₁. (4.8)

When 𝐹(𝑦)<𝐹(𝑥^∗₁), and by (4.3), one has 𝐴𝐹𝑦,𝑥^∗₁=𝐹𝑦−𝐹𝑥^∗₁⩾𝐹𝑥^∗₂−𝐹𝑥^∗₁=𝐴𝐹𝑥^∗₂,𝑥^∗₁. (4.9)

Namely, 𝐴𝐹(𝑦,𝑥^∗₁)⩾𝐴𝐹(𝑥^∗₂,𝑥^∗₁) for all 𝑦∈𝑁(𝑥^∗₂). So, 𝑥^∗₂ is a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁).

Theorem 4.3. Any local minimizer 𝑥 of 𝐴𝐹(𝑥,𝑥^∗₁) is in set 𝑆₂ or a vertex of Ω⋂𝑍^𝑛. If 𝐴𝐹(𝑥,𝑥^∗₁)<0, then 𝑥 is not only a local minimizer of 𝐹(𝑥) which is better than 𝑥^∗₁, but also a local minimizer of 𝑓(𝑥) over 𝑆.

Proof. Let 𝑥 be a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁), and suppose that 𝑥 is not a vertex of Ω⋂𝑍^𝑛, then there exists an integer 𝑖₀∈[1,𝑛] (𝑛 is the dimension of problem (𝑃)), such that 𝑥±𝑒_𝑖0∈Ω⋂𝑍^𝑛. It can be proved that 𝐹(𝑥)<𝐹(𝑥^∗₁).
In fact, suppose that 𝐹(𝑥)⩾𝐹(𝑥^∗₁) holds.
If thereexists𝑦₁∈{𝑥±𝑒_𝑖0} satisfies 𝐹(𝑦₁)<𝐹(𝑥^∗₁), then 𝐴𝐹𝑥,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖>0>𝐹𝑦₁−𝐹𝑥^∗₁=𝐴𝐹𝑦₁,𝑥^∗₁. (4.10)
Otherwise, thereexists𝑦₂∈{𝑥±𝑒_𝑖0} satisfies ‖𝑦₂−𝑥^∗₁‖>‖𝑥−𝑥^∗₁‖, then 𝐴𝐹𝑥,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖>11+‖𝑦₂−𝑥^∗₁‖=𝐴𝐹𝑦₂,𝑥^∗₁, (4.11) which contradicts with the fact that 𝑥 is a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁), so 𝑥∈𝑆₂.
If 𝐴𝐹(𝑥,𝑥^∗₁)<0, we have 𝐹(𝑥)<𝐹(𝑥^∗₁). In fact, suppose that 𝐹(𝑥)⩾𝐹(𝑥^∗₁), then 𝐴𝐹𝑥,𝑥^∗₁=11+‖𝑥−𝑥^∗₁‖>0, (4.12) which contradicts with 𝐴𝐹(𝑥,𝑥^∗₁)<0.
Suppose that 𝑥 is not a local minimizer of 𝐹(𝑥), then thereexists𝑦₃∈𝑁(𝑥) such that 𝐹(𝑥)>𝐹(𝑦₃) holds. By the definition of 𝐴𝐹(𝑥,𝑥^∗₁), one has 𝐴𝐹(𝑥,𝑥^∗₁)=𝐹(𝑥)−𝐹(𝑥^∗₁)>𝐹(𝑦₃)−𝐹(𝑥^∗₁)=𝐴𝐹(𝑦₃,𝑥^∗₁), which contradicts with the fact that 𝑥 is a local minimizer of 𝐴𝐹(𝑥,𝑥^∗₁).
By the definition of 𝐹(𝑥), we know that 𝐹(𝑥)<𝐹(𝑥^∗₁)⩽𝑀, then 𝑥 is a feasible point. By Theorem 3.1, we know that 𝑥 is a local minimizer of 𝑓(𝑥). The proof is completed.

From Theorems 4.2 and 4.3, it can be seen that 𝑆₂ contains a minimizer of 𝐴𝐹(𝑥,𝑥^∗₁), and when the minimizer 𝑥 of 𝐴𝐹(𝑥,𝑥^∗₁) satisfies 𝐴𝐹(𝑥,𝑥^∗₁)<0, it is also a local minimizer of 𝐹(𝑥) and better than the current local minimizer. It can also be seen that even if 𝑥^∗₁ is an infeasible local minimizer, a feasible local minimizer of 𝑓(𝑥) will be found by an iteration, and there will be no infeasible local minimizer in the subsequent iterations.

5. Global Optimization Algorithm Based on New Auxiliary Function

5.1. Global Optimization Algorithm

Based on the theoretical analysis in Section 4, a global optimization algorithm to solve problem (𝑃) is given as follows.

Algorithm 5.1. Step 0. Take an initial point 𝑥₀∈Ω⋂𝑍^𝑛, a sufficiently large number 𝑀.Step 1. Use the local search algorithm in Section 2 to 𝐹(𝑥) at 𝑥₀, obtain a local minimizer 𝑥^∗₁ of 𝐹(𝑥), and set 𝑘∶=1.Step 2. Let 𝐷={𝑑₁,𝑑₂,…,𝑑_2𝑛}, where 𝑑_𝑖=𝑒_𝑖, 𝑑_𝑛+𝑖=−𝑒_𝑖, 𝑖=1,2,…,𝑛.Step 3. Construct the auxiliary function 𝐴𝐹𝑥,𝑥^∗_𝑘=11+‖𝑥−𝑥^∗_𝑘‖×𝑤𝐹(𝑥)−𝐹𝑥^∗_𝑘1+‖𝑥−𝑥^∗_𝑘‖. (5.1) Let 𝑥_𝑘,𝑖=𝑥^∗_𝑘+𝑑_𝑖, set 𝑀𝑆=∅, and 𝑖∶=1.Step 4. Use the local search algorithm in Section 2 to 𝐴𝐹(𝑥,𝑥^∗_𝑘) with the initial point 𝑥_𝑘,𝑖 and obtain a local minimizer 𝑥^∗_𝑘,𝑖 of 𝐴𝐹(𝑥,𝑥^∗_𝑘) over Ω⋂𝑍^𝑛. If 𝐹(𝑥^∗_𝑘,𝑖)<𝐹(𝑥^∗_𝑘), set 𝑀𝑆=𝑀𝑆⋃{𝑥^∗_𝑘,𝑖}. Go to Step 5.Step 5. If 𝑖⩾2𝑛, go to Step 6; otherwise, 𝑖=∶𝑖+1; go to Step 4.Step 6. If 𝑀𝑆=∅, go to Step 7; otherwise, find a point in 𝑀𝑆≠∅ with the smallest function value of 𝐹(𝑥) and denote it as 𝑥^∗_𝑘+1. Set 𝑘∶=𝑘+1, and go back to Step 3.Step 7. Output 𝑥^∗_𝑘.

Remarks on the Algorithm
(1)In Step 4, if a local minimizer 𝑥^∗_𝑘+1 of 𝐴𝐹(𝑥,𝑥^∗_𝑘) is found, then 𝑥^∗_𝑘+1 is a local minimizer of the problem (𝑃). It is different from the discrete filled function method in which the local search algorithm should be employed to 𝐹(𝑥) and 𝐴𝐹(𝑥,𝑥^∗_𝑘) repeatedly. Therefore, the proposed algorithm can reduce the computational cost.(2)From the above theorems, we can see that although 𝑥^∗₁ may be an infeasible point, {𝑥^∗_𝑖,𝑖⩾2} are all feasible local minimizers of 𝑓(𝑥) over 𝑆.(3)Generally, 𝑀 must be a sufficiently large number. In the implementation of the proposed algorithm, we first estimate an upper bound 𝐵_upper of 𝑓(𝑥) over 𝑆, then take 𝑀=𝐵_upper+1 or directly the upper bound 𝐵_upper if it is big enough.

5.2. Numerical Experiment

We apply the proposed algorithm to solve the following test problems.

Problem 1. min𝑓(𝑥)=𝑥₁−10³+𝑥₂−20³,s.t.−𝑥₁−5²−𝑥₂−5²+100⩽0−𝑥₁+10⩽0,−𝑥₂+5⩽0,𝑥∈Ω=𝑥∣0⩽𝑥_𝑖⩽100,𝑖=1,2. (5.2) The box Ω has 10201 points. The global minimizer is 𝑥^∗=[15,5]^𝑇 with 𝑓(𝑥^∗)=−3250.

Problem 2. min𝑓(𝑥)=−2⁴×⁴_𝑖=1𝑥_𝑖,100s.t.⁴_𝑖=1𝑥_𝑖100²=1,𝑥∈Ω=𝑥∣0⩽𝑥_𝑖⩽100,𝑖=1,2,3,4. (5.3) The box Ω has 104060401 points. The global minimizer is 𝑥^∗=[50,50,50,50]^𝑇 with 𝑓(𝑥^∗)=−1.

Problem 3. min𝑓(𝑥)=−25(𝑥₁−2)²−(𝑥₂−2)²−(𝑥₃−1)²−(𝑥₄−4)²−(𝑥₅−1)²−(𝑥₆−4)²,s.t.−(𝑥₃−3)²−𝑥₄+4⩽0,−(𝑥₅−3)²−𝑥₆𝑥+4⩽0,₁−3𝑥₂−2⩽0,−𝑥₁+𝑥₂𝑥−2⩽0,₁+𝑥₂−6⩽0,−𝑥₁−𝑥₂+2⩽0,0⩽𝑥₁⩽6,0⩽𝑥₂⩽8,0⩽𝑥₃⩽5,0⩽𝑥₄⩽6,0⩽𝑥₅⩽10,0⩽𝑥₆⩽10. (5.4) The box Ω has 320166 points. The global minimizer is 𝑥^∗=(5,1,5,0,5,10)^𝑇 with 𝑓(𝑥^∗)=−310.

Problem 4 (Colville’s function). min𝑓(𝑥)=100𝑥₂−𝑥²₁²+1−𝑥₁²+90𝑥₄−𝑥²₃²+1−𝑥₃²+10.1𝑥₂−1²+𝑥₄−1²+19.8𝑥₂−1𝑥₄−1,s.t.−10≤𝑥_𝑖≤10,𝑥_𝑖isinteger,𝑖=1,2,3,4. (5.5)
This box-constrained problem has 1.94481×10⁵ feasible points. The global minimizer is 𝑥^∗=[1,1,1,1]^𝑇 with 𝑓(𝑥^∗)=0.

Problem 5 (Goldstein and Price’s function). min𝑓(𝑥)=𝑔(𝑥)ℎ(𝑥),s.t.𝑥_𝑖=𝑦_𝑖1000,−2000⩽𝑦_𝑖⩽2000,𝑦_𝑖isinteger,𝑖=1,2, (5.6) where 𝑔(𝑥)=1+(𝑥₁+𝑥₂+1)²(19−14𝑥₁+3𝑥²₁−14𝑥₂+16𝑥₁𝑥₂+3𝑥²₂) and ℎ(𝑥)=30+(2𝑥₁−3𝑥₂)²(18−32𝑥₁+12𝑥²₁+48𝑥₂−36𝑥₁𝑥₂+27𝑥²₂).
This box constrained problem has 1.6008001×10⁷ feasible points. The global minimizer is 𝑥^∗=[0,−1]^𝑇 with 𝑓(𝑥^∗)=3.

Problem 6 (Beale’s function). min𝑓(𝑥)=1.5−𝑥₁1−𝑥₂²+2.25−𝑥₁1−𝑥²₂²+2.65−𝑥₁1−𝑥³₂²,s.t.𝑥_𝑖=𝑦_𝑖1000,−10000⩽𝑦_𝑖⩽10000,𝑦_𝑖isinteger,𝑖=1,2. (5.7) This box-constrained problem has 4.00040001×10⁸ feasible points. The global minimizer is 𝑥^∗=[3,0.5]^𝑇 with 𝑓(𝑥^∗)=0.

Problem 7 (Rosenbrock’s function). min𝑓(𝑥)=²⁴_𝑖=1100𝑥_𝑖+1−𝑥²_𝑖²+1−𝑥_𝑖²,s.t.−5⩽𝑥_𝑖⩽5,𝑥_𝑖isinteger,𝑖=1,2,…,25. (5.8)
This box-constrained problem has 1.08347×10²⁶ feasible points. The global minimizer is 𝑥^∗=[1,1,…,1]^𝑇 with 𝑓(𝑥^∗)=0.

We apply the proposed algorithm to solve the above problems, and numerical results are shown in Tables 1 to 5. In these tables, 𝑥₀ denotes the initial point; 𝑥^∗ denotes the last local minimizer of the original objective function; 𝐸_𝑓+𝑔 denotes the total number evaluations of original function and auxiliary (filled) function.

Table 1 

Numerical results for Problems 1–3.

Table 1 gives the results obtained by the proposed algorithm for Problems 1–3.

Problems 1–3 are with nonlinear objective function and nonlinear constraint functions. It is hard for an algorithm to solve. However, it can be seen from Table 1 that for Problems 1–3, the proposed algorithm can find the global optimal solutions for all these three problems using relatively small number of function evaluations.

In order to demonstrate the efficiency of the proposed algorithm, we compare the proposed algorithm with other two discrete filled function methods DFMA and DFMB ([14, 19]) for Problems 4–7. The results are summarized in Tables 2–5, where DFMA and DFMB denote the discrete filled function methods proposed in [14, 19], respectively, and AFM denotes the proposed auxiliary function algorithm. In implementation of the algorithm, 𝐹(𝑥)=𝑓(𝑥) for Problems 4–7.

Table 2 

Numerical results of Problem 4 (𝑀=10⁸).

Table 3 

Numerical results of Problem 5 (𝑀=10⁸).

Table 4 

Numerical results of Problem 6 (𝑀=10⁸).

Table 5 

Numerical results of Problem 7 (𝑀=10⁶).

For Problem 4, it can be seen from Table 2 that all three algorithms can find global optimal solution, but the number of the function evaluations used by proposed algorithm AFM is only about 1/3 of that used by DFMA and only about half of that used by DFMB. For Problems 5 and 6, we can see from Tables 3-4 that all three algorithms can also find global optimal solutions, but the number of function evaluations used by DFMA is more than twice of that used by AFM, and the number of function evaluations used by DFMB is more than 1.5 times of that used by AFM. For Problem 7, it can be seen from Table 5 that the number of function evaluations used by AFM is also much smaller than those used by DFMA and DFMB. These results demonstrate that the proposed algorithm is effective and more efficient than the compared algorithms.

6. Concluding Remarks

In this paper, we propose an auxiliary function method for finding the global minimizer of integer programming problems. The auxiliary function can help to escape from a local minimizer and to get a better one. The minimizer of the auxiliary function also is a feasible minimizer of the original problem and better than the current best one obtained. The numerical results show that the proposed algorithm is more effective and efficient than the compared ones.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 60873099), the Ph.D. Programs Foundation of Education Ministry of China (no. 20090203110005), National Program on Key Basic Research Project of China (973 Program) Grant no. 2010CB328102, and the Fundamental Research Funds for the Central Universities (no. k50510030014).