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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 171697, 11 pages
http://dx.doi.org/10.1155/2011/171697
Research Article

Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming

1Department of Mathematics, Shanghai Second Polytechnic University, Shanghai 201209, China
2Department of Mathematics, Henan University of Science and Technology, Luoyang 471003, China
3Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 27 March 2011; Revised 10 July 2011; Accepted 15 July 2011

Academic Editor: Rigoberto Medina

Copyright © 2011 Wei-Xiang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents a filled function method for finding a global optimizer of integer programming problem. The method contains two phases: the local minimization phase and the filling phase. The goal of the former phase is to identify a local minimizer of the objective function, while the filling phase aims to search for a better initial point for the first phase with the aid of the filled function. A two-parameter filled function is proposed, and its properties are investigated. A corresponding filled function algorithm is established. Numerical experiments on several test problems are performed, and preliminary computational results are reported.

1. Introduction

Consider the following general global nonlinear integer programming: min𝑥𝑋(𝑓(𝑥),P) where 𝑓𝑍𝑛,𝑋𝑍𝑛 is a box set and 𝑍𝑛 is the set of integer points in 𝑅𝑛. The problem (P) is important since lots of real life applications, such as production planning, supply chains, and finance, are allowed to be formulated into this problem.

One of main issues in the global optimization is to avoid being trapped in the basins surrounding local minimizers. Several global optimization solution strategies have been put forward to tackle with the problem (P). These techniques are usually divided into two classes: stochastic method and deterministic method (see [17]). The discrete filled function method is one of the more recently developed global optimization tools for discrete global optimization problems. The first filled function was introduced by Ge and Qin in [8] for continuous global optimization. Papers [6, 7, 911] extend this continuous filled function method to solve integer programming problem. Like the continuous filled function method, the discrete filled function method also contains two phases: local minimization and filling. The local minimization phase uses any ordinary discrete descent method to search for a discrete local minimizer of the problem (P), while the filling phase utilizes an auxiliary function called filled function to find a better initial point for the first phase by minimizing the constructed filled function. The definitions of the filled function proposed in the papers [9, 10] are as follows.

Definition 1.1 (see [9]). 𝑃(𝑥,𝑥) is called a filled function of 𝑓(𝑥) at a discrete local minimizer 𝑥 if 𝑃(𝑥,𝑥) meets the following conditions. (1)𝑃(𝑥,𝑥) has no discrete local minimizers in the set 𝑆1={𝑥𝑋𝑓(𝑥)𝑓(𝑥)}, except a prefixed point 𝑥0𝑆1 that is a minimizer of 𝑃(𝑥,𝑥).(2)If 𝑥 is not a discrete global minimizer of 𝑓(𝑥), then 𝑃(𝑥,𝑥) does have a discrete minimizer in the set 𝑆2={𝑥𝑓(𝑥)<𝑓(𝑥),𝑥𝑋}.

Definition 1.2 (see [10]). 𝑃(𝑥,𝑥) is called a filled function of 𝑓(𝑥) at a discrete local minimizer 𝑥 if 𝑃(𝑥,𝑥) meets the following conditions. (1)𝑃(𝑥,𝑥) has no discrete local minimizers in the set 𝑆1𝑥0, where the prefixed point 𝑥0𝑆1 is not necessarily a local minimizer of 𝑃(𝑥,𝑥).(2)If 𝑥 is not a discrete global minimizer of 𝑓(𝑥), then 𝑃(𝑥,𝑥) has a discrete minimizer in the set 𝑆2.

Although Definitions 1.1 and 1.2 and the corresponding filled functions proposed in the papers [9, 10] have their own advantages, they have some defects in some degree, for example, as the prefixed point 𝑥0 in Definition 1.2 may be a minimizer of the given filled function, which will result in numerical complexity at the iterations or cause the algorithm to fail. To avoid these defects, in this paper, we give a modification of Definitions 1.1 and 1.2 and propose a new filled function.

The rest of this paper is organized as follows. In Section 2, we review some basic concepts of discrete optimization. In Section 3, we propose a discrete filled function and investigate its properties. In Section 4, we state our algorithm and report preliminary numerical results. And, at last, we give our conclusion in Section 5.

2. Basic Knowledge and Some Assumptions

Consider the problem (P). Throughout this paper, we make the following assumptions.

Assumption 2.1. There exists a constant 𝐷>0 satisfying 1𝐷=max𝑥1,𝑥2𝑋,𝑥1𝑥2𝑥1𝑥2<.

Assumption 2.2. There exists a constant 𝐿>0, such that ||||𝑓(𝑥)𝑓(𝑦)𝐿𝑥𝑦(2.1) holds, for any 𝑥, 𝑦𝑥𝑋𝑁(𝑥), where 𝑁(𝑥) is a neighborhood of the point 𝑥 as defined in Definition 2.4.

Most of the existing discrete filled function methods are used for solving a box constrained problem. To an unconstrained global optimization problem (UP):min𝑥𝑅𝑛𝑓(𝑥), if 𝑓(𝑥) satisfies lim𝑥+𝑓(𝑥)=+, then there exists a box set which contains all discrete global minimizers of 𝑓(𝑥). Therefore, (UP) can be turned into an equivalent formulation in (P) and solved by any discrete filled function method.

For convenience, in the following, we recall some preliminaries which will be used throughout this paper.

Definition 2.3 (see [10]). The set of all feasible directions at 𝑥𝑋 is defined by 𝐷𝑥={𝑑𝐷𝑥+𝑑𝑋}, where 𝐷={±𝑒𝑖𝑖=1,2,,𝑛},  𝑒𝑖 is the ith unit vector (the 𝑛-dimensional vector with the ith component equal to one and all other components equal to zero).

Definition 2.4 (see [10]). For any 𝑥𝑍𝑛, the discrete neighborhood of 𝑥 is defined by 𝑁(𝑥)={𝑥,𝑥±𝑒𝑖,𝑖=1,2,,𝑛}.

Definition 2.5 (see [10]). A point 𝑥𝑋 is called a discrete local minimizer of 𝑓(𝑥) over 𝑋 if 𝑓(𝑥)𝑓(𝑥), for all 𝑥𝑋𝑁(𝑥). Furthermore, if 𝑓(𝑥)𝑓(𝑥), for all 𝑥𝑋, then 𝑥 is called a strict discrete local minimizer of 𝑓(𝑥) over 𝑋. If, in addition, 𝑓(𝑥)<𝑓(𝑥), for all (𝑥𝑋{𝑥}), then 𝑥 is called a strict discrete local (global) minimizer of 𝑓(𝑥) over 𝑋.

Algorithm 2.6 (discrete local minimization method). (1)Start from an initial point 𝑥𝑋.(2)If 𝑥 is a local minimizer of 𝑓 over 𝑋, then stop. Otherwise, let 𝑑=argmin𝑑𝑖𝐷𝑥𝑓𝑥+𝑑𝑖𝑓𝑥+𝑑𝑖<𝑓(𝑥).(2.2)(3)Let 𝑥=𝑥+𝑑, and go to Step (2).
Let 𝑥 be a local minimizer of the problem (P). The new definition of the filled function of 𝑓 at 𝑥 is given as follows.

Definition 2.7. 𝑃(𝑥,𝑥) is called a discrete filled function of 𝑓(𝑥) at a discrete local minimizer 𝑥 if 𝑃(𝑥,𝑥) has the following properties. (1)𝑥 is a strict discrete local maximizer of 𝑃(𝑥,𝑥) over 𝑋. (2)𝑃(𝑥,𝑥) has no discrete local minimizers in the region 𝑆1=𝑥𝑥𝑓(𝑥)𝑓𝑥,𝑥𝑋.(2.3)(3)If 𝑥 is not a discrete global minimizer of 𝑓(𝑥), then 𝑃(𝑥,𝑥) does have a discrete minimizer in the region 𝑆2=𝑥𝑥𝑓(𝑥)<𝑓,𝑥𝑋.(2.4)

3. Properties of the Proposed Discrete Filled Function 𝑇(𝑥,𝑥,𝑞,𝑟)

Let 𝑥 denote the current discrete local minimizer of (P). Based on Definition 2.7, a novel filled function is proposed as follows:𝑇𝑥,𝑥=1,𝑞,𝑟𝑞+𝑥𝑥𝜑𝑞𝑥max𝑓(𝑥)𝑓,+𝑟,0(3.1) where𝜑𝑞𝜋(𝑡)=2𝑞arctan𝑡if𝑡0,0if𝑡=0,(3.2) where 𝑟>0 and 𝑞>0 are two parameters and 𝑟 satisfies 0<𝑟<min𝑓(𝑥1)𝑓(𝑥2),𝑥1,𝑥2𝑋|𝑓(𝑥1)𝑓(𝑥2)|.

The following theorems ensure that 𝑇(𝑥,𝑥,𝑞,𝑟) is a filled function under some conditions.

Theorem 3.1. If 0<𝑞<min(𝑟,𝜋/4), then 𝑥 is a strict local maximizer of 𝑇(𝑥,𝑥,𝑞,𝑟).

Proof. Since 𝑥 is a local minimizer of (P), there exists a neighborhood 𝑁(𝑥) of 𝑥 such that 𝑓(𝑥)𝑓(𝑥) and 𝑥𝑥=1 hold, for any 𝑥𝑁(𝑥)𝑋. It follows that 𝑇𝑥,𝑥=1,𝑞,𝑟𝜋𝑞+12𝑞arctan𝑓(𝑥)𝑓(𝑥,𝑇𝑥)+𝑟,𝑥=1,𝑞,𝑟𝑞𝜋2𝑞arctan𝑟.(3.3)
By the condition 0<𝑞<min(𝑟,𝜋/4) and the fact that the inequality arctan𝑎arctan𝑏𝑎𝑏(3.4) holds for any real number 𝑎𝑏, we have Δ=𝑇𝑥,𝑥𝑥,𝑞,𝑟𝑇,𝑥=1,𝑞,𝑟𝑞𝑞(𝑞+1)arctan𝑟𝜋2+1𝑞𝑞+1arctan𝑟𝑞arctan𝑓(𝑥)𝑓(𝑥1)+𝑟𝑞𝜋(𝑞+1)arctan12+𝑞1𝑞+1𝑟1𝑓(𝑥)𝑓(𝑥𝜋)+𝑟=41+1𝑞(𝑞+1)𝑞𝑞+1𝑟𝑥𝑓(𝑥)𝑓𝑓(𝑥)𝑓(𝑥𝜋)+𝑟41+1𝑞(𝑞+1)=1𝑞+1𝜋𝑞(𝑞+1)𝑞4<0.(3.5)
Hence, 𝑇(𝑥,𝑥,𝑞,𝑟)<𝑇(𝑥,𝑥,𝑞,𝑟), which implies that 𝑥 is a strict local maximizer of 𝑇(𝑥,𝑥,𝑞,𝑟).

Lemma 3.2. For every 𝑥𝑋, there exists 𝑑𝐷 such that 𝑥+𝑑𝑥>𝑥𝑥.

For the proof of this lemma, see, for example, [6] or [7].

Theorem 3.3. Suppose that 0<𝑞<min(1,𝑟,((𝜋2)/4(1+𝐷))𝑟). If 𝑓(𝑥)𝑓(𝑥) and 𝑥𝑥, then 𝑥 is not a local minimizer of 𝑇(𝑥,𝑥,𝑞,𝑟).

Proof. For any 𝑥𝑥 with 𝑓(𝑥)𝑓(𝑥), by Lemma 3.2, there exists a direction 𝑑𝐷 with 𝑥+𝑑𝑥𝑋𝑁(𝑥) such that 𝑥+𝑑𝑥>𝑥𝑥. For this 𝑑, we consider the following three cases. Case 1 (𝑓(𝑥+𝑑)𝑓(𝑥)). In this case, by using the given condition and the fact that the inequality arctan𝑎𝑎(3.6) holds for any real number 𝑎0, we have Δ1=𝑇𝑥+𝑑,𝑥,𝑞,𝑟𝑇𝑥,𝑥=1,𝑞,𝑟𝑞+𝑥+𝑑𝑥𝜋2𝑞arctan𝑓(𝑥+𝑑)𝑓(𝑥1)+𝑟𝑞+𝑥𝑥𝜋2𝑞arctan𝑓(𝑥)𝑓(𝑥=𝑞)+𝑟arctan𝑓(𝑥+𝑑)𝑓(𝑥)𝜋+𝑟2𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥)+1𝑞+𝑥𝑥𝑞arctan𝑓(𝑥)𝑓(𝑥𝑞)+𝑟arctan𝑓(𝑥+𝑑)𝑓(𝑥𝑞)+𝑟arctan𝑟𝜋2𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥)+1𝑞+𝑥𝑥𝑞arctan𝑓(𝑥)𝑓(𝑥𝑞)+𝑟+arctan𝑓(𝑥+𝑑)𝑓(𝑥𝑞)+𝑟𝑟𝜋2𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥)+1𝑞+𝑥𝑥𝑞𝑟+𝑞𝑟𝜋12𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥)+1𝑞+𝑥𝑥2𝑞𝑟𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥𝜋)12+2𝑞𝑟𝑞+𝑥+𝑑𝑥𝑥+𝑑𝑥𝑥𝑥.(3.7) Since 𝑞+𝑥+𝑑𝑥1+𝐷 and 𝑥+𝑑𝑥𝑥𝑥1, we have Δ1𝑥+𝑑𝑥𝑥𝑥(𝑞+𝑥+𝑑𝑥)(𝑞+𝑥𝑥𝜋)12+2𝑞𝑟(1+𝐷)<0.(3.8) Hence, in this case, 𝑥 is not a local minimizer of 𝑇(𝑥,𝑥,𝑞,𝑟).Case 2 (𝑓(𝑥+𝑑)<𝑓(𝑥) and 𝑓(𝑥+𝑑)𝑓(𝑥)+𝑟0). In this case, we have Δ2=𝑇𝑥+𝑑,𝑥,𝑞,𝑟𝑇𝑥,𝑥,𝑞,𝑟=𝑇𝑥,𝑥,𝑞,𝑟<0,(3.9) which means the conclusion is true in this case.Case 3 (𝑓(𝑥+𝑑)<𝑓(𝑥) and 𝑓(𝑥+𝑑)𝑓(𝑥)+𝑟>0). In this case, we have 𝑇𝑥+𝑑,𝑥=1,𝑞,𝑟𝑞+𝑥+𝑑𝑥𝜋2𝑞arctan𝑓(𝑥+𝑑)𝑓(𝑥<1)+𝑟𝑞+𝑥+𝑑𝑥𝜋2𝑞arctan𝑟<1𝑞+𝑥𝑥𝜋2𝑞arctan𝑓(𝑥)𝑓(𝑥)+𝑟=𝑇𝑥,𝑥.,𝑞,𝑟(3.10) Hence, in this case, 𝑥 is not a local minimizer of 𝑇(𝑥,𝑥,𝑞,𝑟).
The above discussion implies that 𝑥 is not a discrete local minimizer of 𝑇(𝑥,𝑥,𝑞,𝑟).

Theorem 3.4. Assume that 𝑥 is not a global minimizer of 𝑓(𝑥), then there exists a minimizer 𝑥1 of 𝑇(𝑥,𝑥,𝑞,𝑟) in 𝑆2.

Proof. Since 𝑥 is not a global minimizer of 𝑓(𝑥), there exists 𝑥1𝑆2 such that 𝑓(𝑥1)<𝑓(𝑥)𝑟; it follows that 𝑇(𝑥1,𝑥,𝑞,𝑟)=0. On the other hand, by the structure of 𝑇(𝑥,𝑥,𝑞,𝑟), we have 𝑇(𝑥,𝑥,𝑞,𝑟)0 for any 𝑥𝑋. This shows 𝑥1 is a minimizer of 𝑇(𝑥,𝑥,𝑞,𝑟).

4. Filled Function Algorithm and Numerical Experiments

Based on the theoretical results in the previous section, the filled function method for (P) is described now as follows.

Algorithm 4.1 (discrete filled function method). (1)Input the lower bound of 𝑟, namely, 𝑟𝐿=1𝑒8. Input an initial point 𝑥0(0)𝑋. Let 𝐷={±𝑒𝑖,𝑖=1,2,,𝑛}. (2)Starting from an initial point 𝑥0(0)𝑋, minimize 𝑓(𝑥) and obtain the first local minimizer 𝑥0 of 𝑓(𝑥). Set 𝑘=0,  𝑟=1, and 𝑞=1.(3)Set 𝑥𝑘(0)𝑖=𝑥𝑘+𝑑𝑖, 𝑑𝑖𝐷,  𝑖=1,2,,2𝑛,  𝐽=[1,2,,2𝑛], and 𝑗=1.(4)Set 𝑖=𝐽𝑗 and 𝑥=𝑥𝑘(0)𝑖.(5)If 𝑓(𝑥)<𝑓(𝑥𝑘), then use 𝑥 as initial point for discrete local minimization method to find another local minimizer 𝑥𝑘+1 such that 𝑓(𝑥𝑘+1)<𝑓(𝑥𝑘). Set 𝑘=𝑘+1, and go to (3).(6)Let 𝐷0={𝑑𝐷𝑥+𝑑𝑋}. If there exists 𝑑𝐷0 such that 𝑓(𝑥+𝑑)<𝑓(𝑥𝑘), then use 𝑥+𝑑, where 𝑑=argmin𝑑𝐷0{𝑓(𝑥+𝑑)}, as an initial point for a discrete local minimization method to find another local minimizer 𝑥𝑘+1 such that 𝑓(𝑥𝑘+1)<𝑓(𝑥𝑘). Set 𝑘=𝑘+1, and go to (3).(7)Let 𝐷1={𝑑𝐷0𝑥+𝑑𝑥>𝑥𝑥}. If 𝐷1=, then go to (10).(8)If there exists 𝑑𝐷1 such that 𝑇(𝑥+𝑑,𝑥𝑘,𝑞,𝑟)𝑇(𝑥,𝑥𝑘,𝑞,𝑟), then set 𝑞=0.1𝑞,  𝐽=[𝐽𝑗,,𝐽2𝑛,𝐽1,,𝐽𝑗1],  𝑗=1, and go to (4).(9)Let 𝐷2={𝑑𝐷1𝑓(𝑥+𝑑)<𝑓(𝑥),𝑇(𝑥+𝑑,𝑥𝑘,𝑞,𝑟)<𝑇(𝑥,𝑥𝑘,𝑞,𝑟)}. If 𝐷2, then set 𝑑=argmin𝑑𝐷2{𝑓(𝑥+𝑑)+𝑇(𝑥+𝑑,𝑥𝑘,𝑞,𝑟)}. Otherwise set 𝑑=argmin𝑑𝐷1{𝑇(𝑥+𝑑,𝑥𝑘,𝑞,𝑟)},𝑥=𝑥+𝑑, and go to (6).(10)If 𝑖<2𝑛, then set 𝑖=𝑖+1, and go to (4). (11)Set 𝑟=0.1𝑟. If 𝑟𝑟𝐿, go to (3). Otherwise, the algorithm is incapable of finding a better minimizer starting from the initial points, {𝑥𝑘(0)𝑖𝑖=1,2,,2𝑛}. The algorithm stops, and 𝑥𝑘 is taken as a global minimizer.

The motivation and mechanism behind the algorithm are explained below.

A set of 2𝑛 initial points is chosen in Step (3) to minimize the discrete filled function.

Step (5) represents the situation where the current computer-generated initial point for the discrete filled function method satisfies 𝑓(𝑥)<𝑓(𝑥𝑘). Therefore, we can further minimize the primal objective function 𝑓(𝑥) by any discrete local minimization method starting from 𝑥.

Step (7) aims at selecting a better successor point. If 𝐷2 is not empty, then we get a feasible direction which reduce both the objective function value and filled function value. Otherwise, we can get a descent feasible direction which reduce only filled function value.

In the following, we perform the numerical experiments for five test problems using the above proposed filled function algorithm. All the numerical experiments are programmed in MATLAB 7.0.4. The proposed filled function algorithm succeeds in identifying the global minimizers of the test problems. The computational results are summarized in Table 1, and the symbols used are given as follows:PN: the Nth problem.DN: the dimension of objective function of a problem.IN: the number of iteration cycles.TI: the CPU time in seconds for the algorithm to stop.TN: the number of filled function evaluations for the algorithm to stop.FN: the number of objective function evaluations for the algorithm to stop.

tab1
Table 1

Problem 1. One has 𝑥min𝑓(𝑥)=1002𝑥212+1𝑥12𝑥+904𝑥232+1𝑥32𝑥+10.1212+𝑥412𝑥+19.82𝑥14,1s.t.10𝑥𝑖10,𝑥𝑖isinteger,𝑖=1,2,3,4.(4.1) This problem has 2141.94×105 feasible points where 41 of them are discrete local minimizers but only one of those discrete local minimizers is the discrete global minimum solution: 𝑥global=(1,1,1,1) with 𝑓(𝑥global)=0. We used three initial points in our experiment:(9,6,5,6), (10,10,10,10), (10,10,10,10).

Problem 2. One has min𝑓(𝑥)=𝑔(𝑥)(𝑥),s.t.𝑥𝑖=0.001𝑦𝑖,2000𝑦𝑖2000,𝑦𝑖isinteger,𝑖=1,2,(4.2) where 𝑥𝑔(𝑥)=1+1+𝑥2+121914𝑥1+3𝑥2114𝑥2+6𝑥1𝑥2+3𝑥2,(𝑥)=30+2𝑥13𝑥221832𝑥1+12𝑥21+48𝑥236𝑥1𝑥2+27𝑥22.(4.3) This problem has 400121.60×107 feasible points. More precisely, it has 207 and 2 discrete local minimizers in the interior and the boundary of box 2.00𝑥𝑖2.00,𝑖=1,2, respectively. Nevertheless, it has only one discrete global minimum solution: 𝑥global=(0.000,1.000) with 𝑓(𝑥global)=3. We used three initial points in our experiment: (2000,2000), (2000,2000), (1196,1156).

Problem 3. One has min𝑓(𝑥)=1.5𝑥11𝑥22+2.25𝑥11𝑥222+2.625𝑥11𝑥322,s.t.𝑥𝑖=0.001𝑦𝑖,104𝑦𝑖104,𝑦𝑖isinteger,𝑖=1,2.(4.4) This problem has 2000124.00×108 feasible points and many discrete local minimizers, but it has only one discrete global minimum solution: 𝑥global=(3,0.5) with 𝑓(𝑥global)=0. We used three initial points in our experiment:(9997,6867), (10000,10000), (10000,10000).

Problem 4. One has 𝑥min𝑓(𝑥)=1+10𝑥22𝑥+53𝑥42+𝑥22𝑥34𝑥+101𝑥44,s.t.𝑥𝑖=0.001𝑦𝑖,104𝑦𝑖104,𝑦𝑖isinteger,𝑖=1,2,3,4.(4.5) This problem has 2000141.60×1017 feasible points and many local minimizers, but it has only one global minimum solution: 𝑥global=(0,0,0,0) with 𝑓(𝑥global)=0. We used three initial points in our experiment:(1000,1000,1000,1000), (10000,10000,10000,10000), (10000,,10000).

Problem 5. One has 𝑥min𝑓(𝑥)=112+𝑥𝑛12+𝑛𝑛1𝑖=1𝑥(𝑛𝑖)2𝑖𝑥𝑖+12,s.t.5𝑥𝑖5,𝑥𝑖isinteger,𝑖=1,2,,𝑛.(4.6) This problem has many local minimizers, but it has only one global minimum solution: 𝑥global=(1,,1) with 𝑓(𝑥global)=0.
In this problem, we used initial point (5,,5) in our experiment for 𝑛=25,50,100, respectively.

Problem 6. One has min𝑓(𝑥)=𝑛𝑖=1𝑥4𝑖+𝑛𝑖=1𝑥𝑖2,s.t.5𝑥𝑖5,𝑥𝑖isinteger,𝑖=1,2,,𝑛.(4.7) This problem has many local minimizers, but it has only one global minimum solution: 𝑥global=(1,1,,1) with 𝑓(𝑥global)=0.
In this problem, we used initial point (5,,5) in our experiment for 𝑛=25,50,100, respectively.

5. Conclusions

We have proposed a new two-parameter filled function and presented a corresponding filled function algorithm for the solution of the box constrained global nonlinear integer programming problem. Numerical experiments are also implemented, and preliminary computational results are reported. Our future work is to generalize the discrete filled function techniques to mixed nonlinear integer global optimization problem.

Acknowledgment

This paper was partially supported by the NNSF of China under Grant No. 10571137 and 10971053.

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