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: min๐‘ฅโˆˆ๐‘‹(๐‘“(๐‘ฅ),P) where ๐‘“โˆถ๐‘๐‘›โ†’โ„œ,๐‘‹โŠ‚๐‘๐‘› is a box set and ๐‘๐‘› is the set of integer points in ๐‘…๐‘›. 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 [1โ€“7]). 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, 9โ€“11] 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]). ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) is called a filled function of ๐‘“(๐‘ฅ) at a discrete local minimizer ๐‘ฅโˆ— if ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) meets the following conditions. (1)๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) has no discrete local minimizers in the set ๐‘†1={๐‘ฅโˆˆ๐‘‹โˆถ๐‘“(๐‘ฅ)โ‰ฅ๐‘“(๐‘ฅโˆ—)}, except a prefixed point ๐‘ฅ0โˆˆ๐‘†1 that is a minimizer of ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—).(2)If ๐‘ฅโˆ— is not a discrete global minimizer of ๐‘“(๐‘ฅ), then ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) does have a discrete minimizer in the set ๐‘†2={๐‘ฅโˆฃ๐‘“(๐‘ฅ)<๐‘“(๐‘ฅโˆ—),๐‘ฅโˆˆ๐‘‹}.

Definition 1.2 (see [10]). ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) is called a filled function of ๐‘“(๐‘ฅ) at a discrete local minimizer ๐‘ฅโˆ— if ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) meets the following conditions. (1)๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) has no discrete local minimizers in the set ๐‘†1โงต๐‘ฅ0, where the prefixed point ๐‘ฅ0โˆˆ๐‘†1 is not necessarily a local minimizer of ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—).(2)If ๐‘ฅโˆ— is not a discrete global minimizer of ๐‘“(๐‘ฅ), then ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) has a discrete minimizer in the set ๐‘†2.

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 ๐‘ฅ0 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 ๐ท>0 satisfying 1โ‰ค๐ท=max๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹,๐‘ฅ1โ‰ ๐‘ฅ2โ€–๐‘ฅ1โˆ’๐‘ฅ2โ€–<โˆž.

Assumption 2.2. There exists a constant ๐ฟ>0, such that ||||๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ‰ค๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–(2.1) holds, for any ๐‘ฅ, โ‹ƒ๐‘ฆโˆˆ๐‘ฅโˆˆ๐‘‹๐‘(๐‘ฅ), where ๐‘(๐‘ฅ) is a neighborhood of the point ๐‘ฅ 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):min๐‘ฅโˆˆ๐‘…๐‘›๐‘“(๐‘ฅ), if ๐‘“(๐‘ฅ) satisfies limโ€–๐‘ฅโ€–โ†’+โˆž๐‘“(๐‘ฅ)=+โˆž, then there exists a box set which contains all discrete global minimizers of ๐‘“(๐‘ฅ). 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 ๐‘ฅโˆˆ๐‘‹ is defined by ๐ท๐‘ฅ={๐‘‘โˆˆ๐ทโˆถ๐‘ฅ+๐‘‘โˆˆ๐‘‹}, where ๐ท={ยฑ๐‘’๐‘–โˆถ๐‘–=1,2,โ€ฆ,๐‘›},โ€‰โ€‰๐‘’๐‘– is the ith unit vector (the ๐‘›-dimensional vector with the ith component equal to one and all other components equal to zero).

Definition 2.4 (see [10]). For any ๐‘ฅโˆˆ๐‘๐‘›, the discrete neighborhood of ๐‘ฅ is defined by ๐‘(๐‘ฅ)={๐‘ฅ,๐‘ฅยฑ๐‘’๐‘–,๐‘–=1,2,โ€ฆ,๐‘›}.

Definition 2.5 (see [10]). A point ๐‘ฅโˆ—โˆˆ๐‘‹ is called a discrete local minimizer of ๐‘“(๐‘ฅ) over ๐‘‹ if ๐‘“(๐‘ฅโˆ—)โ‰ค๐‘“(๐‘ฅ), for all ๐‘ฅโˆˆ๐‘‹โˆฉ๐‘(๐‘ฅโˆ—). Furthermore, if ๐‘“(๐‘ฅโˆ—)โ‰ค๐‘“(๐‘ฅ), for all ๐‘ฅโˆˆ๐‘‹, then ๐‘ฅโˆ— is called a strict discrete local minimizer of ๐‘“(๐‘ฅ) over ๐‘‹. If, in addition, ๐‘“(๐‘ฅโˆ—)<๐‘“(๐‘ฅ), for all (๐‘ฅโˆˆ๐‘‹โงต{๐‘ฅโˆ—}), then ๐‘ฅโˆ— is called a strict discrete local (global) minimizer of ๐‘“(๐‘ฅ) over ๐‘‹.

Algorithm 2.6 (discrete local minimization method). (1)Start from an initial point ๐‘ฅโˆˆ๐‘‹.(2)If ๐‘ฅ is a local minimizer of ๐‘“ over ๐‘‹, then stop. Otherwise, let ๐‘‘โˆ—โˆถ=argmin๐‘‘๐‘–โˆˆ๐ท๐‘ฅ๎€ฝ๐‘“๎€ท๐‘ฅ+๐‘‘๐‘–๎€ธ๎€ทโˆถ๐‘“๐‘ฅ+๐‘‘๐‘–๎€ธ๎€พ<๐‘“(๐‘ฅ).(2.2)(3)Let ๐‘ฅโˆถ=๐‘ฅ+๐‘‘โˆ—, and go to Step (2).
Let ๐‘ฅโˆ— be a local minimizer of the problem (P). The new definition of the filled function of ๐‘“ at ๐‘ฅโˆ— is given as follows.

Definition 2.7. ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) is called a discrete filled function of ๐‘“(๐‘ฅ) at a discrete local minimizer ๐‘ฅโˆ— if ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) has the following properties. (1)๐‘ฅโˆ— is a strict discrete local maximizer of ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) over ๐‘‹. (2)๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) has no discrete local minimizers in the region ๐‘†1=๎€ฝ๎€ท๐‘ฅ๐‘ฅโˆฃ๐‘“(๐‘ฅ)โ‰ฅ๐‘“โˆ—๎€ธ๎€ฝ๐‘ฅ,๐‘ฅโˆˆ๐‘‹โงตโˆ—๎€พ๎€พ.(2.3)(3)If ๐‘ฅโˆ— is not a discrete global minimizer of ๐‘“(๐‘ฅ), then ๐‘ƒ(๐‘ฅ,๐‘ฅโˆ—) does have a discrete minimizer in the region ๐‘†2=๎€ฝ๎€ท๐‘ฅ๐‘ฅโˆฃ๐‘“(๐‘ฅ)<๐‘“โˆ—๎€ธ๎€พ,๐‘ฅโˆˆ๐‘‹.(2.4)

3. Properties of the Proposed Discrete Filled Function ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ)

Let ๐‘ฅโˆ— denote the current discrete local minimizer of (P). Based on Definition 2.7, a novel filled function is proposed as follows:๐‘‡๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๐œ‘๐‘ž๎€ท๎€ฝ๎€ท๐‘ฅmax๐‘“(๐‘ฅ)โˆ’๐‘“โˆ—๎€ธ,+๐‘Ÿ,0๎€พ๎€ธ(3.1) where๐œ‘๐‘ž๎ƒฏ๐œ‹(๐‘ก)=2๐‘žโˆ’arctan๐‘กif๐‘กโ‰ 0,0if๐‘ก=0,(3.2) where ๐‘Ÿ>0 and ๐‘ž>0 are two parameters and ๐‘Ÿ satisfies 0<๐‘Ÿ<min๐‘“(๐‘ฅ1)โ‰ ๐‘“(๐‘ฅ2),๐‘ฅ1,๐‘ฅ2โˆˆ๐‘‹|๐‘“(๐‘ฅ1)โˆ’๐‘“(๐‘ฅ2)|.

The following theorems ensure that ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ) is a filled function under some conditions.

Theorem 3.1. If 0<๐‘ž<min(๐‘Ÿ,๐œ‹/4), then ๐‘ฅโˆ— is a strict local maximizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).

Proof. Since ๐‘ฅโˆ— is a local minimizer of (P), there exists a neighborhood ๐‘(๐‘ฅโˆ—) of ๐‘ฅโˆ— such that ๐‘“(๐‘ฅ)โ‰ฅ๐‘“(๐‘ฅโˆ—) and โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–=1 hold, for any ๐‘ฅโˆˆ๐‘(๐‘ฅโˆ—)โˆฉ๐‘‹. It follows that ๐‘‡๎€ท๐‘ฅ,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๎‚ต๐œ‹๐‘ž+12๐‘žโˆ’arctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถ,๐‘‡๎€ท๐‘ฅ)+๐‘Ÿโˆ—,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๐‘ž๎‚€๐œ‹2๐‘žโˆ’arctan๐‘Ÿ๎‚.(3.3)
By the condition 0<๐‘ž<min(๐‘Ÿ,๐œ‹/4) and the fact that the inequality arctan๐‘Žโˆ’arctan๐‘โ‰ค๐‘Žโˆ’๐‘(3.4) holds for any real number ๐‘Žโ‰ฅ๐‘, we have ๎€ทฮ”=๐‘‡๐‘ฅ,๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ,๐‘ž,๐‘Ÿโˆ’๐‘‡โˆ—,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๎‚€๐‘ž๐‘ž(๐‘ž+1)arctan๐‘Ÿโˆ’๐œ‹2๎‚+1๎‚ต๐‘ž๐‘ž+1arctan๐‘Ÿ๐‘žโˆ’arctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถโ‰ค1)+๐‘Ÿ๐‘ž๎‚€๐œ‹(๐‘ž+1)arctan1โˆ’2๎‚+๐‘ž๎‚ต1๐‘ž+1๐‘Ÿโˆ’1๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถ๐œ‹)+๐‘Ÿ=โˆ’41+1๐‘ž(๐‘ž+1)๐‘ž๐‘ž+1๐‘Ÿ๎€ท๐‘ฅ๐‘“(๐‘ฅ)โˆ’๐‘“โˆ—๎€ธ๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๐œ‹)+๐‘Ÿโ‰คโˆ’41+1๐‘ž(๐‘ž+1)=1๐‘ž+1๎‚€๐œ‹๐‘ž(๐‘ž+1)๐‘žโˆ’4๎‚<0.(3.5)
Hence, ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ)<๐‘‡(๐‘ฅโˆ—,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ), which implies that ๐‘ฅโˆ— is a strict local maximizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).

Lemma 3.2. For every ๐‘ฅ๎…žโˆˆ๐‘‹, there exists ๐‘‘โˆˆ๐ท such that โ€–๐‘ฅ๎…ž+๐‘‘โˆ’๐‘ฅโˆ—โ€–>โ€–๐‘ฅ๎…žโˆ’๐‘ฅโˆ—โ€–.

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

Theorem 3.3. Suppose that 0<๐‘ž<min(1,๐‘Ÿ,((๐œ‹โˆ’2)/4(1+๐ท))๐‘Ÿ). If ๐‘“(๐‘ฅ)โ‰ฅ๐‘“(๐‘ฅโˆ—) and ๐‘ฅโ‰ ๐‘ฅโˆ—, then ๐‘ฅ is not a local minimizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).

Proof. For any ๐‘ฅโ‰ ๐‘ฅโˆ— with ๐‘“(๐‘ฅ)โ‰ฅ๐‘“(๐‘ฅโˆ—), by Lemma 3.2, there exists a direction ๐‘‘โˆˆ๐ท with โ‹ƒ๐‘ฅ+๐‘‘โˆˆ๐‘ฅโˆˆ๐‘‹๐‘(๐‘ฅ) such that โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–>โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–. For this ๐‘‘, we consider the following three cases. Case 1 (๐‘“(๐‘ฅ+๐‘‘)โ‰ฅ๐‘“(๐‘ฅโˆ—)). In this case, by using the given condition and the fact that the inequality arctan๐‘Žโ‰ค๐‘Ž(3.6) holds for any real number ๐‘Žโ‰ฅ0, we have ฮ”1๎€ท=๐‘‡๐‘ฅ+๐‘‘,๐‘ฅโˆ—๎€ธ๎€ท,๐‘ž,๐‘Ÿโˆ’๐‘‡๐‘ฅ,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–๎‚ต๐œ‹2๐‘žโˆ’arctan๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถโˆ’1)+๐‘Ÿ๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚ต๐œ‹2๐‘žโˆ’arctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถ=๎‚ต๐‘ž)+๐‘Ÿarctan๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—)โˆ’๐œ‹+๐‘Ÿ2๎‚ถโ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–(๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–)(๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–)+1๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚ต๐‘žarctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๐‘ž)+๐‘Ÿโˆ’arctan๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถโ‰ค๎‚€๐‘ž)+๐‘Ÿarctan๐‘Ÿโˆ’๐œ‹2๎‚โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–(๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–)(๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–)+1๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚ต๐‘žarctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๐‘ž)+๐‘Ÿ+arctan๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถโ‰ค๎‚€๐‘ž)+๐‘Ÿ๐‘Ÿโˆ’๐œ‹2๎‚โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–โ€–(๐‘ž+๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โ€–)(๐‘ž+๐‘ฅโˆ’๐‘ฅโˆ—โ€–)+1๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚€๐‘ž๐‘Ÿ+๐‘ž๐‘Ÿ๎‚โ‰ค๎‚€๐œ‹1โˆ’2๎‚โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–(๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–)(๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–)+1๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–2๐‘ž๐‘Ÿโ‰คโ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–(๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–)(๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—๎‚ต๐œ‹โ€–)1โˆ’2+2๐‘ž๐‘Ÿ๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚ถ.(3.7) Since ๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โ‰ค1+๐ท and โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–โ‰ฅ1, we have ฮ”1โ‰คโ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–โˆ’โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–(๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–)(๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—๎‚ต๐œ‹โ€–)1โˆ’2+2๐‘ž๐‘Ÿ๎‚ถ(1+๐ท)<0.(3.8) Hence, in this case, ๐‘ฅ is not a local minimizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).Case 2 (๐‘“(๐‘ฅ+๐‘‘)<๐‘“(๐‘ฅโˆ—) and ๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—)+๐‘Ÿโ‰ค0). In this case, we have ฮ”2๎€ท=๐‘‡๐‘ฅ+๐‘‘,๐‘ฅโˆ—๎€ธ๎€ท,๐‘ž,๐‘Ÿโˆ’๐‘‡๐‘ฅ,๐‘ฅโˆ—๎€ธ๎€ท,๐‘ž,๐‘Ÿ=โˆ’๐‘‡๐‘ฅ,๐‘ฅโˆ—๎€ธ,๐‘ž,๐‘Ÿ<0,(3.9) which means the conclusion is true in this case.Case 3 (๐‘“(๐‘ฅ+๐‘‘)<๐‘“(๐‘ฅโˆ—) and ๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—)+๐‘Ÿ>0). In this case, we have ๐‘‡๎€ท๐‘ฅ+๐‘‘,๐‘ฅโˆ—๎€ธ=1,๐‘ž,๐‘Ÿ๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–๎‚ต๐œ‹2๐‘žโˆ’arctan๐‘“(๐‘ฅ+๐‘‘)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถ<1)+๐‘Ÿ๐‘ž+โ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–๎‚€๐œ‹2๐‘žโˆ’arctan๐‘Ÿ๎‚<1๐‘ž+โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–๎‚ต๐œ‹2๐‘žโˆ’arctan๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฅโˆ—๎‚ถ๎€ท)+๐‘Ÿ=๐‘‡๐‘ฅ,๐‘ฅโˆ—๎€ธ.,๐‘ž,๐‘Ÿ(3.10) Hence, in this case, ๐‘ฅ is not a local minimizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).
The above discussion implies that ๐‘ฅ is not a discrete local minimizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).

Theorem 3.4. Assume that ๐‘ฅโˆ— is not a global minimizer of ๐‘“(๐‘ฅ), then there exists a minimizer ๐‘ฅโˆ—1 of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ) in ๐‘†2.

Proof. Since ๐‘ฅโˆ— is not a global minimizer of ๐‘“(๐‘ฅ), there exists ๐‘ฅโˆ—1โˆˆ๐‘†2 such that ๐‘“(๐‘ฅโˆ—1)<๐‘“(๐‘ฅโˆ—)โˆ’๐‘Ÿ; it follows that ๐‘‡(๐‘ฅโˆ—1,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ)=0. On the other hand, by the structure of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ), we have ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ)โ‰ฅ0 for any ๐‘ฅโˆˆ๐‘‹. This shows ๐‘ฅโˆ—1 is a minimizer of ๐‘‡(๐‘ฅ,๐‘ฅโˆ—,๐‘ž,๐‘Ÿ).

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 ๐‘Ÿ, namely, ๐‘Ÿ๐ฟ=1๐‘’โˆ’8. Input an initial point ๐‘ฅ0(0)โˆˆ๐‘‹. Let ๐ท={ยฑ๐‘’๐‘–,๐‘–=1,2,โ€ฆ,๐‘›}. (2)Starting from an initial point ๐‘ฅ0(0)โˆˆ๐‘‹, minimize ๐‘“(๐‘ฅ) and obtain the first local minimizer ๐‘ฅโˆ—0 of ๐‘“(๐‘ฅ). Set ๐‘˜=0,โ€‰โ€‰๐‘Ÿ=1, and ๐‘ž=1.(3)Set ๐‘ฅ๐‘˜(0)๐‘–=๐‘ฅโˆ—๐‘˜+๐‘‘๐‘–, ๐‘‘๐‘–โˆˆ๐ท,โ€‰โ€‰๐‘–=1,2,โ€ฆ,2๐‘›,โ€‰โ€‰๐ฝ=[1,2,โ€ฆ,2๐‘›], and ๐‘—=1.(4)Set ๐‘–=๐ฝ๐‘— and ๐‘ฅ=๐‘ฅ๐‘˜(0)๐‘–.(5)If ๐‘“(๐‘ฅ)<๐‘“(๐‘ฅโˆ—๐‘˜), then use ๐‘ฅ as initial point for discrete local minimization method to find another local minimizer ๐‘ฅโˆ—๐‘˜+1 such that ๐‘“(๐‘ฅโˆ—๐‘˜+1)<๐‘“(๐‘ฅโˆ—๐‘˜). Set ๐‘˜=๐‘˜+1, and go to (3).(6)Let ๐ท0={๐‘‘โˆˆ๐ทโˆถ๐‘ฅ+๐‘‘โˆˆ๐‘‹}. If there exists ๐‘‘โˆˆ๐ท0 such that ๐‘“(๐‘ฅ+๐‘‘)<๐‘“(๐‘ฅโˆ—๐‘˜), then use ๐‘ฅ+๐‘‘โˆ—, where ๐‘‘โˆ—=argmin๐‘‘โˆˆ๐ท0{๐‘“(๐‘ฅ+๐‘‘)}, as an initial point for a discrete local minimization method to find another local minimizer ๐‘ฅโˆ—๐‘˜+1 such that ๐‘“(๐‘ฅโˆ—๐‘˜+1)<๐‘“(๐‘ฅโˆ—๐‘˜). Set ๐‘˜=๐‘˜+1, and go to (3).(7)Let ๐ท1={๐‘‘โˆˆ๐ท0โˆถโ€–๐‘ฅ+๐‘‘โˆ’๐‘ฅโˆ—โ€–>โ€–๐‘ฅโˆ’๐‘ฅโˆ—โ€–}. If ๐ท1=โˆ…, then go to (10).(8)If there exists ๐‘‘โˆˆ๐ท1 such that ๐‘‡(๐‘ฅ+๐‘‘,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ)โ‰ฅ๐‘‡(๐‘ฅ,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ), then set ๐‘ž=0.1๐‘ž,โ€‰โ€‰๐ฝ=[๐ฝ๐‘—,โ€ฆ,๐ฝ2๐‘›,๐ฝ1,โ€ฆ,๐ฝ๐‘—โˆ’1],โ€‰โ€‰๐‘—=1, and go to (4).(9)Let ๐ท2โˆถ={๐‘‘โˆˆ๐ท1โˆถ๐‘“(๐‘ฅ+๐‘‘)<๐‘“(๐‘ฅ),๐‘‡(๐‘ฅ+๐‘‘,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ)<๐‘‡(๐‘ฅ,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ)}. If ๐ท2โ‰ โˆ…, then set ๐‘‘โˆ—=argmin๐‘‘โˆˆ๐ท2{๐‘“(๐‘ฅ+๐‘‘)+๐‘‡(๐‘ฅ+๐‘‘,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ)}. Otherwise set ๐‘‘โˆ—=argmin๐‘‘โˆˆ๐ท1{๐‘‡(๐‘ฅ+๐‘‘,๐‘ฅโˆ—๐‘˜,๐‘ž,๐‘Ÿ)},๐‘ฅ=๐‘ฅ+๐‘‘โˆ—, and go to (6).(10)If ๐‘–<2๐‘›, then set ๐‘–=๐‘–+1, and go to (4). (11)Set ๐‘Ÿ=0.1๐‘Ÿ. If ๐‘Ÿโ‰ฅ๐‘Ÿ๐ฟ, go to (3). Otherwise, the algorithm is incapable of finding a better minimizer starting from the initial points, {๐‘ฅ๐‘˜(0)๐‘–โˆถ๐‘–=1,2,โ€ฆ,2๐‘›}. The algorithm stops, and ๐‘ฅโˆ—๐‘˜ is taken as a global minimizer.

The motivation and mechanism behind the algorithm are explained below.

A set of 2๐‘› 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 ๐‘“(๐‘ฅ)<๐‘“(๐‘ฅโˆ—๐‘˜). Therefore, we can further minimize the primal objective function ๐‘“(๐‘ฅ) by any discrete local minimization method starting from ๐‘ฅ.

Step (7) aims at selecting a better successor point. If ๐ท2 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.

Problem 1. One has ๎€ท๐‘ฅmin๐‘“(๐‘ฅ)=1002โˆ’๐‘ฅ21๎€ธ2+๎€ท1โˆ’๐‘ฅ1๎€ธ2๎€ท๐‘ฅ+904โˆ’๐‘ฅ23๎€ธ2+๎€ท1โˆ’๐‘ฅ3๎€ธ2๎‚ƒ๎€ท๐‘ฅ+10.12๎€ธโˆ’12+๎€ท๐‘ฅ4๎€ธโˆ’12๎‚„๎€ท๐‘ฅ+19.82๐‘ฅโˆ’1๎€ธ๎€ท4๎€ธ,โˆ’1s.t.โˆ’10โ‰ค๐‘ฅ๐‘–โ‰ค10,๐‘ฅ๐‘–isinteger,๐‘–=1,2,3,4.(4.1) This problem has 214โ‰ˆ1.94ร—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: ๐‘ฅโˆ—global=(1,1,1,1) with ๐‘“(๐‘ฅโˆ—global)=0. We used three initial points in our experiment:(9,6,5,6), (10,10,10,10), (โˆ’10,โˆ’10,โˆ’10,โˆ’10).

Problem 2. One has min๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ)โ„Ž(๐‘ฅ),s.t.๐‘ฅ๐‘–=0.001๐‘ฆ๐‘–,โˆ’2000โ‰ค๐‘ฆ๐‘–โ‰ค2000,๐‘ฆ๐‘–isinteger,๐‘–=1,2,(4.2) where ๎€ท๐‘ฅ๐‘”(๐‘ฅ)=1+1+๐‘ฅ2๎€ธ+12๎€ท19โˆ’14๐‘ฅ1+3๐‘ฅ21โˆ’14๐‘ฅ2+6๐‘ฅ1๐‘ฅ2+3๐‘ฅ2๎€ธ,๎€ทโ„Ž(๐‘ฅ)=30+2๐‘ฅ1โˆ’3๐‘ฅ2๎€ธ2๎€ท18โˆ’32๐‘ฅ1+12๐‘ฅ21+48๐‘ฅ2โˆ’36๐‘ฅ1๐‘ฅ2+27๐‘ฅ22๎€ธ.(4.3) This problem has 40012โ‰ˆ1.60ร—107 feasible points. More precisely, it has 207 and 2 discrete local minimizers in the interior and the boundary of box โˆ’2.00โ‰ค๐‘ฅ๐‘–โ‰ค2.00,๐‘–=1,2, respectively. Nevertheless, it has only one discrete global minimum solution: ๐‘ฅโˆ—global=(0.000,โˆ’1.000) with ๐‘“(๐‘ฅโˆ—global)=3. We used three initial points in our experiment: (2000,2000), (โˆ’2000,โˆ’2000), (1196,1156).

Problem 3. One has ๎€บmin๐‘“(๐‘ฅ)=1.5โˆ’๐‘ฅ1๎€ท1โˆ’๐‘ฅ2๎€ธ๎€ป2+๎€บ2.25โˆ’๐‘ฅ1๎€ท1โˆ’๐‘ฅ22๎€ธ๎€ป2+๎€บ2.625โˆ’๐‘ฅ1๎€ท1โˆ’๐‘ฅ32๎€ธ๎€ป2,s.t.๐‘ฅ๐‘–=0.001๐‘ฆ๐‘–,โˆ’104โ‰ค๐‘ฆ๐‘–โ‰ค104,๐‘ฆ๐‘–isinteger,๐‘–=1,2.(4.4) This problem has 200012โ‰ˆ4.00ร—108 feasible points and many discrete local minimizers, but it has only one discrete global minimum solution: ๐‘ฅโˆ—global=(3,0.5) with ๐‘“(๐‘ฅโˆ—global)=0. We used three initial points in our experiment:(9997,6867), (10000,10000), (โˆ’10000,โˆ’10000).

Problem 4. One has ๎€ท๐‘ฅmin๐‘“(๐‘ฅ)=1+10๐‘ฅ2๎€ธ2๎€ท๐‘ฅ+53โˆ’๐‘ฅ4๎€ธ2+๎€ท๐‘ฅ2โˆ’2๐‘ฅ3๎€ธ4๎€ท๐‘ฅ+101โˆ’๐‘ฅ4๎€ธ4,s.t.๐‘ฅ๐‘–=0.001๐‘ฆ๐‘–,โˆ’104โ‰ค๐‘ฆ๐‘–โ‰ค104,๐‘ฆ๐‘–isinteger,๐‘–=1,2,3,4.(4.5) This problem has 200014โ‰ˆ1.60ร—1017 feasible points and many local minimizers, but it has only one global minimum solution: ๐‘ฅโˆ—global=(0,0,0,0) with ๐‘“(๐‘ฅโˆ—global)=0. We used three initial points in our experiment:(1000,โˆ’1000,โˆ’1000,1000), (10000,โˆ’10000,โˆ’10000,10000), (โˆ’10000,โ€ฆ,โˆ’10000).

Problem 5. One has ๎€ท๐‘ฅmin๐‘“(๐‘ฅ)=1๎€ธโˆ’12+๎€ท๐‘ฅ๐‘›๎€ธโˆ’12+๐‘›๐‘›โˆ’1๎“๐‘–=1๎€ท๐‘ฅ(๐‘›โˆ’๐‘–)2๐‘–โˆ’๐‘ฅ๐‘–+1๎€ธ2,s.t.โˆ’5โ‰ค๐‘ฅ๐‘–โ‰ค5,๐‘ฅ๐‘–isinteger,๐‘–=1,2,โ€ฆ,๐‘›.(4.6) This problem has many local minimizers, but it has only one global minimum solution: ๐‘ฅโˆ—global=(1,โ€ฆ,1) with ๐‘“(๐‘ฅโˆ—global)=0.
In this problem, we used initial point (5,โ€ฆ,5) in our experiment for ๐‘›=25,50,100, respectively.

Problem 6. One has min๐‘“(๐‘ฅ)=๐‘›๎“๐‘–=1๐‘ฅ4๐‘–+๎ƒฉ๐‘›๎“๐‘–=1๐‘ฅ๐‘–๎ƒช2,s.t.โˆ’5โ‰ค๐‘ฅ๐‘–โ‰ค5,๐‘ฅ๐‘–isinteger,๐‘–=1,2,โ€ฆ,๐‘›.(4.7) This problem has many local minimizers, but it has only one global minimum solution: ๐‘ฅโˆ—global=(1,1,โ€ฆ,1) with ๐‘“(๐‘ฅโˆ—global)=0.
In this problem, we used initial point (5,โ€ฆ,5) in our experiment for ๐‘›=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.