Research Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming

Discrete Dynamics in Nature and Society Volume 20, Article ID 7697, pages doi:0.55/20/7697 Research Article Identifying a Global Optimizer with Filled Function for Nonlinear Integer Programming Wei-Xiang Wang, You-Lin Shang, 2 and Lian-Sheng Zhang 3 Department of Mathematics, Shanghai Second Polytechnic University, Shanghai 20209, China 2 Department of Mathematics, Henan University of Science and Technology, Luoyang 47003, China 3 Department of Mathematics, Shanghai University, Shanghai 200444, China Correspondence should be addressed to Wei-Xiang Wang, zhesx@26.com Received 27 March 20; Revised 0 July 20; Accepted 5 July 20 Academic Editor: Rigoberto Medina Copyright q 20 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 wor is properly cited. 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.. Introduction Consider the following general global nonlinear integer programming: min x X fx, P where f : Z n R,X Z n is a box set and Z n is the set of integer points in R n.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 tacle with the problem P. These techniques are usually divided into two classes: stochastic method and deterministic method see 7. The discrete filled function method is one of the more recently developed global optimization tools for discrete global

2 Discrete Dynamics in Nature and Society optimization problems. The first filled function was introduced by Ge and Qin in 8 for continuous global optimization. Papers 6, 7, 9 extend this continuous filled function method to solve integer programming problem. Lie 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, 0 are as follows. Definition. see 9. Px, x is called a filled function of fx at a discrete local minimizer x if Px, x meets the following conditions. Px, x has no discrete local minimizers in the set S {x X : fx fx }, except a prefixed point x 0 S that is a minimizer of Px, x. 2 If x is not a discrete global minimizer of fx, then Px, x does have a discrete minimizer in the set S 2 {x fx <fx,x X}. Definition.2 see 0. Px, x is called a filled function of fx at a discrete local minimizer x if Px, x meets the following conditions. Px, x has no discrete local minimizers in the set S \ x 0, where the prefixed point x 0 S is not necessarily a local minimizer of Px, x. 2 If x is not a discrete global minimizer of fx, then Px, x has a discrete minimizer in the set S 2. Although Definitions. and.2 and the corresponding filled functions proposed in the papers 9, 0 have their own advantages, they have some defects in some degree, for example, as the prefixed point x 0 in Definition.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. and.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 mae the following assumptions. Assumption 2.. There exists a constant D>0 satisfying D max x,x 2 X,x / x 2 x x 2 <. Assumption 2.2. There exists a constant L>0, such that fx f y L x y 2. holds, for any x, y x X Nx, where Nx is a neighborhood of the point x as defined in Definition 2.4.

Discrete Dynamics in Nature and Society 3 Most of the existing discrete filled function methods are used for solving a box constrained problem. To an unconstrained global optimization problem UP: min x R nfx, if fx satisfies lim x fx, then there exists a box set which contains all discrete global minimizers of fx. 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 0. The set of all feasible directions at x X is defined by D x {d D : x d X}, where D {±e i : i, 2,...,n}, e i 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 0. For any x Z n, the discrete neighborhood of x is defined by Nx {x, x ± e i, i, 2,...,n}. Definition 2.5 see 0. Apointx X is called a discrete local minimizer of fx over X if fx fx, for all x X Nx. Furthermore, if fx fx, for all x X, then x is called a strict discrete local minimizer of fx over X. If, in addition, fx <fx, for all x X \{x }, then x is called a strict discrete local global minimizer of fx over X. Algorithm 2.6 discrete local minimization method. Start from an initial point x X. 2 If x is a local minimizer of f over X, then stop. Otherwise, let d : arg min d i D x { fx di : fx d i <fx }. 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. Px, x is called a discrete filled function of fx at a discrete local minimizer x if Px, x has the following properties. x is a strict discrete local maximizer of Px, x over X. 2 Px, x has no discrete local minimizers in the region S { x fx fx,x X \ {x } }. 2.3 3 If x is not a discrete global minimizer of fx, then Px, x does have a discrete minimizer in the region S 2 { x fx <fx, x X }. 2.4

4 Discrete Dynamics in Nature and Society 3. Properties of the Proposed Discrete Filled Function Tx, 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: T x, x,q,r q x x ϕ { q max fx fx r, 0 }, 3. where π ϕ q t 2 arctan q if t / 0, t 0 if t 0, 3.2 where r>0andq>0 are two parameters and r satisfies 0 <r<min fx / fx 2,x,x 2 X fx fx 2. The following theorems ensure that Tx, x,q,r is a filled function under some conditions. Theorem 3.. If 0 <q<minr, π/4, then x is a strict local maximizer of Tx, x,q,r. Proof. Since x is a local minimizer of P, there exists a neighborhood Nx of x such that fx fx and x x hold,foranyx Nx X. It follows that T x, x,q,r π q T x,x,q,r q 2 arctan q fx fx r π 2 arctan q. r, 3.3 By the condition 0 <q<minr, π/4 and the fact that the inequality arctan a arctan b a b 3.4 holds for any real number a b, we have ΔT x, x,q,r T x,x,q,r q q arctan q r π arctan q 2 q r arctan q fx fx r arctan q q π q 2 q r fx fx r π 4 q q q fx fx q r fx fx r

Discrete Dynamics in Nature and Society 5 π 4 q q q q q q π < 0. 4 3.5 Hence, Tx, x,q,r <Tx,x,q,r, which implies that x is a strict local maximizer of Tx, x,q,r. Lemma 3.2. For every x X, there exists d D such that x d x > x x. For the proof of this lemma, see, for example, 6 or 7. Theorem 3.3. Suppose that 0 <q<min,r,π 2/4 Dr. Iffx fx and x / x, then x is not a local minimizer of Tx, x,q,r. Proof. For any x / x with fx fx,bylemma 3.2, there exists a direction d D with x d x X Nx such that x d x > x x. For this d, we consider the following three cases. Case fx d fx. In this case, by using the given condition and the fact that the inequality arctan a a 3.6 holds for any real number a 0, we have Δ T x d, x,q,r T x, x,q,r q x d x q x x arctan q x x arctan q r π 2 π 2 arctan q π 2 arctan q fx d fx r fx fx r q fx d fx r π x d x x x 2 q x d x q x x arctan q x x arctan q fx fx r arctan x d x x x q x d x q x x q q fx fx arctan r q fx d fx r fx d fx r

6 Discrete Dynamics in Nature and Society q r π x d x x x 2 q x d x q x x q q x x r q r π x d x x x 2 q x d x q x x 2q q x x r x d x x x q x d x q x x π 2 2q r q x d x x d x x x. 3.7 Since q x d x D and x d x x x, we have Δ x d x x x q x d x q x x π 2 2q D < 0. 3.8 r Hence, in this case, x is not a local minimizer of Tx, x,q,r. Case 2 fx d <fx and fx d fx r 0. In this case, we have Δ 2 T x d, x,q,r T x, x,q,r T x, x,q,r < 0, 3.9 which means the conclusion is true in this case. Case 3 fx d <fx and fx d fx r>0. In this case, we have T x d, x,q,r π q x d x 2 arctan q fx d fx r π < q x d x 2 arctan q r π < q x x 2 arctan q fx fx T x, x,q,r. r 3.0 Hence, in this case, x is not a local minimizer of Tx, x,q,r. The above discussion implies that x is not a discrete local minimizer of Tx, x,q,r. Theorem 3.4. Assume that x is not a global minimizer of fx, then there exists a minimizer x of Tx, x,q,r in S 2. Proof. Since x is not a global minimizer of fx, there exists x S 2 such that fx <fx r; it follows that Tx,x,q,r0. On the other hand, by the structure of Tx, x,q,r, we have Tx, x,q,r 0 for any x X. This shows x is a minimizer of Tx, x,q,r.

Discrete Dynamics in Nature and Society 7 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. discrete filled function method. Input the lower bound of r, namely, r L e 8. Input an initial point x 0 0 X. Let D {±e i,i, 2,...,n}. 2 Starting from an initial point x 0 0 X, minimize fx and obtain the first local minimizer x 0 of fx.set 0, r, and q. 3 Set x 0i x d i, d i D, i, 2,...,2n, J, 2,...,2n, andj. 4 Set i J j and x x 0i. 5 If fx <fx, then use x as initial point for discrete local minimization method to find another local minimizer x such that fx <fx.set, and go to 3. 6 Let D 0 {d D : x d X}. If there exists d D 0 such that fx d <fx, then use x d, where d arg min d D0 {fx d}, as an initial point for a discrete local minimization method to find another local minimizer x such that fx < fx.set, and go to 3. 7 Let D {d D 0 : x d x > x x }. IfD, then go to 0. 8 If there exists d D such that Tx d, x,q,r Tx, x,q,r, then set q 0.q, J J j,...,j 2n,J,...,J j, j, and go to 4. 9 Let D 2 : {d D : fx d < fx,tx d, x,q,r < Tx, x,q,r}. If D 2 /, then set d arg min d D2 {fx d Tx d, x,q,r}. Otherwise set d arg min d D {Tx d, x,q,r}, x x d,andgoto6. 0 If i<2n, then set i i, and go to 4. Set r 0.r. Ifr r L,goto3. Otherwise, the algorithm is incapable of finding a better minimizer starting from the initial points, {x 0i : i, 2,...,2n}. The algorithm stops, and x is taen as a global minimizer. The motivation and mechanism behind the algorithm are explained below. Asetof2n 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 fx <fx. Therefore, we can further minimize the primal objective function fx by any discrete local minimization method starting from x. Step 7 aims at selecting a better successor point. If D 2 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, and

8 Discrete Dynamics in Nature and Society Table PN DN IN TI TN FN 4 3 2.436 8087 367 4 3 2.327 7622 3523 4 3 2.4252 9556 3798 2 2 5 35.435 32342 62468 2 2 5 36.2984 326763 65352 2 2 5 36.6879 330835 6667 3 2 5 204.996 558825 3765 3 2 5 206.7242 67823 323564 3 2 5 205.687 59356 3872 4 4 53 3598.3893 3399625 6798325 4 4 53 362.567 34043270 6808654 4 4 53 3574.3248 33933790 6786758 5 25 2 48.863 5867 24496 5 50 2 084.7239 923493 924634 5 00 2 889.984 68965659 536758 6 25 64.265 5246 30673 6 50 24 297.7789 205803 2467864 6 00 50 9045.2396 82897460 7328966 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. One has 2 2 min fx 00 x 2 x 2 x 2 90 x 4 x 2 3 x3 2 [ 0. x 2 2 x 4 2] 9.8x 2 x 4, 4. s.t. 0 x i 0, x i is integer, i, 2, 3, 4. This problem has 2 4.94 0 5 feasible points where 4 of them are discrete local minimizers but only one of those discrete local minimizers is the discrete global minimum solution: x,,, with fx 0. We used three initial points in our global global experiment: 9, 6, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0.

Discrete Dynamics in Nature and Society 9 Problem 2. One has min fx gxhx, s.t. x i 0.00y i, 2000 y i 2000, y i is integer, i, 2, 4.2 where gx x x 2 2 9 4x 3x 2 4x 2 6x x 2 3x 2, hx 30 2x 3x 2 2 8 32x 2x 2 48x 2 36x x 2 27x 2 2. 4.3 This problem has 400 2.60 0 7 feasible points. More precisely, it has 207 and 2 discrete local minimizers in the interior and the boundary of box 2.00 x i 2.00, i, 2, respectively. Nevertheless, it has only one discrete global minimum solution: x global 0.000,.000 with fx 3. We used three initial points in our experiment: 2000, 2000, global 2000, 2000, 96, 56. Problem 3. One has min ] 2 2, fx.5 x x 2 2 [2.25 x x 2 2 [2.625 x x2] 3 s.t. x i 0.00y i, 0 4 y i 0 4, y i is integer, i, 2. 4.4 This problem has 2000 2 4.00 0 8 feasible points and many discrete local minimizers, but it has only one discrete global minimum solution: x global 3, 0.5 with fx 0. We used global three initial points in our experiment: 9997, 6867, 0000, 0000, 0000, 0000. Problem 4. One has min fx x 0x 2 2 5x 3 x 4 2 x 2 2x 3 4 0x x 4 4, s.t. x i 0.00y i, 0 4 y i 0 4, y i is integer, i, 2, 3, 4. 4.5 This problem has 2000 4.60 0 7 feasible points and many local minimizers, but it has only one global minimum solution: x 0, 0, 0, 0 with fx 0. We used three global global initial points in our experiment: 000, 000, 000, 000, 0000, 0000, 0000, 0000, 0000,..., 0000. Problem 5. One has min fx x 2 x n 2 n 2, n n i x 2 i i x s.t. 5 x i 5, x i is integer, i, 2,...,n. i 4.6

0 Discrete Dynamics in Nature and Society This problem has many local minimizers, but it has only one global minimum solution: x global,..., with fx global 0. In this problem, we used initial point 5,...,5 in our experiment for n 25, 50, 00, respectively. Problem 6. One has min fx n n 2 x 4 i x i, i i 4.7 s.t. 5 x i 5, x i is integer, i, 2,...,n. This problem has many local minimizers, but it has only one global minimum solution: x global,,..., with fx global 0. In this problem, we used initial point 5,...,5 in our experiment for n 25, 50, 00, 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 wor is to generalize the discrete filled function techniques to mixed nonlinear integer global optimization problem. Acnowledgment This paper was partially supported by the NNSF of China under Grant No. 05737 and 097053. References C. Mohan and H. T. Nguyen, A controlled random search technique incorporating the simulated annealing concept for solving integer and mixed integer global optimization problems, Computational Optimization and Applications, vol. 4, no., pp. 03 32, 999. 2 O. K. Gupta and A. Ravindran, Branch and bound experiments in convex nonlinear integer programming, Management Science, vol. 3, no. 2, pp. 533 546, 985. 3 F. Körner, A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems, BIT Numerical Mathematics, vol. 28, no. 3, pp. 70 708, 988. 4 D. Li and D. J. White, pth power Lagrangian method for integer programming, Annals of Operations Research, vol. 98, pp. 5 70, 2000. 5 M. W. Cooper and K. Farhangian, Nonlinear integer programming for various forms of constraints, Naval Research Logistics Quarterly, vol. 29, no. 4, pp. 585 592, 982. 6 C.-K. Ng, L.-S. Zhang, D. Li, and W.-W. Tian, Discrete filled function method for discrete global optimization, Computational Optimization and Applications, vol. 3, no., pp. 87 5, 2005. 7 C.-K. Ng, D. Li, and L.-S. Zhang, Discrete global descent method for discrete global optimization and nonlinear integer programming, Global Optimization, vol. 37, no. 3, pp. 357 379, 2007. 8 R. P. Ge and Y. F. Qin, A class of filled functions for finding global minimizers of a function of several variables, Optimization Theory and Applications, vol. 54, no. 2, pp. 24 252, 987.

Discrete Dynamics in Nature and Society 9 W. X. Zhu, A filled function method for integer programming, Acta Mathematicae Applicatae Sinica, vol. 23, no. 4, pp. 48 487, 2000 Chinese. 0 Y.-L. Shang and L.-S. Zhang, A filled function method for finding a global minimizer on global integer optimization, Computational and Applied Mathematics, vol. 8, no., pp. 200 20, 2005. M.-M. Li, Y.-L. Shang, and L.-S. Zhang, A new filled function method for integer programming, Computational Mathematics, vol. 24, no., pp. 25 32, 2006.

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