Penalty method for integer programming

Penalty method for integer programming A continuous approach for integer programming problem with nonlinear constrained Abdelkrim El Mouatasim∗,∗∗ ∗

Department of Mathematics, Faculty of Science - Jazan University P.B. 2097 Jazan Saudi Arabia. ∗∗ Laboratoire d’étude et de recherche en mathématiques appliquées (LERMA). Ecole Mohammadia d’Ingénieurs Rabat -Maroc . [email protected]

RÉSUMÉ. Un problème de programmation en nombres entiers avec contraintes non linéaire transformé en un problème d’optimisation globale avec contraintes non linéaire mondiale. Perturbation aléatoire de la méthode du gradient réduit généralisé est présenté à trouver une solution globale de la problème associé à une optimisation globale avec des contraintes non linéaires. Quelques exemples numériques sont donnés. ABSTRACT. An integer programming problem with nonlinear constrained transformed to a global optimization problem with nonlinear constrained. A random perturbation of generalized reduced gradient method is presented to find a global solution of the associated global optimization problem with nonlinear constraints. Some numerical examples are given. MOTS-CLÉS : Programmation en nombres entiers, optimisation globale, les contraintes non lineaires, perturbation aleatoire, la methode du gradient reduite generalisee. KEYWORDS : Integer programming, global optimization, nonlinear constraints, random perturbation, generalized reduced gradient method.

1. Introduction Integer programming is one of the most interesting and difficult research areas in operations research. Nonlinear integer programming has many applications in real world. Combinatorial optimization has application in water system pump operations [2]. Integer programming with a nonlinear objective function has applications in system reliability design [12], and the procurement problem for a separable inventory system [11]. An integer programming problem with nonlinear constraints can be written as follows :   minimize f (x) subject to g(x) ≤ 0, [1]  x ∈ In where f : Rn −→ R and g : Rn −→ Rm are a continuous functions, and I n is the set of integer points in Rn . The integer programming problem (1) called convex integer programming problem when f (x) and g(x) are convex functions (see for instance, [8, 10]). Solution methods of problem (1) can be categorized into three classes :

The first class of methods transforms a problem (1) into an equivalent continuous global unconstrained optimization problem via penalty function ([1] and [2, 6, 7]), such that the problem can be solved by the approaches of continuous global optimization [4]. The second class of methods has exhaustive characteristics, which include the Branch and Bound (see for instance, [5] and [10, 9]) for a problem with some analytical properties. The third class of approaches is approximate algorithms or heuristic approaches [13]. The idea of a methods in first class, is generalized in this paper for transforming the problem (1) of integer nonlinear programming with constraints to a equivalent continuous global optimization problem with constraints. To make the method efficient, we shall construct a twice continuously differentiable, rather than a nondifferentiable function as used in [4].

2. Associated global optimization problem In this section we consider an integer nonlinear programming under equality nonlinear constrained problem as the following form   Minimize : f (x) Subject to : h(x) = 0, [2]  x ∈ In where f (x) and h(x) are twice continuously differentiable in R n , R EMARQUE. — A n integer programming problem with nonlinear inequality constraints (1) be transferred to the problem with equality constraints (2), where hj (x) = gj (x) − σj , j = 1, .., m and σj is a slack variable. Definition For a point x0 ∈ I n , the set N(x0 ) = {x : kx − x0 k∞ ≤ 51 } is called a 1 -cubic neighborhood of the point x0 . 5

Let S and S 0 be the sets defined by S = {x ∈ Rn : h(x) = 0}

[3]

S 0 = S ∩ I n. Now we construct a global optimization problem  n P   cos 2πxi ,  Minimize : φ(x, k) = f (x) − k i=1

[4]

Subject to : h(x) = 0,    x ∈ Rn ,

where φ is the penalty function for the objective function of an integer nonlinear programming problem (2). The Lagrangian function can be formed as L(x, λ) = φ(x, k) − λT h(x). In the following, we assume that there exist four constants C1 , C2 , C3 and C4 such that k∇f (x)k1 ≤ C1 , k∇2 f (x)k1 ≤ C2 , kJh(x)T λk ≤ C3 and kHh(x)T λk ≤ C4 ∀x ∈ S, [5] where Jh(x) and Hh(x) are the Jacobian matrix and Hessian of the function h(x) = (h1 (x), ..., hm (x))T respectively. Then the following theorems hold. Lemma 1 φ(x, k) has at least one minimizer in the 51 -cubic neighborhood of every point in S 0 , provided that k > max[

C1 + C 3 C2 ]. 2 , 2 2π sin 5 π 4π cos 52 π

[6]

Lemma 2 Any minimizer of problem (4) must be in the 51 -cubic neighborhood of an integer point, provided C1 + C 3 C2 k > max[ ]. [7] 2 , 2 2π sin 5 π 4π sin 25 π Lemma 3 Suppose that y and z are tow different integer points in S, and y 0 = arg and z0 = arg

min

x∈S∩N(z)

min

φ(x, k)

x∈S∩N(y)

φ(x, k). Then if f (y) < f (z),

then φ(y0 , k) < φ(z0 , k),

[8]

provided k > max{

C1 nC12 , 2 }. 2π 2π [f (z) − f (y)]

[9]

Theorem 4 Suppose that x∗ is a global minimum of problem (4) and that k satisfies the inequality k > max(m1 , m2 , m3 )

[10]

where C2 + C 4 nC12 C1 + C 3 2 , m2 = 2 , m3 = 2π 2 m , with m4 = min{f (x2 )−f (x1 )}, 2 2π sin 5 π 4π sin 5 π 4 [11] for all integers y and z in S such that

m1 =

f (z) > f (y) ∗

[12]

1 5-

ˆ ∈ S, then x ˆ is a solution of an cubic neighborhood of an integer point x If x is in a integer nonlinear programming problem (2). Proof : If k > m1 and k > m2 , then (6) and (7) hold and so Lemmas 1 and 2 hold. If k > m1 and k > m3 , then (9) holds for any integer points y and z satisfying (12), and therefore Lemma 3 holds. So if (10) holds, then Lemmas 1 to 3 hold. ˆ ∈ S is not a solution of (2). Then there exists an integer Assume that k satisfies (10), but x point y ∈ S and f (y) < f (ˆ x). Furthermore, Theorems 1 to 3 show that there exists a minimizer y0 ∈ S of φ(x, k) in the 51 - cubic neighborhoods of y such that φ(y0 , k) < φ(x∗ , k). This contradicts the fact that x∗ is a global minimum of integer nonlinear programming problem (4).

3. Numerical examples The algorithm of random perturbation of generalized reduced gradient for global optimization under nonlinear constrained was programmed using Visual Fortran 6.1 see for instance [3]. Concern experiments performed on a workstation HP Intel(R) Celeron(R) M processor 1.30GHz, 224 MB RAM. The row cpu gives the mean CPU time in seconds for one run. Example 1 [14]  min x1 x2 x3 + x1 x4 x5 + x2 x4 x6 + x6 x7 x8 + x2 x5 x7     s.t. 2x1 + 2x4 + 8x8 ≥ 12     11x1 + 7x4 + 13x6 ≥ 41     6x  2 + 9x4 x6 + 5x7 ≥ 60    3x  2 + 5x5 + 7x8 ≥ 42    6x  2 x7 + 9x3 + 5x5 ≥ 53   4x3 x7 + x5 ≥ 13 2x1 + 4x2 + 7x4 + 3x5 + x7 ≤ 69     9x  1 x8 + 6x3 x5 + 4x3 x7 ≤ 47    12x  2 + 8x2 x8 + 2x3 x6 ≤ 73    x + 4x5 + 2x6 + 9x8 ≤ 31  3    0 ≤ x  i ≤ 7, i = 1, 3, 4, 6, 8    0 ≤ x  i ≤ 15, i = 2, 5, 7   xi : integer, i = 1, 2, ..., 8.

If one takes initial point x0 = (7, 6, 0.5, 1, 6.5, 1, 4, 0)T and k = 0.5, then the global minimizer of φ(x, k) with nonlinear constrained is x∗ = (4.901, 4.003, 0.899, 0.982, 5.914, 2.95, 1.969, 0.059)T , ˆ = (5, 4, 1, 1, 6, 3, 2, 0)T and global and so the solution of global discrete optimization is x value is 110. Example 2 [14]  min 100(x2 − x21 )2 + (1 − x1 )2     s.t. 2x21 + x22 ≥ 0.25    − 31 x1 + x2 ≥ 0.1 xi = ji × 10−4     0 ≤ ji ≤ 105    ji : integer, i = 1, 2. If one takes initial point x0 = (0.3, 0.5)T (i.e. j = (3 × 103 , 5 × 103 )) and k = 5, then the global minimizer of φ(x, k) with nonlinear constrained is x∗ = (1.0048, 1.0102)T , ˆ = (1, 1)T and global value is 0.0. and so the solution of global discrete optimization is x Example 3 [14]  min xT Qx   n  P  x2i  s.t. ≤1  9n+i   i=1 n P ixi ≥ n2    i=1    −5 ≤ xi ≤ 5   xi : integer, i = 1, 2, .., n where Q = [Qij ], Qii = 2, Qij = 1 for i 6= j. If one takes initial point x0i = 6, 1 ≤ i ≤ 19, x0i = 5, 20 ≤ i ≤ n and k = 0.1. The ˆ i = 0, i ∈ [1, 10] ∩ I − {9}, and x ˆ9 = 1 solution of global discrete optimization are : x for n = 10, ˆ i = 0, i ∈ [1, 25] ∩ I − {2, 19}, x ˆ 2 = −1 and x ˆ 19 = 1 for n = 25, x ˆ i = 0, i ∈ [1, 50] ∩ I − {3, 19}, x ˆ 3 = −1 and x ˆ 19 = 1 for n = 50, x and global value is 2. Example 1 2 3 3 3

Na 8 2 10 25 50

N Cb 10 2 2 2 2

NIc 90 7 21 8 8

CPU time 6.59 5.73 9.60 3.75 7.40

Tableau 1. Results of numerical examples a

Number of variables

b

Number of constraints

c

Number of iterations.

4. Conclusions In this paper, we proved a new approach for transformed integer programming with nonlinear constraints to the global optimization under nonlinear constraints. The random perturbation of reduced gradient method is proposed for solving associate nonlinear constraints global optimization problem of integer programming with nonlinear constraints. The numerical results show the efficient of this approach.

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