JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 36, No. 1, JANUARY 1982

TECHNICAL NOTE

On the Convergence of Two Branch-and-Bound Algorithms for Nonconvex Programming Problems 1

H. P. BENSON 2

Communicated by G. Leitmann

Abstract, This note presents a new convergence property for each of two branch-and-bound algorithms for nonconvex programming prob- lems (Falk-Soland algorithms and Horst algorithms). For each algorithm, it has been shown previously that, under certain conditions, whenever the algorithm generates an infinite sequence of points, at least one accumulation point of this sequence is a global minimum. We show here that, for each algorithm, in fact, under these conditions, every accumulation point of such a sequence is a global minimum.

Key Words. Nonconvex programming, algorithmic convergence, branch-and-bound procedures, global minimum.

1. Introduction

Given a compact set X C_ R n, consider the problem of finding a vector ~ X which satisfies

f(g) = minf (x ) , subject to x ~ X, (1)

where f is a real-valued function defined on some set Y,

X C Y C R " .

Assume that at least one such vector $ exists. Falk and Soland (Ref. 1) and Horst (Ref. 2) have each developed a branch-and-bound algorithm for solving (1) under different assumptions concerning f and X. However ,

1 The author would like to thank Professor R. M. Soland for his helpful comments concerning this paper.

2 Assistant Professor, College of Business Administration, University of Florida, Gainesville, Florida.

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in each case, f may be a nonconvex function on Y. These algorithms may not terminate after finitely many steps. Instead, either algorithm may generate an infinite sequence {x k} of feasible points for (1). Falk and Soland have shown that, if such a sequence {x k} is generated by their relaxed algorithm, then, under certain conditions, at least one accumulation point of {x k} is a solution of (1) (Ref. 1, Theorems 1' and 2'). If such a sequence {x k} is generated by the Horst algorithm, then, under different conditions, at least one of its accumulation points is also guaranteed to solve (1) [Ref. 2, Theorem (c)]. In practice, however, under these conditions, neither algorithm has been found to generate an infinite sequence of points with a convergent subsequence that tends to a false minimizing point for (1) (Refs. 1 and 3).

In Ref. 4, we prove that, in fact, under the conditions given in Ref. 1 in Theorems 1' and 2', if an infinite sequence {x k} of feasible points for (1) is generated by the relaxed Falk-Soland algorithm, then any accumulation point of {x k} solves (1). Similarly, we prove in Ref. 4 that, if such a sequence {x k} is generated by the Horst algorithm, then, under the conditions given in Ref. 2 in Theorem (c), any accumulation point of {x k} solves (1).

To derive these results, Ref. 4 presents a prototype branch-and-bound algorithm for solving (1) of which the Falk-Soland and Horst algorithms are special cases. A convergence theorem for the prototype algorithm is proven, and, from this theorem, the desired results follow.

In this note, we briefly outline the arguments given in Ref. 4.

2. Prototype Branch-and-Bound Algorithm

As a preliminary to presenting the prototype branch-and-bound algorithm for solving (1), consider the following definition.

Definition 2.1. (Ref. 2). Let G be a compact set in R". A set {G1, G2 . . . . . G~} of finitely many compact subsets G~, i = 1, 2 . . . . . v, of G is a partition of G when

(i) G = U Gi, i~l

(ii) GinGj=OGimOGj, for all i , j~{1, 2 . . . . . v},i#],

where 3G~ denotes the relative boundary of Gi relative to G for each i~{1,2 . . . . . v}.

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Prototype Branch-and-Bound Algorithm for Problem (1)

Step 1

Step i. 1. Choose a compact set G C .R ~ such that X C_ G. Choose a partition {Gll, G12, • • . , Glm} of G. Pick e -> 0.

Step 1.2. Find p~ functions fli: GI~ -+ .R, i = 1, 2 , . . . , p~, such that

(a) fli(x) - f ( x ) , for all x ~ Gli, (b) minfl~(x), subject to x~Xc~Gl i , is computable whenever

Xc~GI~ ~ ~ , for each i = 1, 2 . . . . . p>

Step 1.3. For e a c h i = l , 2 , . . . , p ~ s u c h t h a t X c ~ G a i # ~ , c o m p u t e an optimal solution x li for

rain fl~ (x), subject to x ~ X n GI~.

Step 1.4. Compute

k,,(x )=

where i is restricted to

If

Step 1.5.

{i ~ {1, 2 . . . . . pa}]Xc~Gai# ~}.

Set

x t = x lq and LBl(x 1) =fli~(xlq).

f(x 1) - L B I ( x 1 ) -<E,

then stop the algorithm. Otherwise, set k = 2, and go to Step 2.

Step k, k = 2, 3 , . . .

Step k.1. Assume, without loss of generality, that x k- l~ Gk-~.pk_l. Choose a partition {Ggl, Gk2, • . . , Gkv~,} of Gk-l.vk_~.

Step k.2. Find p'k functions fkj: Gki~R, i = l, 2 . . . . . p~,, such that

(a) fk-l,p~_~(X)<--fki(x)<-f(x), for all x ~ Gki, (b) minfki(x), subject to x6Xc~Gki , is computable whenever

X n Gkl ~ 0 , for each i = 1, 2 . . . . . p},.

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Step k.3. an optimal solution x k~ for

min/ki(x),

Step k.4. Set

for each

Set

For each i = 1, 2 . . . . . p~ such that X c~ Gki ~ 0 , compute

subject to x ~ X ~ Gk~.

X k , P ~ +i ~_ x k - 1 , i,

i e {i e {1, 2 . . . . . Pk-1 - 1}IX ~ Gk-l,~ ~ 0}.

Relabel Gk-l,i with 1, 2 . . . . . pk-1- 1.

Step k.5. Compute

where i is restricted to

If

Step k.6.

Pk = p'k + Pk-1 -- 1.

Gk, v'k+i and fk-l,i with fk,pk'+'~, for each i =

ki k . ki fkik (X ) = m~n fki(X ),

{i ~{1, 2 . . . . . pk} lX n Gk, ~ f~}.

Set k ~ X kik x and LBk(x k) =fklk(xki~).

f (Xk) - -LBk(xk)~E,

then stop the algorithm. Otherwise, set k = k + 1, and go to Step k + 1. This prototype branch-and-bound algorithm is adapted from Horst

(Ref. 2), with certain modifications. It is easily seen that both the Falk- Soland algorithm and the Horst algorithm are special cases of the prototype branch-and-bound algorithm. Notice that, in the algorithm, x k~ Gkik, for each k.

The main convergence theorem proven in Ref. 4 for the prototype algorithm is stated below. It requires the algorithm to satisfy the following condition.

Condition 2.1. Convergence Condition. Assume that the prototype branch-and-bound algorithm generates an infinite sequence {xk}. Suppose that {x ki} is a convergent subsequence of {x k} with limit ~ such that Gkii~

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is a subset of Gki_li~,i_ 1 for each ]. Then,

• k j ~ hm fk,lk~ ( X ) = f(x).

1

Theorem 2.1. Suppose that the prototype branch-and-bound algorithm for (1) is not terminated after finitely many steps. Then, if the prototype algorithm satisfies the convergence condition, it generates a sequence of points {Xk}, any accumulation point of which is a solution for (1).

3. Convergence of the Relaxed Falk-Soland and Horst Algorithms

The Falk-Soland and Horst algorithms for solving (1) operate under different assumptions concerning f and X. The reader is referred to Ref. 4 or to Refs. 1 and 2 for these assumptions.

Falk and Soland give both a nonrelaxed and a relaxed version of their algorithm. In the nonrelaxed version, using the notation of our prototype algorithm, measures are taken to insure that, for each k-> 2, the set Gkik containing x k is an element of the completion of the partition {Gkl, Gk2 . . . . . Gkp~} of G. See Ref. 1 for details. No such requirement is imposed in the relaxed version of the algorithm.

The convergence results proven in Ref. 4 concerning the relaxed Falk-Soland algorithm and the Horst algorithm follow as corollaries of Theorem 2.1. Each corollary is proven in Ref. 4 by showing that, under the conditions of the corollary, the convergence condition holds. These results are as follows. The reader may consult either Ref. 4 or Refs. 1 and 2 for explanations of the strong and weak refining rules, referred to in the first corollary, and of Requirements 1 and 2 of Ref. 2, referred to in the second corollary.

Corollary 3.1. Suppose that the relaxed Falk-Soland algorit~m for (1) is not terminated after finitely many steps. If either (i) the strong refining rule is used or (ii) f is continuous on G and the weak refining rule is used, then the relaxed algorithm generates a sequence {xk}, any accumulation point of which is a solution for (1).

Corollary 3.2. Suppose that the Horst algorithm for (1) is not 'termi- nated after finitely many steps. If Requirements 1 and 2 of Ref 2 are satisfied, then the Horst algorithm generates a sequence {x k}, any accumula- tion point of which is a solution for (1).

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4. Concluding Remarks

Corollary 3.1 shows that the strong convergence results proven pre- viously only for the nonrelaxed Falk-Soland algorithm (Ref. 1, Theorems 1 and 2) also hold for the relaxed algorithm. From this result and the fact that concern with completions of partitions is not present using the relaxed algorithm, it appears that this version of the Falk-Soland algorithm is preferable to the nonrelaxed version. To explore this possibility, computa- tional experiments involving the solution of several nonconvex problems using both the relaxed and the nonrelaxed algorithms should be performed.

References

1. FALK, J. E., and SOLAND, R. M., An Algorithm for Separable Nonconvex Programming Problems, Management Science, Vol. 15, pp. 550-569, i969.

2. HORST, R., An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312-321, 1976.

3. HORST, R., Private Communication, 1980. 4. BENSON, H. P., On the Convergence of Two Branch-and-Bound Algorithms for

Nonconvex Programming Problems, University of Florida, Center for Econometrics and Decision Sciences, Discussion Paper No. 3, 1979.