l)iscretc Applied Mathematics I1 (1985) 265 280 265 North-Holland A BRANCH AND BOUND ALGORITHM FOR EXTREME POINT MATHEMATICAL PROGRAMMING PROBLEMS Suvrajeet SEN ,~v.s/ems and Industrial Engineering Det)arlment, University of Arizona, Tucson, 85721, USA Hanif D. SHERALI l)eparlmenl ~/ Industrial Engineering and Ol)eration,s Re,~'earch, l'irginia Polytechnic Ittstitt~ze attd Slate Univer~ily, Blackshurg, I'A 24061, USA Received 28 October 1983 Rexiscd 29 ,lune 1984 Fhis paper deals with a class of nonconvex mathematical programs called Extreme Point Mathematical Programs. This class is a generalization or" zero-one integer programs and is a special ca~e of the Generalized Lattice Point Problem, and finds applications in various areas such as production scheduling, load balancing, and concave programming. The currenl work ex- isting on this class of problems is limited to certain dual t_~ pes of extreme point ranking melhods (v,'hich do not find a feasible solution until optimality) and some non-convergent cutting plane algorithms. No computational experience exists. This paper develops a finitely con,,ergent branch and bound algorithm for solving the problem. The principles involved in the design of this aleorithm arc quite general and apply to a wider class of mathematical programs inclnditlg the Generalized Lattice F'oint Problem. A random problem generator is described which is capable of genctating problems of varying levels (/1" difficulty. Conq'~tLtational experience on such pt o bletns i,, provided. I. Introduction This paper addresses a class of nonconvex mathematical programming problems called Extreme Point Mathematical Programs (EPMP), which may be stated as follows: (EPMP) minimize {c'x: xeXn V} (1.1) where X {.veR": Ax<_b} (1.2) and V is the set of extreme points or vertices of the set Y- {xeR": Dx<_f, x_>0}. (1.3) This problem is a generalization of zero-one integer programming problems in which the set Y is a unit hypercube in R", and is a special case ot a wider class of problems introduced by Glover and Klingman [9], namely the Generalized Lattice o166 2188 85 $3.30 1985, l:lse~ier Science Publishers B.V. {North-Holland) 266 5;. Sen, tt.D. Sheralt Point Problem (GLPP), which may be mathematically stated as (GLPP) minimize {c-x: xeXQU} (1.4) where X is as defined by (1.2) and U is the set of d (_<n)-dimensional faces of the set Y defined in (1.3). The problem GLPP has several important special cases such as mixed-integer zero-one programming problems and the cardinality constrained linear programming problem, and may itself be cast into the framework of multiple choice constrained programs. All of the above problems fall in the general category of what are known as Disjunctive Programming Problems, and the reader is refer- red to Sherali and Shetty [26] for a detailed taxonomy. It is remarked here that the algorithm presented in this paper can be readily adapted for the wider class of GLPP problems. Beside the host of applications which arise in the special case of zero-one integer programming problems, other applications of EPMP include critical path problems under assignment constraints as in Chandrasekaran, et al. [5], the production scheduling model of Kirby and Scobey [14], the large scale model for assigning teachers to classes as described by Mulvey [21], and the problem of allocating m jobs to n departments in such a way as to obtain an equitable index of loads for each department as presented by Sherali and Shetty [26]. In addition, as suggested by Cabot [3], a sequence of EPMP subproblems may be used to obtain a global op- timum for the nonconvex problem of minimizing a concave function over a polyhedral set. A majority of the algorithms which exist for soving EPMP are extreme point ranking methods. These algorithms are dual procedures in that the first feasible solution encountered is the optimum solution. A first published effort for solving EPMP appears to be due to Kirby, Love and Swamp [12] who exploit the fact that ifxeXD V, then x is a vertex of XD Y. Hence, they suggest the ranking of the ex- treme points of XA Y with respect to c.x, until a ranked extreme point is feasible to V. Subsequently, several variants of this technique followed. The first of these, proposed by the same authors [13], recommended that the extreme points of Y may be ranked until a vertex which also belongs to X (and which is therefore optimal) is obtained. In an effort to improve upon both the above algorithms, Puri and Swarup [22] introduced what they referred to as a 'strong cut procedure'. Here, in the context of the first method [12] above, after ranking all the optimal and the se- cond best extreme points of Y, a cut c.x_> v~ is introduced, where w is the second best objective function value. Thereafter, if further ranking is required, the optimal and second best extreme points of the set }'D{x: c'.v_>v2} are generated (which may include vertices other than those of Y), and this process is repeated until a vertex of Y feasible to Xis encountered. Kumar and Wagner [15] provide a minor variant of [17] in which during the solution of the linear program over the set XC/ Y, only the rows corresponding to the constraints of Y are selected as pivot rows until either the linear program is solved, or it is no longer feasible to continue this selec- tive pivoting, whence the procedure reverts to the scheme of Kirby, et al. [12]. A branch and bound algorithm 267 For all the above methods, no computational experience exists, and one can only conceptualize extreme cases wherein one or the other variant may be obviously preferable. However, it may be worthwhile to briefly indicate the extent of com- putations that may be involved. For example, Kelly and Tolle [11] point out that for a set Y with 15 variables and 15 constraints such that each hyperplane defining Y corresponds to a facet of this set, the expected number of vertices is 162,073. Fur- ther, using the algorithm of Matheiss [18] (which was found superior to other alter- native methods in a recent comparative study by Matheiss and Rubin [19]), the estimated cpu time to enumerate these vertices on an IBM 370/168 computer with coding in Fortran, as given via the regression equation developed in [19], would be approximately 71 minutes. Hence, it is easy to construct examples in which the ef- fort to solve EPMP may vary from a few seconds (for instance, when the linear pro- gramming relaxation solves the problem) to 71 minutes or more for a problem of the above size. The first departure from the above ranking methods was due to Glover and Klingman [9]. They present a (dual) cutting plane algorithm for Problem GLPP which solves a sequence of tighter linear programming relaxations of the problem over Xf'l Y, generating at each stage a convexity cutting plane. These cutting planes are generated by invoking the fact that at a feasible solution to GLPP, at least (n- d) of the inequalities defining Y must be binding. Again, no computational ex- perience was provided. However, as demonstrated by Sen and Sherali [24], an algorithm of this type may generate an infinite sequence of points, with no subse- quence converging to an optimal solution. In a subsequent paper, Cabot [4] presented a procedure to solve GLPP, and hence EPMP, by solving a sequence of nonconvex programs in which a convex function is maximized over a polyhedral set. The special convex function used is such that it assumes larger values for feasible points Xfh V than the other vertices of XD Y. Hence at each stage, a vertex in XD Vis detected and deleted and the process con- tinues until no point in XQ V remains. The difficulty here is that one needs to solve several convex maximization problems. Finally, we mention that a hybrid enumeration and cutting plane algorithm developed for a special class of zero-one mixed integer programs by Glover and Klingman [10] may also be considered applicable to EPMP. This follows by assum- ing that EPMP has the counting property [10], which states that any.re Y for which at least n of the defining inequalities is binding is a vertex of Y, and by transforming EPMP to a zero-one mixed integer program by the usual trick of using binary variables to represent logical inequalities. For a problem of this type, Glover and Klingman [10] use Tui's [29] enumerative cone splitting concept in conjunction with the cutting planes of Young [30] to devise a hybrid algorithm. However, in view of Zwart [31], an algorithm of this type is not necessarily finite because the same cone can be regenerated an infinite number of times. Perhaps, the recent (convergent) adaptation due to Thoai and Tui [27] may be a preferable alternative. Another method one may consider to be indirectly applicable to solving EPMP is Rubin's I23] algorithm. 268 ~. .%en, ILl)..Shera/i The foregoing assessment of available procedures for Problem EPMP, and in particular, the glaring lack of any computational experience for the problem, is in- dicative that this and related important classes of problems deserve more attention than has been forthcoming. The present paper is a step in this direction. In the next section we present the proposed algorithm. Following this, we describe a technique for generating random problems of varying levels of difficulty on which computa- tional experience is provided. Again, we point out that the discussions herein may be readily adapted for the more general Generalized Lattice Point Problem.