An Interior Feasible Direction Method with Constraint Projections for Linear Programming

-A new feasible direction method for linear programming problems is presented. The method is not boundary following. The method proceeds from a feasible interior point in a direction that improves the objective function until a point on a constraint surface is met. At this point searches are initiated in the hyperplane of constant function value by using projections of the bounding constraints until n bounding constraints are identified that yield a vertex as candidate solution. If the vertex is not feasible or feasible with a worse function value, the next iteration is started from the centre of the simplex defined by the identified points on the bounding constraint surfaces. Otherwise the feasible vertex is tested for optimality. If not optimal a perturbed point with improved function value on an edge emanating from the vertex is calculated from which the next iteration is started. The method has successfully been applied to many test problems. 1. I N T R O D U C T I O N For almost 40 years the simplex method of Dantzig [1] was the established and unchallenged method for solving linear programming (LP) problems. Recently the situation has changed with the publication of Karmarkar 's projective method [2] and the claim that the practical behaviour of this polynomial-in-time method is superior to that of the simplex method. At present the question of superiority has not yet been decided although the general impression is that the new method may indeed be superior for specially structured large problems [3]. In any case, the dramatic announcement of the new method has stimulated a marked revival of interest in LP not only in Karmarkar 's method but also in possible new non-simplex ways of approaching the problem. Further motivation for the search for non-simplex methods may be found in the recent publications of Zcleny [4] and Mitra et al. [5]. In particular Zeleny points out the need for parallel optimization algorithms required to take advantage of the emergence of advanced parallel computers in the future. So far, operations research has devoted its effort to sequential algorithms whereas parallel computers require the design of new algorithms with larger "granularity", i.e. characterized by a sufficient number of concurrent tasks in order to keep a large number of processors busy. Most of the older feasible direction alternative methods, such as Zoutendijk's method [6], Roscn's gradient projection method [7], Wolfe's reduced gradient method [8] and Zangwill's convex-simplex method [9] are boundary following methods. It is well-known that these methods are not competitive compared to the performance of the simplex method although they have the advantage that they may be extended to more general non-linear programming problems. In this paper a non-simplex feasible direction method is presented that is not boundary following. The method attempts to identify the optimal bounding set of active constraints by a sequence of steps taken through the interior of the feasible region. In addition it will be shown that the new algorithm possesses significant parallel features compared to the highly sequential simplex method. This should allow the new method to exploit the potential offered by parallel computing. In a typical iteration the new method proceeds from a feasible interior point in a direction that improves the objective function until a constraint surface is met. At this point the negative gradient of the constraint surface is projected onto the hypersurface of constant function value going through this point. Hereby searches are initiated in this hyperplane until n points on n constraint surfaces bounding this constant function value surface have been identified. Solving the system of n linear equations corresponding to the bounding constraints yields a vertex that may or may not