Abstract: Apparatus for optimizing the operational state of a system, which follows the Karmarkar method, and which in the course of carrying out this method, obtains a solution to the linear system of equations AD.sup.2 A.sup.T u=p in accordance with a modified conjugate gradient method that incorporates a preconditioning operation. The preconditioning operation includes pre-drop and post-drop procedures that reduce the number of non-zero terms in the matrix to be preconditioned by judiciously discarding values that are smaller than a certain threshold. Use of the preconditioned conjugate gradient method reduces the processing time required for carrying out each iteration in the Karmarkar method.

Abstract: Method and apparatus for efficiently performing preselected computations on data that, in part, changes seldomly. A recognition that a portion of the data does not change leads to substantial improvements in computation speed through apparatus that tailors the program that executes the computation to the data itself. In the context of multiplication of a matrix that varies seldomly by a vector that changes more often, this is accomplished by reordering the elemental operations so that accesses to memory are reduced, both for purposes of reading information and writing back to memory, and developing code that reflects the required sequence of operations.

Abstract: Method and apparatus for optimizing the operational state of a system employing iterative steps that approximately follow a projective scaling trajectory or an affine scaling trajectory, or curve, in computing from its present state, x.sub.0 to a next state x.sub.1 toward the optimum state. The movement is made in a transformed space where the present (transformed) state of the system is at the center of the space, and the curve approximation is in the form of a power series in the step size. The process thus develops a sequence of tentative states x.sub.1, x.sub.2, x.sub.n . . . . It halts when a selected suitable stopping criterion is satisfied, and assigns the most recent tentative state as the optimized operating state of the system.

Abstract: Apparatus and method for efficient multiplication of a matrix by a vector. The multiplication is realized by rearranging the matrix so that a plurality of adjacent columns form a set, and each set is characterized by a certain pattern in each column. One set contains columns of only a single .+-.1 entry, another set contains columns with two .+-.1 entries, and still another set contains columns with entries that are other than .+-.1. Each of the sets is treated differently during the calculations in accordance with a maximal * cover approach, and a result is developed by multiplying each row in the rearranged matrix by a corresponding element of the vector to form partial results, and accumulating the partial results.

Abstract: A method and apparatus for optimizing resource allocations is disclosed which proceeds in the interior of the solution space polytope instead of on the surface (as does the simplex method), and instead of exterior to the polytope (as does the ellipsoid method). Each successive approximation of the solution point, and the polytope, are normalized such that the solution point is at the center of the normalized polytope. The objective function is then projected into the normalized space and the next step is taken in the interior of the polytope, in the direction of steepest-descent of the objective function gradient and of such a magnitude as to remain within the interior of the polytope. The process is repeated until the optimum solution is closely approximated.

Abstract: Method and apparatus for optimizing the operational state of a system employing iterative steps that approximately follow a projective scaling trajectory or an affine scaling trajectory, or curve, in computing from its present state, x.sub.0 to a next state x.sub.1 toward the optimum state. The movement is made in a transformed space where the present (transformed) state of the system is at the center of the space, and the curve approximation is in the form of a power series in the step size. The process thus develops a sequence of tentative states x.sub.1, x.sub.2, x.sub.n . . . . It halts when a selected suitable stopping criterion is satisfied, and assigns the most recent tentative state as the optimized operating state of the system.