To illustrate the extent of this additional complexity, Fig. Input 6a shows a schematized circuit for a computational unit of method V

Many optimization models of neural networks need constraints to restrict the space of outputs to a subspace which satisfies external criteria. Optimizations using energy methods yield "forces" which act upon the state of the neural network. The penalty method, in which quadratic energy constraints are added to an existing optimization energy, has become popular recently, but is not guaranteed to satisfy the constraint conditions when there are other forces on the neural model or when there are multiple constraints. In this paper, we present the basic differential multiplier method (BDMM), which satisfies constraints exactly; we create forces which gradually apply the constraints over time, using "neurons" that estimate Lagrange multipliers. The basic differential multiplier method is a differential version of the method of multipliers from Numerical Analysis.

To illustrate the extent of this additional complexity, Fig. Input 6a shows a schematized circuit for a computational unit of method V

Many optimization models of neural networks need constraints to restrict the space of outputs to a subspace which satisfies external criteria. Optimizations using energy methods yield "forces" which act upon the state of the neural network. The penalty method, in which quadratic energy constraints are added to an existing optimization energy, has become popular recently, but is not guaranteed to satisfy the constraint conditions when there are other forces on the neural model or when there are multiple constraints. In this paper, we present the basic differential multiplier method (BDMM), which satisfies constraints exactly; we create forces which gradually apply the constraints over time, using "neurons" that estimate Lagrange multipliers. The basic differential multiplier method is a differential version of the method of multipliers from Numerical Analysis.

To illustrate the extent of this additional complexity, Fig. Input 6ashows a schematized circuit for a computational unit of method V-r

We present a new polynomial-time algorithm for linear programming. The running-time of this algorithm is O(n3-5L2), as compared to O(n6L2) for the ellipsoid algorithm. We prove that given a polytope P and a strictly interior point a ε P, there is a projective transformation of the space that maps P, a to P', a' having the following property. The ratio of the radius of the smallest sphere with center a', containing P' to the radius of the largest sphere with center a' contained in P' is O (n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial-time.

In this article we discuss a method for learning useful conditions on the application of operators during heuristic search. Since learning is not attempted until a complete solution path has been found for a problem, credit for correct moves and blame for incorrect moves is easily assigned. We review four learning systems that have incorporated similar techniques to learn in the domains of algebra, symbolic integration, and puzzle-solving. We conclude that the basic approach of learning from solution paths can be applied to any situation in which problems can be solved by sequential search.

The distributive computation of constrained optimization problems by networks of locally interconnected simple processors is examined. A general method for designing such networks is described. The design includes the network of interconnections among the participating processors, as well as the iterative computation to be performed by each processor. The application of these results to relaxation techniques in the processing of visual information is discussed and exemplified.

Zeitschrift fur Operations Research, Serie A, 20 (5), 203-208.

Additive AND/OR graphs are defined as AND/ /ORgraphs without circuits, which can be considered as folded AND/OR trees; i.e. the cost of a common subproblem is added to the cost as many times as the subproblem occurs, but it is computed only once. Additive AND/OR graphs are naturally obtained by reinterpreting the dynamic programming method in the light of the problem-reduction approach. An example of this reduction is given. A top-down and a bottom-up method are proposed for searching additive AND/OR graphs. These methods are, respectively, extensions of the "arrow" method proposed by Nilsson for searching AND/OR trees and Dijkstra's algorithm for finding the shortest path. A proof is given that the two methods find an optimal solution whenever a solution exists. 1) introduction In the literature on artificial intelligence, AND/OR trees have proved to be a good formalism for representing the problem-reduction approach to problem solving. Usually, the search is for any solution tree, but in a paper by Nilsson the problem is presented of finding the best solution tree, where arcs have a given cost, and the cost of a tree is simply the sum of the costs of the arcs. Nilsson gives there an algorithm which assumes available, for each node, an estimate of the cost of the "optimal solution tree rooted at that node.