Expanding constraint theory to determine well-posedness of large mathematical models

UnrestrictedMathematical modeling represents one of the major tools for the conception and management of the ever increasing complexity of systems engineering. Unfortunately, present approaches to math modeling suffer from several theoretical problems which include: model consistency, computational allowability, management of the topologically complex flow of software algorithms, rearrangement of independent and dependent variables, distinction between the model structure and software programs and perhaps the most challenging, the exponential explosions resulting from the management of even medium sized models as well as the large models of the thousands of dimensions necessary to define and manage the complex systems of the future.; Constraint theory was designed to solve the above problems employing a rigorous application of graph theory and attempts to employ the generalizability of mathematics to extend the math model manager’s conceptual understanding from half a dozen dimensions to the desired thousands of dimensions. Constraint theory (CT) went through several stages of detail and maturity, starting with the PhD dissertation in the 1960’s and progressing through several papers and two other PhD research programs. CT’s present frontier can be characterized by the Constraint Theory book written by Dr. George Friedman and published by Springer in 2005.; CT differs from linear programming (LP) in several ways. LP requires a full model of explicit mathematical expressions in linear form whereas CT employs a meta-model based upon relevancy between general (linear or non-linear) relations and variables. CT seeks to determine a model consistency and computational allowability without having to actually solve for a specific solution set whereas LP assumes that the problem is well-posed while attempting to solve for an optimal solution within a given constrained trade space.; As mentioned above, the exponential explosions associated with the management of thousand-dimensional models is truly enormous, on the order of 2N examinations of a model’s N equations are required to determine consistency, for example. CT has converged this process – which would take several universe lifetimes even with nanosecond computer cycle times – by a factor of trillions. This convergence is based on an ordered series of graph theoretic steps involving connected-ness, tree-ness, circuit-ness and cluster-ness. A constraint theoretic structure called the “Basic Nodal Square” (BNS) is identified as the kernel of constraint in a math model. An n × n BNS is essentially a complete sub-system of n relations and n variables within the overall model. However, Friedman’s book effectively stops at the identification of circuit clusters of approximately 30 relations, and suggests that BNS within these clusters can be found by modern computers in a few hours. It is suggested that perhaps more research can employ the topological property of adjacency to converge the search for BNS within larger circuit clusters.; The central contribution resultant from this investigation realizes and improves the computational efficiency of BNS search by factors of trillions, asymptotically (see Figure 10-2). This improvement has been accomplished by innovative application of graph theory, topology, algorithm analysis and linear algebra, which were not addressed to sufficient depth in the original efforts. Leveraging research results in graph theory since the early 1970’s, several BNS search methods, based on nodal adjacency, circuit adjacency and nodal degree, have been developed and compared against the baseline (brute-force) approach of 2N. Primary key research findings and enablers include:; a. Decomposition of a model graph into its connected components by employing the graph-theoretic concept of a spanning tree and applying the depth-first search algorithm. -- b. Innovation of the edge-centric method, over the legacy approach of vertex-centricity, to identify, and remove, internal trees (or bridges) within a connected component. -- c. Further isolation of circuit clusters containing potential BNS by using the graph-theoretic concept of articulation point (or separating vertex). -- d. Rigorous proof of circuit vector-based theorems to simplify the computational complexity of constructing unions of adjacent circuits, and thus reduce BNS search space. -- e. Application of vectorial dot product to detect adjacency among circuits, and overlapping among nodes or BNSs. -- f. Development and demonstration of a meta-meta-model graph to represent overlapping among nodes and to reduce the solution-time for BNS search, from exponential to polynomial.; Additional accomplishments further improve the utility of CT by developing an integrated set of efficient computing algorithms to determine model consistency and computational allowability. The output of these algorithms can also advise the model builder of repair alternatives to correct any model inconsistency detected. Such algorithms are necessary to bridge the gap between theoretical abstracts and practical realization of CT in terms of an effective computer-assisted tool for math model management