Qualitative Reasoning About Constraint Activity Using Monotonic Influence Diagrams

N. F. Michelena and A. M. Agogino

Monotonic influence diagrams (MID) are proposed for representing and manipulating qualitative and mathematical relationships between variables and constraints in order to design from physical principles. The theory of MID’s is based on a graph-theoretic representation of an optimization problem which can be topologically transformed as a means of solving the problem and exploring variable-objective-constraint relationships. Monotonic influence diagrams are a synthesis of influence diagrams and monotonicity analysis. Formally, a monotonic influence diagram is a directed graph consisting of nodes and arcs. The nodes represent design variables and the arcs reveal their relationships. Nodes in a MID can represent either deterministic or uncertain quantities. A deterministic qualitative relation between two variables is given by the sign of the partial derivative of the function defining one of the variables with respect to the other variable. A probabilistic qualitative relation is defined in terms of a constraint on the joint probability distribution of the variables. Only deterministic quantities and relationships will be addressed in this paper. Topological transformations such as arc reversal and node removal allow us to determine qualitative relations between constrained design variables and the objective function to be minimized or maximized. In this sense, MID’s provide a reasoning mechanism about constraint activity which entails explicit reasoning about inequality constraints, so candidates for active constraints or flaws in the problem formulation can be detected.

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