Abstract: A classification system that includes a first input module, a marker module, unclassified Lagrange multiplier calculation module, and a classification module. The unclassified vector describes a physical condition. The marker module assigns a marker to the unclassified vector. The marker should place the unclassified vector into one of two classes. The unclassified Lagrange multiplier calculation module calculates a classifying Lagrange multiplier for the unclassified vector using the marker and a classification rule. The classification rule may be generated using a training system for classifying the physical condition. The classification module is configured to determine that the marker places the unclassified vector in the proper class if the classifying Lagrange multiplier is small.

Abstract: A classification system that includes a first input module, a marker module, unclassified Lagrange multiplier calculation module, and a classification module. The unclassified vector describes a physical condition. The marker module assigns a marker to the unclassified vector. The marker should place the unclassified vector into one of two classes. The unclassified Lagrange multiplier calculation module calculates a classifying Lagrange multiplier for the unclassified vector using the marker and a classification rule. The classification rule may be generated using a training system for classifying the physical condition. The classification module is configured to determine that the marker places the unclassified vector in the proper class if the classifying Lagrange multiplier is small.

Abstract: A system for finding a solution to a constrained optimization problem is disclosed. The system uses a mathematical formulation describing the constrained optimization problem. A transformer builds a specific function using the mathematical formulation, variables; an objective function; at least one constraint; a class of transformation functions with a predefined set of properties; and Lagrange multipliers. The specific function may include a nonlinear resealing part; and an augmented Lagrangian part. An iterative solver uses the specific function, and Lagrange multipliers to generate a solution.