Vector-Processing Algorithms for Extreme-Scale Geometric Programming

The patent segments the extreme-scale geometric programming problem into manageable components by exploiting the hidden separability property. The objective function and constraints are decomposed into separable terms that can be processed independently, allowing the problem to be solved in parallel across multiple processors. This segmentation reduces the computational burden from handling the entire problem at once to processing individual separable components, making extreme-scale problems solvable.

The patent transforms the high-dimensional extreme-scale problem into a different computational dimension by reformulating it as a conjugate dual problem. This dimensionality change allows the use of vector-processing algorithms that operate efficiently in the transformed space, converting an intractable optimization problem into a series of simpler linear algebra operations that can be executed in real-time on parallel-processing supercomputers.

The patent replaces traditional iterative optimization algorithms with direct vector-processing methods based on conjugate duality. Instead of using mechanical iterative procedures that require many sequential steps, the invention substitutes a direct mathematical transformation approach that computes the solution through explicit formulas involving vector operations. This substitution eliminates the iterative loop and achieves real-time solutions through direct computation on parallel architectures.

The patent changes the fundamental parameters of the problem representation by transforming from the primal geometric programming formulation to the conjugate dual formulation. This parameter change involves redefining the decision variables, objective function, and constraints in terms of dual variables and functions. The transformed parameters enable the use of efficient vector-processing algorithms that operate in linear time, achieving real-time solution capability for extreme-scale problems.