Flexible vector-processing algorithms for numerically solving extreme-scale, linear and non-linear, predictive and prescriptive, problems in science and engineering, on parallel-processing super computers

Abstract

A computer-implemented method for numerical solution of a geometric programming problem is described, including the computer-implemented steps of: reformulating the geometric programming problem as an equivalent generalized geometric programming optimization problem with only linear constraints, and solving the equivalent generalized geometric programming optimization problem by vector processing, including determining by computer-implemented numerical computation a solution for an unconstrained objective function whose independent vector variable is the generalized geometric programming conjugate dual of a primal decision vector variable of the geometric programming problem, and includes a variable linear combination of fixed vectors enabling the vector processing. Also described are computer-readable storage devices, computer program products, and computer systems for such numerical solution methodology.

What is claimed is: 
     
         1 . A computer-implemented method for numerical solution of a geometric programming problem, comprising the computer-implemented steps of:
 reformulating the geometric programming problem as an equivalent generalized geometric programming optimization problem with only linear constraints, and   solving the equivalent generalized geometric programming optimization problem by vector processing, comprising determining by computer-implemented numerical computation a solution for an unconstrained objective function whose independent vector variable is the generalized geometric programming conjugate dual of a primal decision vector variable of the geometric programming problem, and comprises a variable linear combination of fixed vectors enabling the vector processing.   
     
     
         2 . The method of  claim 1 , wherein the geometric programming problem comprises at least one of design, modeling, and optimization of a system. 
     
     
         3 . The method of  claim 2 , wherein the system comprises an exa-scale system. 
     
     
         4 . The method of  claim 1 , wherein the geometric programming problem comprises a stochastic linear programming optimization problem. 
     
     
         5 . The method of  claim 1 , wherein the geometric programming problem comprises multi-scale climate modeling. 
     
     
         6 . The method of  claim 1 , wherein the geometric programming problem comprises modeling of controlled nuclear fusion. 
     
     
         7 . The method of  claim 1 , wherein the geometric programming problem comprises a problem selected from the group consisting of: economic equilibration, physical equilibration, profit maximization, and cost minimization. 
     
     
         8 . The method of  claim 1 , wherein the geometric programming problem comprises a problem selected from the group consisting of: modeling of physiological processes; modeling of nanoscale electronic devices and microelectromechanical systems (MEMS); pharmaceutical drug design; design of materials selected from the group consisting of high temperature superconductors, quantum computing components, and structural nanomaterials; modeling of planetary, interstellar, and galactic systems; design of particle accelerators; modeling of population dynamics; design of pandemic intervention models; optimization of global supply chains; design of multinational disaster response systems; modeling of seismological activity and events; design and optimization of robotic systems; optimization of resource allocation in resource-constrained environments; political forecasting; modeling of combustion energy systems; modeling of dark energy and dark matter interactions and environments; modeling of exaJoule-level energy flows; design and optimization of exaFLOP computing systems; and modeling of nanoscale sensor monitoring systems. 
     
     
         9 . The method of  claim 1 , wherein said solving comprises convex optimization of a non-convex geometric programming problem. 
     
     
         10 . The method of  claim 1 , comprising Rockafellar bi-function programming. 
     
     
         11 . The method of  claim 1 , comprising fuzzy optimization. 
     
     
         12 . The method of  claim 1 , wherein the vector processing produces an optimal solution that is used to solve a deterministic primal problem of the geometric programming problem by a linear programming method. 
     
     
         13 . The method of  claim 1 , wherein said computer-implemented steps are conducted on a parallel processing computer. 
     
     
         14 . A computer-readable storage device embodying a non-transitory program of machine-readable instructions executable by a digital processing apparatus to perform a method of numerical solution of a geometric programming problem, the method comprising:
 reformulating the geometric programming problem as an equivalent generalized geometric programming optimization problem with only linear constraints, and   solving the equivalent generalized geometric programming optimization problem by vector processing, comprising determining by computer-implemented numerical computation a solution for an unconstrained objective function whose independent vector variable is the generalized geometric programming conjugate dual of a primal decision vector variable of the geometric programming problem, and comprises a variable linear combination of fixed vectors enabling the vector processing.   
     
     
         15 . The computer-readable storage device of  claim 14 , selected from the group consisting of random access memory, magnetic data storage diskettes, CD discs, hard drives, RAID arrays, magnetic tape, electronic read-only memory, and optical storage devices. 
     
     
         16 . A computer program product comprising a computer program for enabling numerical solution of a geometric programming problem, the computer program comprising:
 instructions for reformulating the geometric programming problem as an equivalent generalized geometric programming optimization problem with only linear constraints, and   instructions for solving the equivalent generalized geometric programming optimization problem by vector processing, comprising instructions for determining by computer-implemented numerical computation a solution for an unconstrained objective function whose independent vector variable is the generalized geometric programming conjugate dual of a primal decision vector variable of the geometric programming problem, and comprises a variable linear combination of fixed vectors enabling the vector processing.   
     
     
         17 . A computer system comprising a memory and parallel processors that are programmatically adapted by the computer program product of  claim 16  to generate the numerical solution of the geometric programming problem. 
     
     
         18 . The computer system of  claim 17 , wherein the geometric programming problem comprises at least one of design, modeling, and optimization of an exa-scale system. 
     
     
         19 . The computer system of  claim 18 , wherein the geometric programming problem comprises a stochastic linear programming optimization problem. 
     
     
         20 . A computer-implemented method for generating a solution output on a graphical user interface or other output component of a programmed vector parallel-processing computer to a mixed integer linearly constrained convex programming (MILCCP) application, wherein the programmed vector parallel-processing computer is programmed to reduce the MILCCP application to an unconstrained convex programming application, and wherein said method comprises automatic operation of the programmed vector parallel-processing computer to reformulate the MILCCP application as an equivalent convex generalized geometric programming (GGP) problem whose corresponding conjugate dual problem is unconstrained, and automatic operation of the programmed computer to solve the GGP problem with an unconstrained convex programming algorithm, by vector parallel-processing. 
     
     
         21 . The computer-implemented method of  claim 20 , wherein the MILCCP application comprises Markowitz portfolio selection.