A KKT Simplex Method for Efficiently Solving Linear Programs for Grasp Analysis Based on the Identification of Nonbinding Constraints
Un método simplex KKT para resolver eficientemente programas lineales para análisis de la sujeción basado en la identificación de restricciones no atadas
Alejo Mosso-Vázquez1, David Juárez-Romero1, Marco Antonio Cruz-Chávez1, and Luis Enrique Sucar2
1 Centro de Investigación en Ingeniería y Ciencias Aplicadas, Cuernavaca, Morelos, Mexico alejo.mosso@gmail.com, djuarez7@gmail.com, mcruz@uaem.mx
]]> 2 Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), Tonantzintla, Puebla, Mexico esucar@inaoep.mx
Abstract
A one-phase efficient method to solve linear programming (LP) problems for grasp analysis of robotic hands is proposed. Our method, named as KKT Simplex method, processes free variables directly while choosing the entering and leaving variables, which makes it a one-phase method able to start at any point of the set of feasible solutions. Besides, the proposed method lowers the number of simplex steps by an angular pricing strategy to choose the entering variable. Moreover, the method reduces the size of an LP problem by the identification of nonbinding constraints that preserves the Karush-Kuhn-Tucker (KKT) cone. We developed the KKT Simplex method by incorporating to the well-known revised simplex method the following components: a method to process free variables, a pricing strategy, and an identification method. We solve LP problems of grasp analysis to test the efficiency and the one-phase nature of the proposed method.
Keywords: KKT Simplex method, linear programming, grasp analysis, nonbinding constraints.
Resumen
Se propone un método eficiente de una fase para resolver problemas de programación lineal (LP) para análisis de la sujeción por manos robóticas. El método, nombrado como método Simplex KKT, procesa variables libres directamente mientras selecciona las variables entrante y saliente, lo que lo convierte en un método de una fase que es capaz de iniciar en cualquier punto del conjunto de soluciones factibles. Además, el método disminuye el número de pasos simplex por una estrategia angular de costo para seleccionar la variable entrante. Aún más importante, el método reduce el tamaño del problema LP por identificación de restricciones no atadas que preserva el cono Karush-Kuhn-Tucker (KKT). Desarrollamos el método Simplex KKT por la incorporación al bien conocido método simplex revisado de los siguientes componentes: un método para procesar variables libres, una estrategia de costo, y un método de identificación. Resolvemos problemas LP de análisis de la sujeción para probar la eficiencia y la naturaleza de una fase del método propuesto.
Palabras clave: Método Simplex KKT, programación lineal, análisis de la sujeción, restricciones no atadas.
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