Abstract: The primary objective of this study is to propose the use of nonlinear functions to find a feasible interior point in a region defined by the constraints of linear programming problems with the advantage of not having to use slack or surplus variables. The feasible region of linear programming problems is defined by convex sets named polyhedra that have the form Ax ≤ b and that can be bounded or unbounded. Finding an interior point (even extreme or frontier point) is not a trivial issue but is a necessary condition to initialize algorithms for solving a linear programming problem. To achieve this, unrestricted nonlinear penalty functions are applied and optimized. As a result, it is demonstrated that the optimal solution point corresponds to an interior point of the original polyhedron. It is concluded that the proposed algorithmic procedure possesses features that provide advantages for solving linear programming problems.

Resumen: El presente trabajo tiene como objetivo presentar una propuesta para encontrar un punto interior factible en la región definida por las restricciones de problemas de programación lineal, con la ventaja de no tener que recurrir al empleo de variables de holgura o exceso. La región factible en los problemas de programación lineal es el conjunto solución del sistema de desigualdades lineales Ax ≤ b, las cuales definen un poliedro que puede ser acotado o no acotado. Encontrar un punto interior o en la frontera de esta no es trivial y es un aspecto necesario para iniciar los algoritmos de solución de problemas de programación lineal. Para lograrlo, se recurre a funciones irrestrictas de penalización no lineales que deben ser optimizadas. Como resultado se demuestra que la propuesta converge a un punto interior del poliedro original o se establece que el mismo no es factible. Se concluye que el procedimiento algorítmico propuesto tiene características ventajosas para resolver problemas de programación lineal.

In this work we study how to solve the SVM optimization problem by using the Spectral Projected Gradient (SPG) method with three different strategies for computing the projection onto the constrained set. One of the strategies is based on Dykstra's alternating projection algorithm since there is not a mathematical equation for the projection onto the whole constrained set but the projection on each restriction is easy to compute with exact formulations. We present another strategy based on the Karush-Kunh-Tucker optimality conditions, we call it the Projected-KKT algorithm. We compare these strategies with a third one proposed by Dai and Fletcher. The three schemes are low computational cost and their use within the SPG algorithm leads to a solution of the SVM problem. We study the computational performance of the three strategies when solving randomly generated as well as real life SVM problems. The numerical results show that Projected-KKT is competitive in general with the Dai and Fletcher algorithm, and it is more efficient for some specific problems. They both outperform Dykstra's algorithm in all the tests.