Study On The Projection Algorithm For Circular Cone Programming

Circular cone programming is a special nonsymmetric cone programming,which plays an important role in the field of cone programming.There are linear circular cone programming and convex quadratic circular cone programming.Because the circular cone is a nonsymmetric cone,it is difficult to design some algorithms for the circular cone programming.The existed algorithms overcome these difficulties by using the relationship between circular cone and second-order cone,but there are still some shortcomings in complexity and convergence.In order to overcome these shortcomings,the different projection algorithms are proposed for two circular cone programming problems in this paper.To solve the linear optimization over circular cones,a projection function is introduced firstly.Then a Full-Newton step projection algorithm with better performance is proposed.Based on the algebraic relationship between circular cone and second order cone,a projection equation that is equivalent to the complementary condition is introduced.Then a suitable constant coefficient matrix is constructed to solve the problem as a linear system of equations.Unlike the existed methods,the algorithm does not directly convert the circular cone programming into second-order cone programming,avoiding the truncation error that may be caused by the direct conversion.Because the projection equation is simple and the Full-Newton step is used in each iteration without performing any line search,the complexity of the algorithm can be greatly reduced.Moreover,starting from any initial point,the algorithm converges to the global optimal solution without strict complementarity of the problem.Therefore the algorithm has strong convergence.In order to verify the performance of the algorithm,a large number of numerical experiments have been done.Furthermore the proposed algorithm is compared with the classical interior point algorithm through some numerical experiments.For the convex quadratic optimization over circular cones,a convex quadratic circular cone programming model of force optimization problems is established based on the dynamic formulas and friction constraints.Then the optimal conditions of the problem is transformed into a new projection system of equations.According to the characteristics of the projection system of equations,a set of equivalent linear equations is constructed.A Full-Newton step projection algorithm solving the convex quadratic circular cone programming is proposed.In order to verify the efficiency of the algorithm,a large number of different numerical experiments are carried out.At the same time,the algorithm is applied to the grasping force optimization problem for multi-fingered arm robots and contact force optimization problem.The trajectories of minimum grasping force and contact force are obtained.Moreover the proposed algorithm is compared with the classical interior point algorithm through some simulation experiments.The numerical results show that,compared with the existed algorithms,the proposed algorithm is simpler,more efficient and has higher convergence.Furthermore the simulation results of force optimization problems demonstrate that the trajectories of optimal solution obtained by the two algorithms are approximate under same accuracy requirement,but the proposed algorithm is more rapid at optimization speed and more suitable for dealing with large-scale problems in practical engineering.