Study On Smoothing Newton Algorithm For Circular Cone Programming And Circular Cone Complementarity Problems

Circular cone programming(CCP)and circular cone complementarity problems(CCCPs)are an important class of optimization problems.CCP is a class of optimization problem which minimizes or maximizes a linear function over the intersection of an affine linear manifold with the Cartesian product of circular cones.CCCPs are a class of equilibrium optimization problems.CCP and CCCPs are widely used in engineering problems,for example,the optimal grasping manipulation for multifingered hand-arm robots,contact force optimization problem and force optimization for the legs of a quadruped robot.However,the circular cone generally is non-self-dual under the standard inner product.Thus there is little work about algorithms for solving CCP and CCCPs.In this paper,we present smoothing Newton methods for solving CCP and CCCPs.The main results of this paper are stated as follows:1.Based on a smoothing function and the relationship between the circular cone and the second-order cone,we propose a smoothing Newton method for solving CCP.The global and local quadratic convergence of the algorithm are analyzed by the Euclidean Jordan algebra theory.Finally,numerical results of the force optimization problem for a quadruped robot and random generated CCP illustrate the effectiveness of the algorithm.2.A nonmonotone smoothing Newton method is presented for solving CCP.The proposed algorithm uses a nonmonotone line search scheme,which can improve the convergence speed.Under suitable assumptions,we prove that the algorithm is globally and locally quadratically convergent.Some numerical results illustrate that the proposed algorithm can solve CCP with little CPU time and iteration number,and it is relatively stable,thus indicate the effectiveness of the method.3.We reformulate the circular cone complementarity problem as a nonlinear system of equations by a one-parametric class of smoothing function,and then propose a nonmonotone smoothing Newton method for solving CCCPs.The proposed algorithm uses a nonmonotone line search scheme,which can obtain better numerical results.Under suitable assumptions,the coerciveness of the merit function,the global and local quadratic convergence of the proposed algorithm are proved.Numerical results illustrate the effectiveness of the proposed algorithm.