Research On The Smoothing Algorithms For Second-order Cone Programming And Second-order Cone Complementarity Problems

The second-order cone programming(SOCP) problem is a special case of symmetric cone programming and has broad application in communication engineering, control optimization, facility location, engineering design, antenna array design, portfolio investment optimization. In the mathmatical programming, linear programming, robust least squares and quadratically constrained convex quadratic programming, norm minimization problems can be recast as SOCP problems.The second-order cone complementarity problems are a class of equilibrium problems which is to find a vector satisfying a system of equations and a complementarity condition defined on the Cartesian product of second-order cones. In recent years, based on the technique of Euclidean Jordan algebra, many researchers have achieved a breakthrough in the study of second-order cone complementarity problems and made them attract lots of attention, them contains: the existence and properties of the solution, merit function and error bound, many smoothing method and optimization method and real applications.In Chapter 2, we consider linear second-order cone programming: a new smoothing function corresponding to second-order cone is proposed, and its properties are studied.Based on the smoothing function, a modified predictor-corrector smoothing method for SOCP is given. we proof that the modified method’s initial point is arbitrary. Without any strict complementarity assumptions, we obtain the global and the local quadratic convergence of the algorithm. Numerical results indicate that the algorithm is e?ective.In Chapter 3, we consider we consider second-order cone complementarity problems:a new class of merit function is proposed for SOCCP. With some suitable assumptions,a global error for the solution to the SOCCP is established and the level of every merit function is bounded based on these merit functions. Based on the smoothing function in chapter, a smoothing Newton method SOCCP is proposed, and under some suitable assumptions, the global and the local quadratic convergence of the algorithm can be obtained. Numerical results indicate that the algorithm is e?ective. In Chapter 4, finally,the summary and prospect are listed in this chapter.