Several Algorithms For Second-order Cone Optimization Problems

Second-order cone related optimization problems are widely used in game theory,data processing,machine learning,network design and many other fields.In this dissertation we investigate two specific classes of these problems:most widely applied second-order cone constrained problem(SOCP)and not-well studied stochastic mathematical programs with second-order cone complementarity constraints.The main results we obtained are listed as follows.1.In Chapter 2,we present some preliminary concepts and theorems.Specifically,we examine the differences between definitions on stationarities proposed by Zhang et al.and Ye et al.,and acquired that weak,weak C-,C-and M-stationary conditions are getting stronger one by one.An example is presented to illustrate that weak C-stationarity may not indicate Cstationarity.A sufficient condition that ensures the equivalence of weak C-and C-stationarities is also provided.2.In Chapter 3,we suggest a proximal method of multipliers(PMM)for a nonlinear secondorder cone constrained optimization problem.The objective function of PMM is constructed by adding a proximal term to the classical augmented Lagrangian function.Under the assumptions of constraint nondegeneracy,strict complementarity and second-order sufficient condition,we obtain an inequality involving decision variables and corresponding multipliers,by which we conclude that proposed PMM converges linearly or superlinearly,depending on parameter selection.3.In Chapter 4,we employ the smoothing technique and the sample average technique to develop a smoothing SAA approach for solving the SSOCMPCC.If SOCMPCC-LICQ holds,then stationary points of smoothing SAA sub-problems converge to the C-stationary points of SSOCMPCC almost surely.In addition,if strictly complementarity and the second order sufficient condition also hold,then almost sure convergence to M-stationary points is ensured.We apply this approach to a stochastic inverse quadratic SOCP problem,which is a sub-class of SSOCMPCC.Because of the special structure of the inverse problem,almost surely convergence is acquired with few assumptions.At the end of this chapter,a simple numerical example is proposed to illustrate the proposed smoothing SAA approach is applicable.4.In Chapter 5,we suggest a relaxation SAA approach for SSOCMPCC.We prove that if SOCMPCC-LICQ holds,then all cluster points of the sub-problem stationary points are weak C-stationary points of SSOCMPCC almost surely.