Xiaoni Chi

A non-interior continuation method for second-order cone programming

We extend the smoothing function proposed by Huang, Han and Chen [Journal of Optimization Theory and Applications, 117 (2003), pp. 39–68] for the non-linear complementarity problems to the second-order cone programming (SOCP). Based on this smoothing function, a non-interior continuation method is presented for solving the SOCP. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. It is shown that our algorithm is globally and locally superlinearly convergent in absence of strict complementarity at the optimal solution. Numerical results indicate the effectiveness of the algorithm.

The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.

A Two-Parametric Class of Merit Functions for the Second-Order Cone Complementarity Problem

We propose a two-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) based on the one-parametric class of complementarity functions. By the new class of merit functions, the SOCCP can be reformulated as an unconstrained minimization problem. The new class of merit functions is shown to possess some favorable properties. In particular, it provides a global error bound if and have the joint uniform Cartesian -property. And it has bounded level sets under a weaker condition than the most available conditions. Some preliminary numerical results for solving the SOCCPs show the effectiveness of the merit function method via the new class of merit functions.

A power penalty method for second-order cone linear complementarity problems

We propose a power penalty method for solving the second-order cone linear complementarity problems (SOCLCPs), which is an extension of Wang and Yangs research (Wang and Yang, 2008). Using this method, the SOCLCP is converted to asymptotic power penalty equations (PPEs). We prove that the solution sequence of the asymptotic PPEs converges to the solution of the SOCLCP at an exponential rate under a mild assumption. Numerical results are reported to examine the efficiency of the proposed method.

A power penalty method for second-order cone nonlinear complementarity problems

A power penalty method for solving nonlinear second-order cone complementarity problems (SOCCPs) is proposed. By using this method, the nonlinear SOCCP is converted to asymptotic nonlinear equations. The merit of this method shows that the solution sequence of the asymptotic nonlinear equations converges to the solution of the nonlinear SOCCP at an exponential rate when the penalty parameter tends to positive infinity under mild assumptions. An algorithm is constructed and numerical examples indicate the feasibility of our method.

The models of bilevel programming with lower level second-order cone programs

Robust optimization is an effective method for dealing with the optimization problems under uncertainty. When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel programming problem with lower level second-order cone program (SOCBLP). In this paper, we present the mathematical models of the SOCBLP, and we give some basic concepts, such as constraint region, inducible region, and optimal solution. It is illustrated that the SOCBLP is generally a nonconvex and nondifferentiable optimization problem, whose feasible set may be not connected in some cases and the constraint region is generally not polyhedral. Finally under suitable conditions we propose the optimality conditions for several models of the SOCBLP in the optimistic case.MSC:90C30.

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.

The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra

A weighted complementarity problem is to find a pair of vectors belonging to the intersection of a manifold and a cone such that the product of the vectors in a certain algebra equals a given weight vector. If the weight vector is zero, we get a complementarity problem. Examples of such problems include the Fisher market equilibrium problem and the linear programming and weighted centering problem. In this paper we consider the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and establish some existence and uniqueness results. For a pair of linear transformations on a Euclidean Jordan algebra, we introduce the concepts of

A one-step smoothing Newton method for second-order cone programming

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