A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function

Abstract

Abstract.A new smoothing function of the well known Fischer-Burmeister function is given.Based on this new function, a smoothing Newton-type method is proposed for solving secondorder cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.Mathematical subject classification: 90C25, 90C30, 90C51, 65K05, 65Y20.

Analysis of a non-monotone smoothing-type algorithm for the second-order cone programming

“…With an SOC complementarity function, one (e.g., [2], [3], [5], [6], [18], [19]) can rewrite (1.1) as:…”

“…Step 5: A similar algorithmic framework has been extensively studied for solving the SOCP (see, e.g., [2], [3], [5], [6], [18], [19]). However, all papers mentioned above discussed this kind of algorithms based on a monotone line search.…”

“…In recent years, the SOCP has been studied extensively due to its various applications in many fields (see [1], [15]). Various methods have been developed for solving the SOCP, where the smoothing-type algorithms (e.g., [2], [3], [5], [6], [18], [19]) are one of the most effective methods. This kind of algorithms reformulates the system (1.1) as a family of parameterized smooth equations and solve the smooth equations approximately by using Newton's method at each iteration.…”

“…By driving the parameter to converge to zero, one can expect to find a solution to the SOCP. In general, the smoothing-type algorithms for the SOCP are designed based on a monotone line search (e.g., [2], [3], [5], [6], [18], [19]). It is well known that the non-monotone line search rule has many advantages, especially in the case of iterates trapped in a narrow curved valley of objective functions (e.g., [8], [20]).…”

“…The proposed algorithm is based on a non-monotone line search scheme, which was first introduced by Zhang and Hager [20], and includes the usual monotone line search (e.g., see [2], [3], [5], [6], [18], [19]). By using the theory of Euclidean Jordan algebras, we prove the global and local quadratical convergence of the proposed algorithm under suitable assumptions.…”