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

Abstract

A new one-step smoothing newton method for second-order cone programmingApplications of Mathematics, Vol. 57 (2012) Abstract. In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets of strictly feasible solutions. Without requiring strict complementarity at the SOCP solution, the proposed algorithm is shown to be globally and locally quadratically convergent under suitable assumptions. Numerical experiments demonstrate the feasibility and efficiency of our algorithm.,

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.…”