Interior proximal methods and central paths for convex second-order cone programming

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second-order cones

National Natural Science Foundation of China (11101248, 71101140), Shandong Province Natural Science Foundation (ZR2010AQ026), and Young Teacher

Abstract. In this paper, we deal with the semi-infinite complementarity problems (SICP), in which several important issues are covered, such as solvability, semismoothness of residual functions, and error bounds. In particular, we characterize the solution set by investigating the relationship between SICP and the classical complementarity problem. 1 Furthermore, we show that the SICP can be equivalently reformulated as a typical semiinfinite min-max programming problem by employing NCP functions. Finally, we study the concept of error bounds and introduce its two variants, ε-error bounds and weak error bounds, where the concept of weak error bounds is highly desirable in that the solution set is not restricted to be nonempty. Key words. semi-infinite complementarity problem, semidifferentiable and semismooth, error bounds, weak error bounds

On Set-Valued Complementarity Problems

This paper investigates the set-valued complementarity problems (SVCP) which poses rather different features from those that classical complementarity problems hold, due to tthe fact that he index set is not fixed, but dependent on . While comparing the set-valued complementarity problems with the classical complementarity problems, we analyze the solution set of SVCP. Moreover, properties of merit functions for SVCP are studied, such being as level bounded and error bounded. Finally, some possible research directions are discussed

Symmetric cone monotone functions and symmetric cone convex functions

Abstract. Symmetric cone (SC) monotone functions and SC-convex functions are real scalar valued functions which induce Löwner operators associated with a simple Euclidean Jordan algebra to preserve the monotone order and convex order, respectively. In this paper, for a general simple Euclidean Jordan algebra except for octonion case, we show that the SC-monotonicity (respectively, SC-convexity) of order r is implied by the matrix monotonicity (respectively, matrix convexity) of some fixed order r (≥ r). As a consequence, we draw the conclusion that (except for octonion case) a function i

Differentiability v.s. convexity for complementarity functions

The J.-S. Chen's work is supported by Ministry of Science and Technology, Taiwan. The work of J. E. Martinez-Legaz has been supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and under Australian Research Council's Discovery Projects funding scheme (project number DP140103213). He is affiliated to MOVE (Markets, Organizations and Votes in Economics).It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity problem (SOCCP) as special cases. Moreover, there is also so-called generalized complementarity problem (GCP) in infinite dimensional space. Among the existing NCP-functions, it was observed that there are no differentiable and convex NCP-functions. In particular, Miri and Effati (J Optim Theory Appl 164:723-730, 2015) show that convexity and differentiability cannot hold simultaneously for an NCP-function. In this paper, we further establish that such result also holds for general complementarity functions associated with the GCP

Constructing new complementarity functions from existing ones (Study on Nonlinear Analysis and Convex Analysis)

In this short report, we survey concepts and properties of various types of complementarity functions, including NCP-functions, SOCCP-fucntions, and SCCPfunctions. In addition, we provide an idea for constructing new complementarity functions from existing ones