On strong semismoothness and superlinear convergence of complementarity problems over homogeneous cones

• Main Author: Nguyen, Hai Ha
• Other Authors: Chua Chek Beng
• Format: Theses and Dissertations
• Language: English
• Published: 2018
• Subjects: DRNTU::Science::Mathematics
• Online Access: http://hdl.handle.net/10356/74464
• Institution: Nanyang Technological University

In Chapter 1, we first review several literature and relevant results that lead to the ideas of the main problems discussed within the thesis. The subsequent parts provide the basic notations and de nitions for basic concepts regarding to the main classes of cones we consider in the thesis, including positive semi-de nite (PSD) cones, symmetric cones and second-order cones (SOCs). Especially, for the class of symmetric cones, beside defi ning the symmetric cone via using the concept of homogeneous cone, we also introduce the closely related concepts like Euclidean Jordan algebra, Jordan frame, Pierce decomposition, etc. In the last section of this chapter, we take a glance over the main contributions, discussed in Chapers 2 and Chapter 3. We start Chapter 2 by recalling several concepts about differentiability, semismoothness and strong semismoothness. In the next section, we revise the method of verifying the strong semismoothness of projection onto the closed convex cone K in the vector space X given in the article "On the Semismoothness of Projection Mappings and Maximum Eigenvalues Function" by M. Goh and F. Meng, and divide the method into four steps. The next parts of Chapter 2 discuss the application of the method for adjusting the strong semismoothness of projection onto second-order cones, then give a couple of counter examples to see the important things we need to notice when doing this method. Chapter 3 mentions the smoothing Newton continuation algorithm firstly given in the article "A combined smoothing and regularization method for monotone second-order cone complementarity problems" by S. Hayashi, N. Yamashita and M. Fukushima (Algorithm 2) to solve the SOC complementarity problems. C.B. Chua and L. T. K. Hien, in their article "A superlinearly convergent smoothing Newton continuation algorithm for variational inequalities over de nable set", give the criterion for this algorithm to converge superlinearly when being applied to solve the smoothing natural map equation. The follow up sections of Chapter 3 give the proof for a lemma that ensure the sufficient condition for one of the criterion, applied for the case of PSD cones, then generalize to symmetric cones (in the paper of Chua and Hien, the lemma is applied for the epigraph of nuclear norm). The method used for the proofs is based on the explicit formular for the smoothing approximations and application of Lowner's operator for the spectral decomposition. Chapter 4 sums up the works of Chapter 2 and Chapter 3. It also points out the diffculties we may encounter for doing the method discussed in Chapter 2. Finally, we consider the possible way of generalize the lemma in Chapter 3 to the case of homogeneous cones, when we cannot get the implicit formula for the smoothing approximation, by using the graphical convergence of monotone mappings.