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Inexact sequential injective algorithm for weakly univalent vector equation and its application to regularized smoothing Newton algorithm for mixed second-order cone complementarity problems

The author is supported by JSPS KAKENHI Grant No. JP19K11836

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  • It is known that the complementarity problems and the variational inequality problems are reformulated equivalently as a vector equation by using the natural residual or Fischer-Burmeister function. In this paper, we first propose an inexact sequential injective algorithm (ISIA) for a vector equation, and show the global convergence under weak univalence assumption. Roughly speaking, the ISIA generates the sequence of inexact solutions of approximate vector equations, which consist of the injectives converging to the original vector-valued function. Although the ISIA is simple and conceptual, it can be a prototype to many other algorithms such as a smoothing Newton algorithm, semismooth Newton algorithm, etc. Next, we apply the ISIA prototype to the regularized smoothing Newton algorithm (ReSNA) for mixed second-order cone complementarity problems (MSOCCPs). Exploiting the ISIA convergence scheme, we prove that the ReSNA is globally convergent under Cartesian $ P_0 $ assumption.

    Mathematics Subject Classification: Primary: 90C33; Secondary: 65K06, 49M15.


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  • [1] S. BiS. Pan and J.-S. Chen, The same growth of FB and NR symmetric cone complementarity functions, Optimization Letters, 6 (2012), 153-162.  doi: 10.1007/s11590-010-0257-z.
    [2] J.-S. Chen, A new merit function and its related properties for the second-order cone complementarity problem, Pacific Journal of Optimization, 2 (2006), 167-179. 
    [3] J.-S. ChenSOC Functions and Their Applications, Springer, 2019. 
    [4] J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Mathematical Programming, 104 (2005), 293-327.  doi: 10.1007/s10107-005-0617-0.
    [5] J.-S. Chen and S. Pan, A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs, Pacific Journal of Optimization, 8 (2012), 33-74. 
    [6] S. ChenL.-P. Pang and D. Li, An inexact semismooth Newton method for variational inequality with symmetric cone constraints, Journal of Industrial and Management Optimization, 11 (2015), 733-746.  doi: 10.3934/jimo.2015.11.733.
    [7] F. Facchinei and  J.-S. PangFinite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003. 
    [8] J. Faraut and  A. KorányiAnalysis on Symmetric Cones, Clarendon Press, New York, 1994. 
    [9] M. FukushimaZ.-Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems, SIAM Journal on Optimization, 12 (2001), 436-460.  doi: 10.1137/S1052623400380365.
    [10] M. S. Gowda and R. Sznajder, Weak univalence and connectedness of inverse images of continuous functions, Mathematics of Operations Research, 24 (1999), 255-261.  doi: 10.1287/moor.24.1.255.
    [11] P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming, 48 (1990), 161-220.  doi: 10.1007/BF01582255.
    [12] S. HayashiN. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems, SIAM Journal on Optimization, 15 (2005), 593-615.  doi: 10.1137/S1052623403421516.
    [13] S. HayashiN. Yamashita and M. Fukushima, Robust Nash equilibria and second-order cone complementarity problems, Journal of Nonlinear and Convex Analysis, 6 (2005), 283-296. 
    [14] Y. Ito, Robust Wardrop Equilibria in the Traffic Assignment Problem with Uncertain Data, Master's thesis, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, 2011.
    [15] L. KongJ. Sun and N. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems, SIAM Journal on Optimization, 19 (2008), 1028-1047.  doi: 10.1137/060676775.
    [16] Y. NarushimaN. Sagara and H. Ogasawara, A smoothing Newton method with Fischer-Burmeister function for second-order cone complementarity problems, Journal of Optimization Theory and Applications, 149 (2011), 79-101.  doi: 10.1007/s10957-010-9776-0.
    [17] Y. Narushima, H. Ogasawara and S. Hayashi, A smoothing method with appropriate parameter control based on Fischer-Burmeister function for second-order cone complementarity problems, Abstract and Applied Analysis, 2013 (2013), Article ID 830698, 16 pages. doi: 10.1155/2013/830698.
    [18] R. NishimuraS. Hayashi and M. Fukushima, Robust Nash equilibria in ${N}$-person noncooperative games: Uniqueness and reformulation, Pacific Journal of Optimization, 5 (2009), 237-259. 
    [19] L. Qi and D. Sun, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems, Journal of Optimization Theory and Applications, 113 (2002), 121-147.  doi: 10.1023/A:1014861331301.
    [20] D. Sun and L. Qi, Solving variational inequality problems via smoothing-nonsmooth reformulations, Journal of Computational and Applied Mathematics, 129 (2001), 37-62.  doi: 10.1016/S0377-0427(00)00541-0.
    [21] J. Zhang and K. Zhang, An inexact smoothing method for the monotone complementarity problem over symmetric cones, Optimization Methods and Software, 27 (2012), 445-459.  doi: 10.1080/10556788.2010.534164.
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