# American Institute of Mathematical Sciences

2011, 1(1): i-v. doi: 10.3934/naco.2011.1.1i

## Preface

 1 University of Würzburg, Institute of Mathematics, Am Hubland, 97074 Würzburg 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631 3 Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501

Published  February 2011

It is our great pleasure and honor to dedicate the first issue of “Numerical Algebra, Control and Optimization” to Professor Masao Fukushima on the occasion of his 60th birthday. The papers contributed to this issue have been written by his old friends, colleagues and former students, and bring up various topics on optimization, which represent Professor Fukushima's wide-ranging interest in all aspects of optimization.

Citation: Christian Kanzow, Dong-Hui Li, Nobuo Yamashita. Preface. Numerical Algebra, Control & Optimization, 2011, 1 (1) : i-v. doi: 10.3934/naco.2011.1.1i
##### References:
 [1] H. Mine, K. Ohno and M. Fukushima, A "conjugate" interior penalty method for certain convex programs,, SIAM Journal on Control and Optimization, 15 (1977), 747.  doi: 10.1137/0315047.  Google Scholar [2] H. Mine and M. Fukushima, A minimization method for the sum of a convex function and a continuously differentiable function,, Journal of Optimization Theory and Applications, 33 (1981), 9.  doi: 10.1007/BF00935173.  Google Scholar [3] M. Fukushima and H. Mine, A generalized proximal point algorithm for certain nonconvex minimization problems,, International Journal of Systems Science, 12 (1981), 989.  doi: 10.1080/00207728108963798.  Google Scholar [4] M. Fukushima, An outer approximation algorithm for solving general convex programs,, Operations Research, 31 (1983), 101.  doi: 10.1287/opre.31.1.101.  Google Scholar [5] M. Fukushima, A fixed point approach to certain convex programs with applications in stochastic programming,, Mathematics of Operations Research, 8 (1983), 517.  doi: 10.1287/moor.8.4.517.  Google Scholar [6] M. Fukushima, On the convergence of a class of outer approximation algorithms for convex programs,, Journal of Computational and Applied Mathematics, 10 (1984), 147.  doi: 10.1016/0377-0427(84)90051-7.  Google Scholar [7] M. Fukushima, A modified Frank-Wolfe algorithm for solving the traffic assignment problem,, Transportation Research, 18B (1984), 169.  doi: 10.1016/0191-2615(84)90029-8.  Google Scholar [8] M. Fukushima, On the dual approach to the traffic assignment problem,, Transportation Research, 18B (1984), 235.  doi: 10.1016/0191-2615(84)90034-1.  Google Scholar [9] M. Fukushima, A nonsmooth optimization approach to nonlinear multicommodity network flow problems,, Journal of the Operations Research Society of Japan, 27 (1984), 151.   Google Scholar [10] M. Fukushima, A descent algorithm for nonsmooth convex optimization,, Mathematical Programming, 30 (1984), 163.  doi: 10.1007/BF02591883.  Google Scholar [11] M. Fukushima, A relaxed projection method for variational inequalities,, Mathematical Programming, 35 (1986), 58.  doi: 10.1007/BF01589441.  Google Scholar [12] M. Fukushima, A successive quadratic programming algorithm with global and superlinear convergence properties,, Mathematical Programming, 35 (1986), 253.  doi: 10.1007/BF01580879.  Google Scholar [13] E. Yamakawa, M. Fukushima and T. Ibaraki, An efficient trust region algorithm for minimizing nondifferentiable composite functions,, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 562.  doi: 10.1137/0910036.  Google Scholar [14] M. Fukushima, A conjugate gradient algorithm for sparse linear inequalities,, Journal of Computational and Applied Mathematics, 30 (1990), 329.  doi: 10.1016/0377-0427(90)90283-6.  Google Scholar [15] M. Fukushima, A successive quadratic programming method for a class of constrained nonsmooth optimization problems,, Mathematical Programming, 49 (1991), 231.  doi: 10.1007/BF01588789.  Google Scholar [16] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.  doi: 10.1007/BF01585696.  Google Scholar [17] M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Computational Optimization and Applications, 1 (1992), 93.  doi: 10.1007/BF00247655.  Google Scholar [18] K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities,, Mathematical Programming, 58 (1993), 369.  doi: 10.1007/BF01581276.  Google Scholar [19] N. Yamashita and M. Fukushima, On stationary points of the implicit Lagrangian for nonlinear complementarity problems,, Journal of Optimization Theory and Applications, 84 (1995), 653.  doi: 10.1007/BF02191990.  Google Scholar [20] M. Fukushima, The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem,, Mathematical Programming, 72 (1996), 1.  doi: 10.1007/BF02592328.  Google Scholar [21] P. Tseng, N. Yamashita and M. Fukushima, Equivalence of complementarity problems to differentiable minimization: A unified approach,, SIAM Journal on Optimization, 6 (1996), 446.  doi: 10.1137/0806024.  Google Scholar [22] M. Fukushima, Merit functions for variational inequality and complementarity problems,, in, (1996), 155.   Google Scholar [23] K. Taji and M. Fukushima, A new merit function and a successive quadratic programming algorithm for variational inequality problems,, SIAM Journal on Optimization, 6 (1996), 704.  doi: 10.1137/S1052623494271199.  Google Scholar [24] M. Fukushima and L. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization,, SIAM Journal on Optimization, 6 (1996), 1106.  doi: 10.1137/S1052623494278839.  Google Scholar [25] N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of variational inequality problems,, Journal of Optimization Theory and Applications, 92 (1997), 439.  doi: 10.1023/A:1022660704427.  Google Scholar [26] H. Jiang, M. Fukushima, L. Qi and D. Sun, A trust region method for solving generalized complementarity problems,, SIAM Journal on Optimization, 8 (1998), 140.  doi: 10.1137/S1052623495296541.  Google Scholar [27] M. Fukushima, Z. Q. Luo and J. S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 10 (1998), 5.  doi: 10.1023/A:1018359900133.  Google Scholar [28] M. Fukushima, Parallel variable transformation in unconstrained optimization,, SIAM Journal on Optimization, 8 (1998), 658.  doi: 10.1137/S1052623496309879.  Google Scholar [29] M. Fukushima and J. S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 8 (1998), 673.  doi: 10.1137/S105262349731577X.  Google Scholar [30] C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,, Mathematical Programming, 83 (1998), 55.  doi: 10.1007/BF02680550.  Google Scholar [31] J.-S. Pang and M. Fukushima, Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints,, Computational Optimization and Applications, 13 (1999), 111.  doi: 10.1023/A:1008656806889.  Google Scholar [32] X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization,, Mathematical Programming, 85 (1999), 313.  doi: 10.1007/s101070050059.  Google Scholar [33] M. Fukushima and J. S. Pang, Convergence of a smoothing continuation method for mathematical programs with complementarity constraints,, in, 477 (1999), 99.   Google Scholar [34] N. Yamashita and M. Fukushima, The proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem,, SIAM Journal on Optimization, 11 (2001), 364.  doi: 10.1137/S105262349935949X.  Google Scholar [35] D. Li and M. Fukushima, On the global convergence of the BFGS method for nonconvex unconstrained optimization problems,, SIAM Journal on Optimization, 11 (2001), 1054.  doi: 10.1137/S1052623499354242.  Google Scholar [36] M. Fukushima, Z.-Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems,, SIAM Journal on Optimization, 12 (2001), 436.  doi: 10.1137/S1052623400380365.  Google Scholar [37] M. Fukushima and P. Tseng, An implementable active-set algorithm for computing a B-stationary point of the mathematical program with linear complementarity constraints,, SIAM Journal on Optimization, 12 (2002), 724.  doi: 10.1137/S1052623499363232.  Google Scholar [38] M. Fukushima, Z.-Q. Luo and P. Tseng, A sequential quadratically constrained quadratic programming method for differentiable convex minimization,, SIAM Journal on Optimization, 13 (2003), 1098.  doi: 10.1137/S1052623401398120.  Google Scholar [39] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Computational Management Science, 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar [40] S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593.  doi: 10.1137/S1052623403421516.  Google Scholar [41] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [42] P. Zhong and M. Fukushima, Second order cone programming formulations for robust multi-class classification,, Neural Computation, 19 (2007), 258.  doi: 10.1162/neco.2007.19.1.258.  Google Scholar [43] M. Fukushima, A class of gap functions for quasi-variational inequality problems,, Journal of Industrial and Management Optimization, 3 (2007), 165.   Google Scholar [44] M. A. Majig, A. R. Hedar and M. Fukushima, Hybrid evolutionary algorithm for solving general variational inequality problems,, Journal of Global Optimization, 38 (2007), 637.  doi: 10.1007/s10898-006-9102-4.  Google Scholar [45] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [46] G. H. Lin, X. Chen and M. Fukushima, Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization,, Mathematical Programming, 116 (2009), 343.  doi: 10.1007/s10107-007-0119-3.  Google Scholar [47] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar [48] C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM Journal on Optimization, 20 (2009), 297.  doi: 10.1137/060657662.  Google Scholar [49] S. S. Zhu and M. Fukushima, Worst-case conditional Value-at-Risk with application to robust portfolio management,, Operations Research, 57 (2009), 1155.  doi: 10.1287/opre.1080.0684.  Google Scholar [50] G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey,, Pacific Journal of Optimization, 6 (2010), 455.   Google Scholar

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##### References:
 [1] H. Mine, K. Ohno and M. Fukushima, A "conjugate" interior penalty method for certain convex programs,, SIAM Journal on Control and Optimization, 15 (1977), 747.  doi: 10.1137/0315047.  Google Scholar [2] H. Mine and M. Fukushima, A minimization method for the sum of a convex function and a continuously differentiable function,, Journal of Optimization Theory and Applications, 33 (1981), 9.  doi: 10.1007/BF00935173.  Google Scholar [3] M. Fukushima and H. Mine, A generalized proximal point algorithm for certain nonconvex minimization problems,, International Journal of Systems Science, 12 (1981), 989.  doi: 10.1080/00207728108963798.  Google Scholar [4] M. Fukushima, An outer approximation algorithm for solving general convex programs,, Operations Research, 31 (1983), 101.  doi: 10.1287/opre.31.1.101.  Google Scholar [5] M. Fukushima, A fixed point approach to certain convex programs with applications in stochastic programming,, Mathematics of Operations Research, 8 (1983), 517.  doi: 10.1287/moor.8.4.517.  Google Scholar [6] M. Fukushima, On the convergence of a class of outer approximation algorithms for convex programs,, Journal of Computational and Applied Mathematics, 10 (1984), 147.  doi: 10.1016/0377-0427(84)90051-7.  Google Scholar [7] M. Fukushima, A modified Frank-Wolfe algorithm for solving the traffic assignment problem,, Transportation Research, 18B (1984), 169.  doi: 10.1016/0191-2615(84)90029-8.  Google Scholar [8] M. Fukushima, On the dual approach to the traffic assignment problem,, Transportation Research, 18B (1984), 235.  doi: 10.1016/0191-2615(84)90034-1.  Google Scholar [9] M. Fukushima, A nonsmooth optimization approach to nonlinear multicommodity network flow problems,, Journal of the Operations Research Society of Japan, 27 (1984), 151.   Google Scholar [10] M. Fukushima, A descent algorithm for nonsmooth convex optimization,, Mathematical Programming, 30 (1984), 163.  doi: 10.1007/BF02591883.  Google Scholar [11] M. Fukushima, A relaxed projection method for variational inequalities,, Mathematical Programming, 35 (1986), 58.  doi: 10.1007/BF01589441.  Google Scholar [12] M. Fukushima, A successive quadratic programming algorithm with global and superlinear convergence properties,, Mathematical Programming, 35 (1986), 253.  doi: 10.1007/BF01580879.  Google Scholar [13] E. Yamakawa, M. Fukushima and T. Ibaraki, An efficient trust region algorithm for minimizing nondifferentiable composite functions,, SIAM Journal on Scientific and Statistical Computing, 10 (1989), 562.  doi: 10.1137/0910036.  Google Scholar [14] M. Fukushima, A conjugate gradient algorithm for sparse linear inequalities,, Journal of Computational and Applied Mathematics, 30 (1990), 329.  doi: 10.1016/0377-0427(90)90283-6.  Google Scholar [15] M. Fukushima, A successive quadratic programming method for a class of constrained nonsmooth optimization problems,, Mathematical Programming, 49 (1991), 231.  doi: 10.1007/BF01588789.  Google Scholar [16] M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.  doi: 10.1007/BF01585696.  Google Scholar [17] M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Computational Optimization and Applications, 1 (1992), 93.  doi: 10.1007/BF00247655.  Google Scholar [18] K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities,, Mathematical Programming, 58 (1993), 369.  doi: 10.1007/BF01581276.  Google Scholar [19] N. Yamashita and M. Fukushima, On stationary points of the implicit Lagrangian for nonlinear complementarity problems,, Journal of Optimization Theory and Applications, 84 (1995), 653.  doi: 10.1007/BF02191990.  Google Scholar [20] M. Fukushima, The primal Douglas-Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem,, Mathematical Programming, 72 (1996), 1.  doi: 10.1007/BF02592328.  Google Scholar [21] P. Tseng, N. Yamashita and M. Fukushima, Equivalence of complementarity problems to differentiable minimization: A unified approach,, SIAM Journal on Optimization, 6 (1996), 446.  doi: 10.1137/0806024.  Google Scholar [22] M. Fukushima, Merit functions for variational inequality and complementarity problems,, in, (1996), 155.   Google Scholar [23] K. Taji and M. Fukushima, A new merit function and a successive quadratic programming algorithm for variational inequality problems,, SIAM Journal on Optimization, 6 (1996), 704.  doi: 10.1137/S1052623494271199.  Google Scholar [24] M. Fukushima and L. Qi, A globally and superlinearly convergent algorithm for nonsmooth convex minimization,, SIAM Journal on Optimization, 6 (1996), 1106.  doi: 10.1137/S1052623494278839.  Google Scholar [25] N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of variational inequality problems,, Journal of Optimization Theory and Applications, 92 (1997), 439.  doi: 10.1023/A:1022660704427.  Google Scholar [26] H. Jiang, M. Fukushima, L. Qi and D. Sun, A trust region method for solving generalized complementarity problems,, SIAM Journal on Optimization, 8 (1998), 140.  doi: 10.1137/S1052623495296541.  Google Scholar [27] M. Fukushima, Z. Q. Luo and J. S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 10 (1998), 5.  doi: 10.1023/A:1018359900133.  Google Scholar [28] M. Fukushima, Parallel variable transformation in unconstrained optimization,, SIAM Journal on Optimization, 8 (1998), 658.  doi: 10.1137/S1052623496309879.  Google Scholar [29] M. Fukushima and J. S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints,, SIAM Journal on Optimization, 8 (1998), 673.  doi: 10.1137/S105262349731577X.  Google Scholar [30] C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,, Mathematical Programming, 83 (1998), 55.  doi: 10.1007/BF02680550.  Google Scholar [31] J.-S. Pang and M. Fukushima, Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints,, Computational Optimization and Applications, 13 (1999), 111.  doi: 10.1023/A:1008656806889.  Google Scholar [32] X. Chen and M. Fukushima, Proximal quasi-Newton methods for nondifferentiable convex optimization,, Mathematical Programming, 85 (1999), 313.  doi: 10.1007/s101070050059.  Google Scholar [33] M. Fukushima and J. S. Pang, Convergence of a smoothing continuation method for mathematical programs with complementarity constraints,, in, 477 (1999), 99.   Google Scholar [34] N. Yamashita and M. Fukushima, The proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem,, SIAM Journal on Optimization, 11 (2001), 364.  doi: 10.1137/S105262349935949X.  Google Scholar [35] D. Li and M. Fukushima, On the global convergence of the BFGS method for nonconvex unconstrained optimization problems,, SIAM Journal on Optimization, 11 (2001), 1054.  doi: 10.1137/S1052623499354242.  Google Scholar [36] M. Fukushima, Z.-Q. Luo and P. Tseng, Smoothing functions for second-order-cone complementarity problems,, SIAM Journal on Optimization, 12 (2001), 436.  doi: 10.1137/S1052623400380365.  Google Scholar [37] M. Fukushima and P. Tseng, An implementable active-set algorithm for computing a B-stationary point of the mathematical program with linear complementarity constraints,, SIAM Journal on Optimization, 12 (2002), 724.  doi: 10.1137/S1052623499363232.  Google Scholar [38] M. Fukushima, Z.-Q. Luo and P. Tseng, A sequential quadratically constrained quadratic programming method for differentiable convex minimization,, SIAM Journal on Optimization, 13 (2003), 1098.  doi: 10.1137/S1052623401398120.  Google Scholar [39] J.-S. Pang and M. Fukushima, Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games,, Computational Management Science, 2 (2005), 21.  doi: 10.1007/s10287-004-0010-0.  Google Scholar [40] S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593.  doi: 10.1137/S1052623403421516.  Google Scholar [41] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [42] P. Zhong and M. Fukushima, Second order cone programming formulations for robust multi-class classification,, Neural Computation, 19 (2007), 258.  doi: 10.1162/neco.2007.19.1.258.  Google Scholar [43] M. Fukushima, A class of gap functions for quasi-variational inequality problems,, Journal of Industrial and Management Optimization, 3 (2007), 165.   Google Scholar [44] M. A. Majig, A. R. Hedar and M. Fukushima, Hybrid evolutionary algorithm for solving general variational inequality problems,, Journal of Global Optimization, 38 (2007), 637.  doi: 10.1007/s10898-006-9102-4.  Google Scholar [45] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [46] G. H. Lin, X. Chen and M. Fukushima, Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization,, Mathematical Programming, 116 (2009), 343.  doi: 10.1007/s10107-007-0119-3.  Google Scholar [47] X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.  doi: 10.1007/s10107-007-0163-z.  Google Scholar [48] C. Kanzow, I. Ferenczi and M. Fukushima, On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity,, SIAM Journal on Optimization, 20 (2009), 297.  doi: 10.1137/060657662.  Google Scholar [49] S. S. Zhu and M. Fukushima, Worst-case conditional Value-at-Risk with application to robust portfolio management,, Operations Research, 57 (2009), 1155.  doi: 10.1287/opre.1080.0684.  Google Scholar [50] G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey,, Pacific Journal of Optimization, 6 (2010), 455.   Google Scholar
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