The Tikhonov regularization for vector equilibrium problems

We consider vector equilibrium problems in real Banach spaces and study their regularized problems. Based on cone continuity and generalized convexity properties of vector-valued mappings, we propose general conditions that guarantee existence of solutions to such problems in cases of monotonicity and nonmonotonicity. First, our study indicates that every Tikhonov trajectory converges to a solution to the original problem. Then, we establish the equivalence between the problem solvability and the boundedness of any Tikhonov trajectory. Finally, the convergence of the Tikhonov trajectory to the least-norm solution of the original problem is discussed.

[1]  I. V. Konnov,et al.  On the convergence of a regularization method for variational inequalities , 2006 .

[2]  Paulo Roberto Oliveira,et al.  A Tikhonov-type regularization for equilibrium problems in Hilbert spaces , 2013 .

[3]  Z. Chbani,et al.  Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities , 2000 .

[4]  M. Seetharama Gowda,et al.  Regularization of P[sub 0]-Functions in Box Variational Inequality Problems , 2000, SIAM J. Optim..

[5]  S. Schaible,et al.  Vector Equilibrium Problems with Generalized Monotone Bifunctions , 1997 .

[6]  Alfredo N. Iusem,et al.  On certain conditions for the existence of solutions of equilibrium problems , 2008, Math. Program..

[7]  Qamrul Hasan Ansari,et al.  A generalization of vectorial equilibria , 1997, Math. Methods Oper. Res..

[8]  C. S. Lalitha,et al.  Vector variational inequalities with cone-pseudomonotone bifunctions , 2005 .

[9]  Szilárd László.  Vector Equilibrium Problems on Dense Sets , 2016, J. Optim. Theory Appl..

[10]  Nikolaus von Daniels,et al.  Tikhonov regularization of control-constrained optimal control problems , 2017, Comput. Optim. Appl..

[11]  Massimo Pappalardo,et al.  Nonlinear Programming Techniques for Equilibria , 2018 .

[12]  Paul Tseng,et al.  Exact Regularization of Convex Programs , 2007, SIAM J. Optim..

[13]  József Kolumbán,et al.  Some regularities for parametric equilibrium problems , 2009, J. Glob. Optim..

[14]  Jen-Chih Yao,et al.  Vector Variational Inequalities and Vector Optimization , 2018 .

[15]  Igor V. Konnov,et al.  On penalty methods for non monotone equilibrium problems , 2014, J. Glob. Optim..

[16]  F. Browder.  Existence and approximation of solutions of nonlinear variational inequalities. , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Michaël Gaydu.  Stability properties of the Tikhonov regularization for nonmonotone inclusions , 2012, J. Glob. Optim..

[18]  Gábor Kassay,et al.  Existence results for vector equilibrium problems given by a sum of two functions , 2015, J. Glob. Optim..

[19]  Tamaki Tanaka,et al.  Generalized Semicontinuity and Existence Theorems for Cone Saddle Points , 1997 .

[20]  Igor V. Konnov,et al.  Nonmonotone equilibrium problems: coercivity conditions and weak regularization , 2011, J. Glob. Optim..

[21]  Do Van Luu,et al.  Optimality Condition for Local Efficient Solutions of Vector Equilibrium Problems via Convexificators and Applications , 2016, J. Optim. Theory Appl..

[22]  D. Russell Luke,et al.  Lagrange multipliers, (exact) regularization and error bounds for monotone variational inequalities , 2014, Math. Program..

[23]  Michael C. Ferris,et al.  QPCOMP: A quadratic programming based solver for mixed complementarity problems , 1997, Math. Program..

[24]  Lea Fleischer,et al.  Regularization of Inverse Problems , 1996 .

[25]  Suliman Al-Homidan,et al.  Existence of Solutions and Algorithms for Bilevel Vector Equilibrium Problems: An Auxiliary Principle Technique , 2017, J. Optim. Theory Appl..

[26]  Vicentiu D. Radulescu,et al.  The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities , 2015, Optim. Lett..

[27]  César Gutiérrez,et al.  Ekeland Variational Principles in Vector Equilibrium Problems , 2017, SIAM J. Optim..

[28]  Igor V. Konnov.  Equilibrium formulations of relative optimization problems , 2019, Math. Methods Oper. Res..

[29]  Angelia Nedic,et al.  On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems , 2017, Math. Program..

[30]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[31]  Dang Van Hieu.  New subgradient extragradient methods for common solutions to equilibrium problems , 2017, Comput. Optim. Appl..

[32]  Lam Quoc Anh,et al.  Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces , 2007, J. Glob. Optim..

[33]  G. D. Reddy,et al.  A class of parameter choice rules for stationary iterated weighted Tikhonov regularization scheme , 2019, Appl. Math. Comput..

[34]  L. Muu,et al.  Convergence of an adaptive penalty scheme for finding constrained equilibria , 1992 .

[35]  Yu Deng,et al.  Optimal control in first-order Sobolev spaces with inequality constraints , 2019, Comput. Optim. Appl..

[36]  Igor V. Konnov.  Regularized Penalty Method for General Equilibrium Problems in Banach Spaces , 2015, J. Optim. Theory Appl..

[37]  L. Muu,et al.  The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions , 2011 .