zbMATH — the first resource for mathematics

Variational inequality formulation of circular cone eigenvalue complementarity problems. (English) Zbl 06563710
Summary: In this paper, we study the circular cone eigenvalue complementarity problem (CCEiCP) by variational inequality technique, prove the existence of a solution to CCEiCP, and investigate different nonlinear programming formulations of the symmetric and asymmetric CCEiCP, respectively. We reduce CCEiCP to a variational inequality problem on a compact convex set, which guarantees that CCEiCP has at least one solution. Based on the variational inequality formulation of CCEiCP, the symmetric CCEiCP can be reformulated as a nonlinear programming problem NLP$$_{1}$$, and solved by computing a stationary point of the Rayleigh quotient function on a compact set. We formulate the asymmetric CCEiCP as another nonlinear programming problem NLP$$_{2}$$, and show that any global minimum of NLP$$_{2}$$ with an objective function value equal to zero is a solution of the asymmetric CCEiCP. Moreover, a stationary point of NLP$$_{2}$$ is a solution of the asymmetric CCEiCP, if and only if the Lagrange multipliers associated with the equalities in NLP$$_{2}$$ are equal to zero. The different formulations of CCEiCP provide alternative approaches for solving CCEiCP, which will play an important role in designing efficient algorithms to solve CCEiCP.

MSC:
 47 Operator theory 54 General topology
PATH Solver
Full Text:
References:
 [1] Martins, JAC; Barbarin, S; Raous, M; Pinto da Costa, A, Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction, Comput. Methods Appl. Mech. Eng., 177, 289-328, (1999) · Zbl 0943.74023 [2] Martins, JAC; Pinto da Costa, A, Stability of finite-dimensional nonlinear elastic systems with unilateral contact and friction, Int. J. Solids Struct., 37, 2519-2564, (2000) · Zbl 0959.74048 [3] Martins, JAC; Pinto da Costa, A; Figueiredo, IN; Júdice, JJ, The directional instability problem in systems with frictional contacts, Comput. Methods Appl. Mech. Eng., 193, 357-384, (2004) · Zbl 1075.74596 [4] Fernandesa, LM; Fukushima, M; Júdice, JJ; Sherali, HD, The second-order cone eigenvalue complementarity problem, Optim. Methods Softw., 31, 24-52, (2016) · Zbl 1338.90206 [5] Adly, S; Rammal, H, A new method for solving eigenvalue complementarity problems, Comput. Optim. Appl., 55, 703-731, (2013) · Zbl 1296.90124 [6] Adly, S; Seeger, A, A nonsmooth algorithm for cone-constrained eigenvalue problems, Comput. Optim. Appl., 49, 299-318, (2011) · Zbl 1220.90128 [7] Seeger, A; Vicente-Perez, J, On cardinality of Pareto spectra, Electron. J. Linear Algebra, 22, 758-766, (2011) · Zbl 1254.15015 [8] Dirkse, SP; Ferris, MC, The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156, (1995) [9] Júdice, JJ; Raydan, M; Rosa, S; Santos, S, On the solution of symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm, Numer. Algorithms, 47, 391-407, (2008) · Zbl 1144.65042 [10] Pindo da Costa, A; Seeger, A, Cone-constrained eigenvalue problems: theory and algorithms, Comput. Optim. Appl., 45, 25-57, (2010) · Zbl 1193.65039 [11] Adly, S; Rammal, H, A new method for solving second-order cone eigenvalue complementarity problems, J. Optim. Theory Appl., 165, 563-585, (2015) · Zbl 1321.90137 [12] Seeger, A; Torki, M, On eigenvalues induced by a cone constraint, Linear Algebra Appl., 372, 181-206, (2003) · Zbl 1046.15008 [13] Alizadeh, F; Goldfarb, D, Second-order cone programming, Math. Program., 95, 3-51, (2003) · Zbl 1153.90522 [14] Lobo, M; Vandenberghe, L; Boyd, S; Lebret, H, Applications of second-order cone programming, Linear Algebra Appl., 284, 193-228, (1998) · Zbl 0946.90050 [15] Wang, GQ; Bai, YQ, A new full Nesterov-Todd step primal-dual path-following interior-point algorithm for symmetric optimization, J. Optim. Theory Appl., 154, 966-985, (2012) · Zbl 1256.90036 [16] Che, HT, A smoothing and regularization predictor-corrector method for nonlinear inequalities, J. Inequal. Appl., 2012, (2012) · Zbl 1282.90192 [17] Chen, JW; Liou, YC; Wan, Z; Yao, JC, A proximal point method for a class of monotone equilibrium problems with linear constraints, Oper. Res. Int. J., 15, 275-288, (2015) [18] Pinto Da Costa, A; Seeger, A, Numerical resolution of cone-constrained eigenvalue problems, Comput. Appl. Math., 28, 37-61, (2009) · Zbl 1171.65027 [19] Zhou, JC; Chen, JS; Hung, HF, Circular cone convexity and some inequalities associated with circular cones, J. Inequal. Appl., 2013, (2013) · Zbl 1297.26025 [20] Bai, YQ; Gao, XR; Wang, GQ, Primal-dual interior-point algorithms for convex quadratic circular cone optimization, Numer. Algebra Control Optim., 5, 211-231, (2015) · Zbl 1317.90193 [21] Dattorro, J: Convex Optimization and Euclidean Distance Geometry. Meboo Publishing, Palo Alto (2005) [22] Zhou, JC; Chen, JS, Properties of circular cone and spectral factorization associated with circular cone, J. Nonlinear Convex Anal., 14, 807-816, (2013) · Zbl 1294.49007 [23] Faraut, U, Korányi, A: Analysis on Symmetric Cones. Oxford University Press, New York (1994) · Zbl 0841.43002 [24] Júdice, JJ; Sherali, H; Ribeiro, IM; Rosa, S, On the asymmetric eigenvalue complementarity problem, Optim. Methods Softw., 24, 549-586, (2009) · Zbl 1177.90386 [25] Facchinei, F, Pang, J: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) · Zbl 1062.90002 [26] Bazaraa, MS, Sherali, HD, Shetty, CM: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006) · Zbl 1140.90040 [27] Brás, C; Fukushima, M; Júdice, J; Rosa, S, Variational inequality formulation of the asymmetric eigenvalue complementarity problem and its solution by means of gap functions, Pac. J. Optim., 8, 197-215, (2012) · Zbl 1241.65056 [28] Queiroz, MG; Júdice, JJ; Humes, JC, The symmetric eigenvalue complementarity problem, Math. Comput., 73, 1849-1863, (2004) · Zbl 1119.90059 [29] Seeger, A; Torki, M, Local minima of quadratic forms on convex cones, J. Glob. Optim., 44, 1-28, (2009) · Zbl 1179.90255
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.