A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets
- Jong-Shi Pang
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This paper studies the sensitivity of a parametric nonsmooth equation using degree theory and set-valued analysis. Under the assumption of existence of an approximating function of a certain kind, we investigate the existence, continuity, and differentiability of the set-valued solutions of the parametric equation. Our analysis is a synthesis of the work of several authors (King and Rockafellar, 1992; Gowda and Pang, 1991; Robinson, 1991). Applications of the derived results to the convergence of Newton's method and to the sensitivity analysis of parametric complementarity problems and variational inequalities will be discussed.
- J.-P. Aubin and H. Frankowska,Set-Valued Analysis (Birkhäuser, Boston, MA, 1990).
- R.W. Cottle, J.S. Pang, and R.E. Stone,The Linear Complementarity Problem (Academic Press, Boston, MA, 1992).
- T. Fujisawa and E.S. Kuh, “Piecewise-linear theory of resistive networks,”SIAM Journal of Applied Mathematics 22 (1972) 307–328.
- M.S. Gowda, “On the continuity of the solution map in linear complementarity problems,” to appear in:SIAM Journal on Optimization.
- M.S. Gowda and J.S. Pang, “Stability analysis of variational inequalities and nonlinear complementarity problems, via the mixed linear complementarity problem and degree theory,” manuscript, Department of Mathematical Sciences, The Johns Hopkins University (Baltimore, MD, 1991).
- S.P. Han, J.S. Pang and N. Rangaraj, “Globally convergent Newton methods for nonsmooth equations,”Mathematics of Operations Research 17 (1992) 586–607.
- A.J. King and R.T. Rockafellar, “Sensitivity analysis for nonsmooth generalized equations,”Mathematical Programming 55 (1992) 193–212.
- N.G. Lloyd,Degree Theory (Cambridge University Press, Cambridge, 1978).
- O.L. Mangasarian, “Locally unique solutions of quadratic programs, linear and nonlinear complementarity problem,”Mathematical Programming 19 (1980) 200–212.
- J. Moré and W.C. Rheinboldt, “OnP- andS-functions and related classes ofn-dimensional nonlinear mappings,”Linear Algebra and its Applications 6 (1973) 45–68.
- J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
- J.S. Pang, “Solution differentiability and continuation of Newton's method for variational inequality problems over polyhedral sets,”Journal of Optimization Theory and Applications 66 (1990) 121–135.
- J.S. Pang, “Newton's method for B-differentiable equations,”Mathematics of Operations Research 15 (1990) 311–341.
- J.S. Pang, “Convergence of splitting and Newton methods for complementarity problems: An application of some sensitivity results,”Mathematical Programming 58 (1993) 149–160.
- J.S. Pang and S.A. Gabriel, “NE/SQP: A robust algorithm for the nonlinear complementarity problem,”Mathematical Programming 60 (1993) 295–337.
- J.S. Pang and L. Qi, “Nonsmooth equations: motivation and algorithms,” to appear in:SIAM Journal on Optimization.
- D. Ralph, “Global convergence of damped Newton's method for nonsmooth equations, via the path search,” to appear in:Mathematics of Operations Research.
- A. Reinoza, “The strong positivity conditions,”Mathematics of Operations Research 10 (1985) 54–62.
- S.M. Robinson, “Strongly regular generalized equations,”Mathematics of Operations Research 5 (1980) 43–62.
- S.M. Robinson, ‘Some continuity properties of polyhedral multifunctions,”Mathematical Programming Study 14 (1981) 206–214.
- S.M. Robinson, “Newton's method for a class of nonsmooth functions,” manuscript, Department of Industrial Engineering, University of Wisconsin (Madison, WI, 1988).
- S.M. Robinson, “An implicit-function theorem for a class of nonsmooth functions,”Mathematics of Operations Research 16 (1991) 292–309.
- S.M. Robinson, “Normal maps induced by linear transformations,”Mathematics of Operations Research 17 (1992) 691–714.
- S.M. Robinson, “Homeomorphsim conditions for normal maps of polyhedra,” in: A. Ioffe, M. Marcus and S. Reich, eds.,Optimization and Nonlinear Analysis (Longman, London, 1992), forthcoming
- R.T. Rockafellar, “Lipschitzian properties of multifunctions,”Nonlinear Analysis, Theory, Methods & Applications 9 (1985) 867–885.
- R.T. Rockafellar, “First and second-order epidifferentiability with applications to optimization”,Transactions of the American Mathematical Society 307 (1988) 75–108.
- R.T. Rockafellar, “Proto-differentiability of set-valued mappings and its applications in optimization,” in: H. Attouch, J.-P. Aubin, F.H. Clarke, I. Ekeland, eds.,Analyse Non Linéaire (Gauthier-Villars, Paris, 1989) pp. 449–482.
- A degree-theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets
Volume 62, Issue 1-3 , pp 359-383
- Cover Date
- Print ISSN
- Online ISSN
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- Nonsmooth equations
- degree theory
- set-valued analysis
- sensitivity analysis
- complementarity problem
- variational inequality
- Newton's method
- Industry Sectors
- Jong-Shi Pang (1)
- Author Affiliations
- 1. Department of Mathematical Sciences, The Johns Hopkins University, 21218, Baltimore, MD, USA