Error bounds in mathematical programming

Mathematical Programming, Series B

Volumn 79, Issue 1-3, 1997, Pages 299-332

  Pang, Jong Shi ^a  

^a JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

Complementarity problems; Convergence of algorithms; Error bounds; Inequality systems; Metric regularity; Penalty function; Variational inequalities

Indexed keywords

EID: 0000150038     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02614322     Document Type: Article

References (163)

1. J. Abadie, On the Kuhn-Tucker theorem, in: J. Abadie, ed., Nonlinear Programming (North-Holland, Amsterdam, 1967) 19-36.
2. G. Auchmuty, Variational principles for variational inequalities, Numerical Functional Analysis and Optimization 10 (1989) 863-874.
3. A. Auslender, Optimisation: Méthodes Numériques (Masson, Paris, 1976).
4. A. Auslender, Convergence of stationary sequences for variational inequalities with maximal monotone operators, Applied Mathematics and Optimization 28 (1993) 161-172.
5. A. Auslender, R. Cominetti and J.-P. Crouzeix, Convex functions with unbounded level sets and applications to duality theory, SIAM Journal on Optimization 3 (1993) 669-687.
6. A. Auslender and J.P. Crouzeix, Global regularity theorems, Mathematics of Operations Research 13 (1988) 243-253.
7. A. Auslender and J.P. Crouzeix, Well behaved asymptotical convex functions, Analyse Non-linéare (1989) 101-122.
8. H.H. Bauschke and J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review 38 (1996) 367-426.
9. H.H. Bauschke, J.M. Borwein and A.S. Lewis, The method of cyclic projections for closed convex sets in Hilbert Space, in: Y. Censor and S. Reisch, eds., Proc. Special Session on Optimization and Nonlinear Analysis, Jerusalem (1995), to appear.
10. C. Bergthaller and I. Singer, The distance to a polyhedron, Linear Algebra and its Applications 169 (1992) 111-129.
11. D.P. Bertsekas, Nonlinear Programming (Athena Scientific, Massachusetts, 1995).
12. D.P. Bertsekas and E. Gafni, Projection methods for variational inequalities with application to the traffic assignment problem, Mathematical Programming Study 12 (1982) 139-159.
13. D.P. Bertsekas and P. Tseng, Partial proximal minimization algorithms for convex programming, SIAM Journal on Optimization 4 (1994) 551-572.
14. E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Institut des Hautes Etudes Scientifiques, Publications Mathématiques 67 (1988) 5-42.
15. J.M. Borwein, Stability and regular points of inequality systems, Journal of Optimization Theory and Applications 48 (1986) 9-52.
16. J.M. Borwein and D.M. Zhang, Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, Journal of Mathematical Analysis and Applications 134 (1988) 441-459.
17. D. Boukary and A.V. Fiacco, Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993, Optimization 31 (1995) 301-334.
18. J.V. Burke, On the identification of active constraints II: The nonconvex case, SIAM Journal on Numerical Analysis 27 (1990) 1081-1102.
19. J.V. Burke, An exact penalization viewpoint of constrained optimization problem, SIAM Journal on Control and Optimization 29 (1991) 968-998.
20. J.V. Burke and M.C. Ferris, Weak sharp minima in mathematical programming, SIAM Journal on Control and Optimization 31 (1993) 1340-1359.
21. J.V. Burke and J.J. Moré, On the identification of active constraints, SIAM Journal on Numerical Analysis 25 (1988) 1197-1211.
22. J.V. Burke and J.J. Moré, Exposing constraints, SIAM Journal on Optimization 4 (1994) 573-595.
23. J.V. Burke and P. Tseng, A unified analysis of Hoffman's bound via Fenchel duality, SIAM Journal on Optimization 6 (1996) 265-282.
24. C.C. Chou, K.F. Ng, and J.S. Pang, Minimizing and stationary sequences of optimization problems, SIAM Journal on Control and Optimization, revision under review.
25. F.H. Clarke, Optimization and Nonsmooth Analysis (John Wiley and Sons, New York, 1983).
26. W. Cook, A.M.H. Gerards, A. Schrijver and E. Tardös, Sensitivity theorems in integer linear programming, Mathematical Programming 34 (1986) 251-264.
27. R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem (Academic Press, Boston, MA, 1992).
28. J.P. Dedieu, Penalty functions in subanalytic optimization, Optimization 26 (1992) 27-32.
29. S. Deng, Perturbation analysis of a condition number for convex systems, Manuscript, Dept. of Mathematics, Northern Illinois University (DeKalb, revised September 1995).
30. S. Deng, Global error bounds for convex inequality systems in Banach Spaces, Manuscript, Dept. of Mathematics, Northern Illinois University (DeKalb, October 1995).
31. S. Deng, Computable error bounds for convex inequality systems in reflexive Banach Spaces, SIAM Journal on Optimization 7 (1997) 274-279.
32. S. Deng and H. Hu, Computable error bounds for semidefinite programming, Manuscript, Department of Mathematics, Northern Illinois University (DeKalb, revised June 1996).
33. G. Di Pillo and F. Facchinei, Exact barrier functions methods for Lipschitz programs, Applied Mathematics and Optimization 32 (1995) 1-31.
34. G. Di Pillo and L. Grippo, Exact penalty functions in constrained optimization, SIAM Journal on Control and Optimization 27 (1989) 1333-1360.
35. A.L. Dontchev and R.T. Rockafellar, Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM Journal on Optimization 6 (1996) 1087-1105.
36. A.S. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Mathematics, Vol. 1543 (Springer, Berlin, 1991).
37. J.C. Dunn, Global and asymptotic convergence rate estimates for a class of projected gradient processes, SIAM Journal on Control and Optimization 19 (1981) 368-400.
38. A.S. El-Bakry, R.A. Tapia and Y. Zhang, A study of indicators for identifying zero variables in interior-point methods, SIAM Review 36 (1994) 45-72.
39. I.I. Eremin, The method of penalties in convex programming, Doklady Akademy Nauk. USSR 173 (1967) 748-751.
40. F. Facchinei, A. Fischer and C. Kanzow, On the accurate identification of active constraints, Manuscript, Dipartimento di Informatica e Sistemistica, Università di Roma La Sapienza, Roma, 1996.
41. F. Facchinei and S. Lucidi, Convergence to second order stationary points in inequality constrained optimization, manuscript, Dipartimento di Informatica e Sistemistica, Università di Roma La Sapienza, Roma, 1996.
42. M.C. Ferris, Weak sharp minima and penalty functions in mathematical programming, Tech. Rept. 779, Computer Science Dept., University of Wisconsin, Madison, 1988.
43. M.C. Ferris, Finite termination of the proximal point algorithm, Mathematical Programming 50 (1991) 359-366.
44. M.C. Ferris and O.L. Mangasarian, Minimum principle sufficiency, Mathematical Programming 57 (1992) 1-14.
45. M.C. Ferris and O.L. Mangasarian, Error bounds and strong upper semicontinuity for monotone affine variational inequalities, Annals of Operations Research 47 (1993) 293-305.
46. M.C. Ferris and J.S. Pang, Nondegenerate solutions and related concepts in affine variational inequalities, SIAM Journal on Control and Optimization 34 (1996) 244-263.
47. M.C. Ferris and J.S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, to appear.
48. M.C. Ferris and J.S. Pang, eds., Complementarity and Variational Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995 (SIAM Publications, Philadelphia, PA, 1997).
49. A. Fischer, A special Newton-type optimization method, Optimization 24 (1992) 269-284.
50. A. Fischer, An NCP-function and its use for the solution of complementarity problems, in: D.Z. Du, L. Qi and R.S. Womersley, eds., Recent Advances in Nonsmooth Optimization (World Scientific, Singapore, 1995) 88-105.
51. A. Fischer and C. Kanzow, On finite termination of an iterative method for linear complementarity problems, Mathematical Programming 74 (1996) 279-292.
52. M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical Programming 53 (1992) 99-110.
53. M. Fukushima, Merit functions for variational inequality and complementarity problems, in: G. Di Pillo and F. Giannessi, eds., Nonlinear Optimization and Applications (Plenum Press, New York, 1996) 155-170.
54. M. Fukushima and J.S. Pang, Minimizing and stationary sequences of merit functions for complementarity problems and variational inequalities, in: M.C. Ferris and J.S. Pang, eds., Variational and Complementarity Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995 (SIAM Publications, Philadelphia, PA, 1997) 91-104.
55. S.A. Gabriel and J.S. Pang, An inexact NE/SQP method for solving the nonlinear complementarity problem, Computational Optimization and Applications 1 (1992) 67-92.
56. M.S. Gowda, On the continuity of the solution map in linear complementarity problems, SIAM Journal on Optimization 2 (1992) 619-634.
57. M.S. Gowda, An analysis of zero set and global error bound properties of a piecewise affine function via its recession function, SIAM Journal on Matrix Analysis 17 (1996) 594-609.
58. M.S. Gowda and J.S. Pang, On the boundedness and stability of solutions to the affine variational inequality problem, SIAM Journal on Control and Optimization 32 (1994) 421-441.
59. M.S. Gowda and R. Sznajder, On the Lipschitzian properties of polyhedral multifunctions, Mathematical Programming 74 (1996) 267-278.
60. M.S. Gowda and R. Sznajder, On the pseudo-Lipschitzian behavior of the inverse of a piecewise affine function, in: M.C. Ferris and J.S. Pang, eds., Variational and Complementarity Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995 (SIAM Publications, Philadelphia, PA, 1997) 117-131.
61. O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization 29 (1991) 403-419.
62. O. Güler, A.J. Hoffman and U.G. Rothblum, Approximations to solutions to systems of linear inequalities, SIAM Journal on Matrix Analysis and Applications 16 (1995) 688-696.
63. 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.
64. D.W. Hearn, The gap function of a convex program, Operations Research Letters 1 (1982) 67-71.
65. A.J. Hoffman, On approximate solutions of systems of linear inequalities, Journal of Research of the National Bureau of Standards 49 (1952) 263-265.
66. A. Hoffmann, The distance to the intersection of two convex sets expressed by the distances to each of them, Mathematische Nachrichten 157 (1992) 81-98.
67. L. Hörmander, On the division of distributions by polynomials, Arkiv for Mathematik 3 (1958) 555-568.
68. A.N. Iusem, On dual convergence and the rate of primal convergence of Bregman's convex programming method, SIAM Journal on Optimization 1 (1991) 401-423.
69. A.N. Iusem and M. Teboulle, Convergence rate analysis of nonquadratic proximal methods for convex and linear programming, Mathematics of Operations Research 20 (1995) 657-677.
70. C. Kanzow and M. Fukushima, Equivalence of the generalized complementarity problem to differentiable unconstrained minimization, Journal of Optimization Theory and Applications 90 (1996) 581-603.
71. C. Kanzow, N. Yamashita and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications (1997), to appear.
72. D. Klatte, Lipschitz stability and Hoffman's error bounds for convex inequality systems, in: Proc. Conf. on Parametric Optimization and Related Topics IV, Enschede (1995), to appear.
73. D. Klatte, Hoffman's error bound for systems of convex inequalities, in: A.V. Fiacco, ed., Mathematical Programming with Data Perturbations (Marcel Dekker, New York), to appear.
74. D. Klatte and W. Li, Asymptotic constraint qualifications and global error bounds for convex inequalities, manuscript, Dept. of Mathematics and Statistics, Old Dominion University, Norfolk, 1996.
75. M. Kojima, S. Shindoh and S. Hara, Interior-point methods for the monotone linear complementarity problem in symmetric matrices, SIAM Journal on Optimization 7 (1997) 86-125.
76. G.M. Korpelevich, The extragradient method for finding saddle points and other problems, Matecon 12 (1976) 747-756.
77. B.W. Kort and D.P. Bertsekas, Combined primal-dual and penalty methods for convex programming SIAM Journal on Control and Optimization 14 (1976) 268-294.
78. B. Lemaire, Bonne position, conditionnement, et bon comportement asymptotique, Exposé No. 5, Seminaire D'Analyse Convexe, Université de Montpellier, Montpellier, 1992.
79. E.S. Levitin and B.T. Polyak, Convergence of minimizing sequences in conditional extremum problems, Soviet Mathematics Doklady 7 (1966) 764-767.
80. A.S. Lewis and J.S. Pang, Error bounds for convex inequality systems, in J.P. Crouzeix, ed., Proc. 5th Symp. on Generalized Convexity, Luminy-Marseille, (1996), to appear.
81. W. Li, The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program, Linear Algebra and its Applications 187 (1993) 15-40.
82. Wu Li, Remarks on convergence of the matrix splitting algorithm for the symmetric linear complementarity problem, SIAM Journal on Optimization 3 (1993) 155-164.
83. Wu Li, Error bounds for piecewise convex quadratic programs and applications, SIAM Journal on Control and Optimization 33 (1995) 1510-1529.
84. W. Li, Abadie's constraint qualification, metric regularity and error bounds for differentiable convex inequalities, SIAM Journal on Optimization 7 (1997).
85. W. Li and I. Singer, Global error bounds for convex multifunctions and applications, Manuscript, Dept. of Mathematics and Statistics, Old Dominion University, Norfolk, 1996.
86. M.S. Lojasiewicz, Sur le problème de la division. Studia Mathematica 18 (1959) 87-136.
87. A.V. Lotov, An estimate of solution set perturbations for a system of linear inequalities, Optimization Methods and Software 6 (1995) 1-24.
88. X.D. Luo and Z.Q. Luo, Extensions of Hoffman's error bound to polynomial systems, SIAM Journal on Optimization 4 (1994) 383-392.
89. X.D. Luo and P. Tseng, On a global projection-type error bound for the linear complementarity problem, Linear Algebra and its Applications (1996).
90. Z.Q. Luo, Convergence analysis of primal-dual interior point algorithms for convex quadratic programs, in: R.P. Agarwal, ed., Recent Trends in Optimization Theory and Applications (World Scientific, Singapore, 1995).
91. Z.Q. Luo, O.L. Mangasarian, J. Ren and M.V. Solodov, New error bounds for the linear complementarity problem, Mathematics of Operations Research 19 (1994) 880-892.
92. Z.Q. Luo and J.S. Pang, Error bounds for analytic systems and their applications, Mathematical Programming 67 (1994) 1-28.
93. Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge, MA, 1996).
94. Z.Q. Luo, J.S. Pang, D. Ralph and S.Q. Wu, Exact penalization and stationarity conditions of mathematical programs with equilibrium, Mathematical Programming 75 (1996) 19-76.
95. Z.Q. Luo, J.F. Sturm and S.Z. Zhang, Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming, Manuscript, Econometric Institute, Erasmus University, Rotterdam, 1996.
96. Z.Q. Luo and P. Tseng, On the linear convergence of descent methods for convex essentially smooth minimization, SIAM Journal on Control and Optimization 30 (1992) 408-425.
97. Z.Q. Luo and P. Tseng, On the convergence of coordinate descent method for convex differentiable minimization, Journal of Optimization Theory and Applications 72 (1992) 7-35
98. Z.Q. Luo and P. Tseng, Error bound and the convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM Journal on Optimization 2 (1992) 43-54.
99. Z.Q. Luo and P. Tseng, On the convergence of the affine scaling algorithm, Mathematical Programming 56 (1992) 301-319.
100. Z.Q. Luo and P. Tseng, Error bounds and the convergence analysis of feasible descent methods: a general approach, Annals of Operations Research 46 (1993) 157-178.
101. Z.Q. Luo and P. Tseng, Error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set, SIAM Journal on Optimization 3 (1993) 43-59.
102. Z.Q. Luo and P. Tseng, On the convergence rate of dual ascent methods for linearly constrained convex minimization, Mathematics of Operations Research (1993) 846-867.
103. Z.Q. Luo and P. Tseng, Perturbation analysis of a condition number for linear systems, SIAM Journal on Matrix Analysis and Applications 15 (1994) 636-660.
104. Z.Q. Luo and P. Tseng, On the rate of convergence of a distributed asynchronous routing algorithm, IEEE Transactions on Automatic Control 39 (1994) 1123-1129.
105. Z.Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in: M.C. Ferris and J.S. Pang, eds., Variational and Complementarity Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995 (SIAM Publications, Philadelphia, 1997) 204-225.
106. F.J. Luque, Asymptotic convergence analysis of the proximal point algorithm, SIAM Journal on Control and Optimization 22 (1984) 277-293.
107. O.L. Mangasarian, A condition number of linear inequalities and equalities, in: G. Bamber and O. Optiz, eds., Proc. 6th Symp. über Operations Research, Universität Augsburg, 1981, Methods of Operations Research, Vol. 43 (Athennäum/Hain/Scriptor/Hanstein, Konigstein, 1981) 3-15.
108. O.L. Mangasarian, A condition number for differentiable convex inequalities, Mathematics of Operations Research 10 (1985) 175-179.
109. O.L. Mangasarian, Error bounds for nondegenerate monotone linear complementarity problems, Mathematical Programming 48 (1990) 437-445.
110. O.L. Mangasarian, Global error bounds for monotone affine variational inequality problems, Linear Algebra and its Applications 174 (1992) 153-163.
111. O.L. Mangasarian, Error bounds for inconsistent linear inequalities and programs, Operations Research Letters 15 (1994) 187-192.
112. O.L. Mangasarian, Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification, Mathematical Programming Tech. Rept. 96-04, Computer Sciences Dept., University of Wisconsin, Madison, 1996.
113. O.L. Mangasarian, The ill-posed linear complementarity problem, in: M.C. Ferris and J.S. Pang, eds., Variational and Complementarity Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995, (SIAM Publications, Philadelphia, PA, 1997) 226-233.
114. O.L. Mangasarian and J.S. Pang, Exact penalty functions for mathematical programs with linear complementarity constraints, Optimization, to appear.
115. O.L. Mangasarian and J. Ren, New improved error bounds for the linear complementarity problem, Mathematical Programming 66 (1994) 241-257.
116. O.L. Mangasarian and T.-H. Shiau, Error bounds for monotone linear complementarity problems, Mathematical Programming 36 (1986) 81-89.
117. O.L. Mangasarian and T.-H. Shiau, Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems, SIAM Journal on Control and Optimization 25 (1987) 583-595.
118. O.L. Mangasarian and M.V. Solodov, Nonlinear complementarity as unconstrained and constrained minimization, Mathematical Programming 62 (1993) 277-297.
119. R. Mathias and J.S. Pang, Error bounds for the linear complementarity problem with a P-matrix, Linear Algebra and its Applications 132 (1990) 123-136.
120. R.D.C. Monteiro and J.S. Pang, On two interior-point mappings for nonlinear semidefinite complementarity problems, Manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, 1996.
121. R.D.C. Monteiro and J.S. Pang, A potential reduction Newton method for constrained equations, Manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, 1997.
122. R.D.C. Monteiro and T. Tsuchiya, Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem, Mathematics of Operations Research 21 (1996) 793-814.
123. R.D.C. Monteiro and S.J. Wright, Local convergence of interior-point algorithms for degenerate monotone LCP, Computational Optimization and Applications 3 (1994) 131-156.
124. B. Mordukhovich, Complete characterization of openness, metric regularity, and Lipschitzian properties, Transactions of the American Mathematical Society 340 (1993) 1-36.
125. G.S.R. Murthy, T. Parthasarathy and M. Sabatini, Lipschitzian Q-matrices are P-matrices, Mathematical Programming 74 (1996) 55-58.
126. A.M. Odlyzko, Asymptotic enumeration methods, in: R.L. Graham, M. Grötschel and L. Lovász, eds., Handbook of Combinatorics (The MIT Press, North-Holland, Cambridge, 1995), Chapter 22, 1063-1230.
127. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
128. J.S. Pang, Inexact Newton methods for the nonlinear complementarity problems, Mathematical Programming 36 (1986) 54-71.
129. J.S. Pang, A posteriori error bound for the linearly-constrained variational inequality problem, Mathematics of Operations Research 12 (1987) 474-484.
130. J.S. Pang, Complementarity problems, chapter in: R. Horst and P. Pardalos, eds., Handbook of Global Optimization (Kluwer Academic Publishers, Dordrecht, 1995) 271-338
131. J.S. Pang and S.A. Gabriel, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Mathematical Programming 60 (1993) 295-338.
132. J.S. Pang and D. Ralph, Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Mathematics of Operations Research 21 (1996) 401-426.
133. J.S. Pang and J.M. Yang, Two-stage parallel iterative methods for the symmetric linear complementarity problem, Annals of Operations Research 14 (1988) 61-75.
134. J.M. Peng, Equivalence of variational inequality problems to unconstrained optimization, Tech. Rept., State Key Laboratory of Scientific and Engineering Computing, Academia Sinica, Beijing, 1995.
135. J.P. Penot, Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Analysis, Theory, Methods and Applications 13 (1989) 629-643.
136. D. Ralph, A new proof of Robinson's homeomorphism theorem for piecewise linear maps, Linear Algebra and its Applications 178 (1993) 249-260.
137. S.M. Robinson, An application of error bounds for convex programming in a linear space, SIAM Journal on Control 13 (1975) 271-273.
138. S.M. Robinson, Regularity and stability for convex multivalued functions, Mathematics of Operations Research 1 (1976) 130-143.
139. S.M. Robinson, Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM Journal on Numerical Analysis 13 (1976) 497-513.
140. S.M. Robinson, Strongly regular generalized equations, Mathematics of Operations Research 5 (1980) 43-62.
141. S.M. Robinson, Some continuity properties of polyhedral multifunction, Mathematical Programming Study 14 (1981) 206-214.
142. S.M. Robinson, Normal maps induced by linear transformations, Mathematics of Operations Research 16 (1992) 292-309.
143. S.M. Robinson, Homeomorphism conditions for normal maps of polyhedra, in: A. Ioffe, M. Marcus and S. Reich, eds., Optimization and Nonlinear Analysis (Longman, London, 1992) 691-714.
144. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
145. H. Scheel and S. Scholtes, Mathematical programs with equilibrium constraints: Stationarity, optimality and sensitivity, Manuscript, Dept. of Engineering, University of Cambridge, Cambridge, 1996.
146. S. Scholtes, Introduction to piecewise differentiable equations, Habilitation Thesis, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, 1994.
147. S. Scholtes, Homeomorphism conditions for coherently oriented piecewise affine mappings, Research Rept., Institut für Statistik unde Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, 1994.
148. S. Scholtes and M. Stöhr, Exact penalisation of mathematical programs with equilibrium constraints, Manuscript, Institut für Statistik und Mathematische, Universität Karlsruhe, Karlsruhe, 1996.
149. R.E. Stone, Lipschitzian matrices are nondegenerate INS-matrices, in: M.C. Ferris and J.S. Pang, eds., Complementarity and Variational Problems: State of the Art, Proc. Internat. Conf. on Complementarity Problems, 1995 (SIAM Publications, Philadelphia, PA, 1997) 440-451.
150. J.F. Sturm, Superlinear convergence of an algorithm for monotone linear complementarity problems when no strictly complementary solution exists, Manuscript, Econometric Institute, Erasmus University Rotterdam, Rotterdam, 1996.
151. R. Sznajder and M.S. Gowda, Nondegeneracy concepts for zeros of piecewise smooth functions, Mathematics of Operations Research, to appear.
152. K. Taji, M. Fukushima and T. Ibaraki, A globally convergent Newton method for solving strongly monotone variational inequalities, Mathematical Programming 58 (1993) 369-383.
153. P. Tseng, On the rate of convergence of a partially asynchronous gradient projection algorithm, SIAM Journal on Optimization 1 (1991) 603-319.
154. P. Tseng, On linear convergence of iterative methods for the variational inequality problem, Journal of Computational and Applied Mathematics 60 (1995) 237-252.
155. C. Ursescu, Multifunctions with closed convex graph, Czechoslovak Mathematics Journal 25 (1975) 438-441.
156. D.W. Walkup and R.J.B. Wets, A Lipschitzian characterization of convex polyhedra, Proceedings of the American Mathematical Society 20 (1969) 167-173.
157. T. Wang and J.S. Pang, Global error bounds for convex quadratic inequality systems, Optimization 31 (1994) 1-12.
158. J. Warga, A necessary and sufficient condition for a constrained minimum, SIAM Journal on Optimization 2 (1992) 665-667.
159. S.J. Wright, Identifiable surfaces in constrained optimization, SIAM Journal on Control and Optimization 31 (1993) 1063-1079.
160. N. Yamashita and M. Fukushima, Equivalent unconstrained minimization and global error bounds for variational inequality problems, SIAM Journal on Control and Optimization 35 (1997) 273-284.
161. N. Yamashita, K. Taji and M. Fukushima, Unconstrained optimization reformulations of variational inequality problems, Journal of Optimization Theory and Applications 92 (1997) 439-456.
162. Y. Ye, A further result on the potential reduction algorithm for the P-matrix linear complementarity problem, in: P. Pardalos ed., Advances in Optimization and Parallel Computing (North-Holland, New York, 1992) 310-316.
163. W.I. Zangwill, Non-linear programming via penalty functions, Management Science 13 (1967) 344-358.