F. Alizadeh, Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization, SIAM Journal on Optimization, vol.5, issue.1, pp.13-51, 1995.
DOI : 10.1137/0805002

F. Alizadeh and D. Goldfarb, Second-order cone programming, Mathematical Programming, pp.3-51, 2003.
DOI : 10.1007/s10107-002-0339-5

P. Apkarian and D. Noll, A prototype primal-dual LMI-interior algorithm for nonconvex robust control problems, 2001.

A. Auslender, Penalty and Barrier Methods: A Unified Framework, SIAM Journal on Optimization, vol.10, issue.1, pp.211-230, 1999.
DOI : 10.1137/S1052623497324825

A. Auslender, Variational inequality over the cone of semidefinite positive symmetric matrices and over the Lorentz cone. Optimization Methods and Software, pp.1-18, 2003.

A. Auslender, R. Cominetti, and M. Haddou, Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming, Mathematics of Operations Research, vol.22, issue.1, pp.43-62, 1997.
DOI : 10.1287/moor.22.1.43

A. Auslender, H. Ramírez, and C. , Penalty and barrier methods in nonlinear semidefinite programming. SoumisàSoumisà Zeitschrift für Operations Research. Ce travail sera presentépresentéà la, French-Latin American Conferences on Applied Mathematics " (FLACAM), 2005.
DOI : 10.1007/s00186-005-0054-0

P. T. Boggs and J. W. Tolle, Sequential Quadratic Programming, Acta Numerica, vol.1991, pp.1-50, 1995.
DOI : 10.1007/BF01580366

J. F. Bonnans, R. Cominetti, and A. Shapiro, Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets, SIAM Journal on Optimization, vol.9, issue.2, pp.466-492, 1999.
DOI : 10.1137/S1052623496306760

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.70.6862

J. F. Bonnans, J. Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimization, 2002.
DOI : 10.1007/978-3-662-05078-1

J. F. Bonnans, H. Ramírez, and C. , A note on strong regularity for semidefinite programming. Presenté au colloque d'optimisation " French-German- Spanish Conferences on Optimization, 2004.

J. F. Bonnans, H. Ramírez, and C. , Perturbation analysis of second-order cone programming problems, Mathematical Programming, vol.38, issue.2-3, 2004.
DOI : 10.1007/s10107-005-0613-4

URL : https://hal.archives-ouvertes.fr/inria-00070707

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

J. F. Bonnans and A. Sulem, Pseudo power expansions of solutions of generalized equations and constrained optimization problems, Mathematical Programming, pp.123-148, 1995.

R. Correa, H. Ramírez, and C. , A Global Algorithm for Nonlinear Semidefinite Programming, SIAM Journal on Optimization, vol.15, issue.1, 2002.
DOI : 10.1137/S1052623402417298

URL : https://hal.archives-ouvertes.fr/inria-00071913

A. L. Dontchev and R. T. Rockafellar, Characterization of Lipschitzian stability in nonlinear programming, Mathematical Programming with Data Perturbations, pp.65-82, 1998.

B. Fares, P. Apkarian, and D. Noll, An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, International Journal of Control, vol.74, issue.4, pp.348-360, 2001.
DOI : 10.1080/00207170010010605

B. Fares, D. Noll, and P. Apkarian, Robust Control via Sequential Semidefinite Programming, SIAM Journal on Control and Optimization, vol.40, issue.6, pp.1791-1820, 2002.
DOI : 10.1137/S0363012900373483

R. W. Freund and F. Jarre, A sensivity analysis and a convergence result for a sequential semidefinite programming method, 2003.

M. X. Goemans and D. P. Williamson, 878-Approximation Algorithms for MAX CUT and MAX 2SAT, ACM Symposium on Theory of Computing (STOC), 1994.

C. C. Gonzaga and R. A. Castillo, A nonlinear programming algorithm based on non-coercive penalty functions, Mathematical Programming, pp.87-101, 2003.
DOI : 10.1007/s10107-002-0332-z

F. Jarre, An interior method for nonconvex semidefinite programs, Optimization and Engineering, vol.1, issue.4, pp.347-372, 2000.
DOI : 10.1023/A:1011562523132

C. Helmberg, F. Rendl, R. Vanderbei, and H. , Wolkowicz An interior point method for semidefinite programing, SIAM Journal on Optimization, vol.6, pp.673-696, 1996.

N. K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, vol.244, issue.S, pp.373-395, 1984.
DOI : 10.1007/BF02579150

L. G. Khachiyan, Polynomial algorithms in linear programming, USSR Computational Mathematics and Mathematical Physics, vol.20, issue.1, pp.1093-1096, 1979.
DOI : 10.1016/0041-5553(80)90061-0

K. Krishnan and T. Terlaky, Interior Point and Semidefinite Approaches in Combinatorial Optimization, AdvOl Report, vol.2, 2004.
DOI : 10.1007/0-387-25592-3_5

Y. N. Nesterov and A. S. Nemirovski, Interior Point Polynomial Algorithms in Convex Programing, 1994.

S. M. Robinson, Strongly Regular Generalized Equations, Mathematics of Operations Research, vol.5, issue.1, pp.43-62, 1980.
DOI : 10.1287/moor.5.1.43

URL : http://www.dtic.mil/get-tr-doc/pdf?AD=ADA064030

S. M. Robinson, Generalized equations and their solutions, part II: Applications to nonlinear programming, Math Programming Stud, vol.19, pp.200-221, 1982.
DOI : 10.1007/BFb0120989

R. T. Rockafellar and R. Wets, Variational analysis, 1998.
DOI : 10.1007/978-3-642-02431-3

C. Sim and G. Zhao, A note on treating second-order cone problems as a special case of semidefinite problems

M. J. Todd, A study of search directions in primal-dual interior-point methods for semidefinite programming. Optimization Methods and Software, pp.1-46, 1999.

M. J. Todd, Semidefinite programming, Acta Numerica, vol.10, pp.515-560, 2001.

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003

Y. Zhang, On Extending Some Primal--Dual Interior-Point Algorithms From Linear Programming to Semidefinite Programming, SIAM Journal on Optimization, vol.8, issue.2, pp.365-386, 1998.
DOI : 10.1137/S1052623495296115

M. Centre and . Modelling, University of Chile-CNRS, casilla 170-3

P. Apkarian, D. Noll, and D. Tuan, A prototype primal-dual LMI-interior algorithm for nonconvex robust control problems, 2001.

J. F. Bonnans, R. Cominetti, and A. Shapiro, Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets, SIAM Journal on Optimization, vol.9, issue.2, pp.466-492, 1999.
DOI : 10.1137/S1052623496306760

J. F. Bonnans, J. Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimisation, 2002.

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

R. Cominetti, Metric regularity, tangent sets, and second-order optimality conditions, Applied Mathematics & Optimization, vol.25, issue.1, pp.265-287, 1990.
DOI : 10.1007/BF01445166

R. Correa and A. Seeger, Directional derivative of a minimax function, Nonlinear Analysis: Theory, Methods & Applications, vol.9, issue.1, pp.13-22, 1985.
DOI : 10.1016/0362-546X(85)90049-5

B. Fares, P. Apkarian, and D. Noll, An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory, International Journal of Control, vol.74, issue.4, pp.348-360, 2001.
DOI : 10.1080/00207170010010605

B. Fares, D. Noll, and P. Apkarian, Robust Control via Sequential Semidefinite Programming, SIAM Journal on Control and Optimization, vol.40, issue.6, pp.1791-1820, 2002.
DOI : 10.1137/S0363012900373483

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.154.8358

M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, 1973.
DOI : 10.1007/978-1-4615-7904-5

S. P. Han, A globally convergent method for nonlinear programming, Journal of Optimization Theory and Applications, vol.5, issue.3, pp.297-309, 1977.
DOI : 10.1007/BF00932858

URL : http://ecommons.cornell.edu/bitstream/1813/6209/1/75-257.pdf

J. Hiriart-urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, 1993.
DOI : 10.1007/978-3-662-02796-7

R. Horn and C. Johnson, Matrix Analysis, 1985.

S. Kurcyusz, On the existence and nonexistence of Lagrange multipliers in Banach spaces, Journal of Optimization Theory and Applications, vol.17, issue.1, pp.81-110, 1976.
DOI : 10.1007/BF00933349

J. Nocedal and S. J. Wright, Numerical optimization. Springer series in operations research, 1999.

D. Noll, M. Torki, and P. Apkarian, Partially Augmented Lagrangian Method for Matrix Inequality Constraints, SIAM Journal on Optimization, vol.15, issue.1, 2002.
DOI : 10.1137/S1052623402413963

S. M. Robinson, First Order Conditions for General Nonlinear Optimization, SIAM Journal on Applied Mathematics, vol.30, issue.4, pp.597-607, 1976.
DOI : 10.1137/0130053

S. M. Robinson, Stability Theory for Systems of Inequalities, Part II: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, vol.13, issue.4, pp.497-513, 1976.
DOI : 10.1137/0713043

S. M. Robinson, Generalized equations and their solutions, part II: Applications to nonlinear programming, Math Programming Stud, vol.19, pp.200-221, 1982.
DOI : 10.1007/BFb0120989

R. T. Rockafellar and R. Wets, Variational analysis, 1998.
DOI : 10.1007/978-3-642-02431-3

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, pp.301-320, 1997.
DOI : 10.1007/BF02614439

C. M. Theobald, An inequality for the trace of the product of two symmetric matrices, Mathematical Proceedings of the Cambridge Philosophical Society, vol.18, issue.02, pp.77-265, 1975.
DOI : 10.1016/0022-5096(68)90031-8

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003

J. Neumann, Some matrix inequalities and metrization of matrixspace, Collected Works, pp.286-300, 1937.

H. Wolkowitz, R. Saigal, and L. Vandenberghe, Handbook of Semidefinite Programming: Theory, Algorithms and Applications. Kluwer's International Series in Operations Research and Management Science, 2000.
DOI : 10.1007/978-1-4615-4381-7

A. Auslender, Penalty and Barrier Methods: A Unified Framework, SIAM Journal on Optimization, vol.10, issue.1, pp.211-230, 1999.
DOI : 10.1137/S1052623497324825

A. Auslender, Variational inequalities over the cone of semidefinite positive matrices and over the Lorentz cone, Optimization Methods and Software, pp.1-18, 2003.

A. Auslender, R. Cominetti, and M. Haddou, Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming, Mathematics of Operations Research, vol.22, issue.1, pp.43-62, 1997.
DOI : 10.1287/moor.22.1.43

A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities, 2003.

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

A. Ben-tal and A. Nemirowskii, Lectures on Modern Convex Optimization, Analysis, Algorithms, and Engineering Applications, MPS-SIAM Series on Optimization, 2002.
DOI : 10.1137/1.9780898718829

A. Ben-tal and M. Zibulevsky, Penalty/Barrier Multiplier Methods for Convex Programming Problems, SIAM Journal on Optimization, vol.7, issue.2, pp.347-366, 1997.
DOI : 10.1137/S1052623493259215

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, vol.19, issue.no. 4, pp.97-138, 1996.
DOI : 10.1007/BF00249052

R. Cominetti and J. P. Dussault, A stable exponential penalty method with superlinear convergence, pp.285-309, 1994.

D. D. Hertzog, C. Roos, and T. Terlaky, Inverse barrier method for linear programming

K. R. Frisch, The logarithmic potential method of convex programming, 1995.

D. Goldfarb and K. Scheinberg, Interior Point Trajectories in Semidefinite Programming, SIAM Journal on Optimization, vol.8, issue.4, pp.871-886, 1998.
DOI : 10.1137/S105262349630009X

C. Gonzaga and R. A. Castillo, A nonlinear programming algorithm based on non-coercive penalty functions, Mathematical Programming, vol.96, issue.1, pp.87-101, 2003.
DOI : 10.1007/s10107-002-0332-z

L. M. Graña-drummond and Y. Peterzil, The central path in smooth convex semidefinite programming, Optimization, pp.207-233, 2002.

T. Kato, Perturbation Theory for Linear Operators, 1970.

A. S. Lewis, Convex Analysis on the Hermitian Matrices, SIAM Journal on Optimization, vol.6, issue.1, pp.164-177, 1996.
DOI : 10.1137/0806009

R. A. Polyak, Modified barrier functions (theory and methods), Mathematical Programming, vol.12, issue.2, pp.177-222, 1992.
DOI : 10.1007/BF01586050

R. T. Rockafellar, Convex Analysis, 1970.
DOI : 10.1515/9781400873173

A. Seeger, Convex Analysis of Spectrally Defined Matrix Functions, SIAM Journal on Optimization, vol.7, issue.3, pp.679-696, 1997.
DOI : 10.1137/S1052623495288866

A. E. Xavier, Hyperbolic Penalization, 1992.

P. Sydoco and B. P. Inria-rocquencourt, Le Chesnay, France, e-mail : Frederic .Bonnans@inria.fr. 3. Department of Mathematical Engineering, [1] F. Alizadeh and D. Goldfarb. Second-order cone programming. Mathematical Programming, 95:Ser. B, pp.3-51, 2003.

J. F. Bonnans, R. Cominetti, and A. Shapiro, Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets, SIAM Journal on Optimization, vol.9, issue.2, pp.466-492, 1999.
DOI : 10.1137/S1052623496306760

J. F. Bonnans, J. Ch, C. Gilbert, C. Lemaréchal, and . Sagastizábal, Numerical Optimization: theoretical and numerical aspects, 2004.
DOI : 10.1007/978-3-662-05078-1

J. F. Bonnans, H. Ramírez, and C. , A note on strong regularity for semidefinite programming

J. F. Bonnans and A. Shapiro, Nondegeneracy and Quantitative Stability of Parameterized Optimization Problems with Multiple Solutions, SIAM Journal on Optimization, vol.8, issue.4, pp.940-946, 1998.
DOI : 10.1137/S1052623497316518

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

J. F. Bonnans and A. Sulem, Pseudopower expansion of solutions of generalized equations and constrained optimization problems, Mathematical Programming, pp.123-148, 1995.
DOI : 10.1007/BF01585932

URL : https://hal.archives-ouvertes.fr/inria-00074717

R. Correa, H. Ramírez, and C. , A Global Algorithm for Nonlinear Semidefinite Programming, SIAM Journal on Optimization, vol.15, issue.1, 2002.
DOI : 10.1137/S1052623402417298

URL : https://hal.archives-ouvertes.fr/inria-00071913

A. L. Dontchev and R. T. Rockafellar, Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets, SIAM Journal on Optimization, vol.6, issue.4, pp.1087-1105, 1996.
DOI : 10.1137/S1052623495284029

M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, Applications of second-order cone programming, Linear Algebra and its Applications, vol.284, issue.1-3, pp.193-228, 1998.
DOI : 10.1016/S0024-3795(98)10032-0

Y. Nesterov and A. Nemirovsky, Interior-point polynomial methods in convex programming, Studies in Applied Mathematics, SIAM, vol.13, 1994.

S. M. Robinson, Generalized equations and their solutions, part II: Applications to nonlinear programming, Math Programming Stud, vol.19, pp.200-221, 1982.
DOI : 10.1007/BFb0120989

S. M. Robinson, Strongly Regular Generalized Equations, Mathematics of Operations Research, vol.5, issue.1, pp.43-62, 1980.
DOI : 10.1287/moor.5.1.43

R. T. Rockafellar, Convex Analysis, 1970.
DOI : 10.1515/9781400873173

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, pp.301-320, 1997.
DOI : 10.1007/BF02614439

A. Shapiro and A. Nemirovski, Duality of linear conic problems, 2004.

C. Sim and G. Zhao, A note on treating second-order cone problems as a special case of semidefinite problems

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003

J. F. Bonnans, R. Cominetti, and A. Shapiro, Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets, SIAM Journal on Optimization, vol.9, issue.2, pp.466-492, 1999.
DOI : 10.1137/S1052623496306760

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, 2000.
DOI : 10.1007/978-1-4612-1394-9

J. F. Bonnans and A. Sulem, Pseudo power expansions of solutions of generalized equations and constrained optimization problems, Mathematical Programming, pp.123-148, 1995.

R. Correa, H. Ramírez, and C. , A Global Algorithm for Nonlinear Semidefinite Programming, SIAM Journal on Optimization, vol.15, issue.1, 2002.
DOI : 10.1137/S1052623402417298

URL : https://hal.archives-ouvertes.fr/inria-00071913

A. L. Dontchev, A Proof of the Necessity of Linear Independence Condition and Strong Second-Order Sufficient Optimality Condition for Lipschitzian Stability in Nonlinear Programming, Journal of Optimization Theory and Applications, vol.95, issue.2, pp.467-473, 1998.
DOI : 10.1023/A:1022649803808

A. L. Dontchev and R. T. Rockafellar, Characterization of Lipschitzian stability in nonlinear programming, Mathematical Programming with Data Perturbations, pp.65-82, 1998.

R. W. Freund and F. Jarre, A sensivity analysis and a convergence result for a sequential semidefinite programming method, Report, 2003.

M. R. Hestenes, Optimization theory, the finite dimensional case, 1975.

S. M. Robinson, Generalized equations and their solutions, part II: Applications to nonlinear programming, Math Programming Stud, vol.19, pp.200-221, 1982.
DOI : 10.1007/BFb0120989

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, pp.301-320, 1997.
DOI : 10.1007/BF02614439

A. Shapiro, Sensitivity analysis of generalized equations, Journal of Mathematical Science, vol.20, pp.116-127, 2002.

A. Shapiro and M. K. Fan, On Eigenvalue Optimization, SIAM Journal on Optimization, vol.5, issue.3, pp.552-569, 1995.
DOI : 10.1137/0805028

L. Vandenberghe and S. Boyd, Semidefinite Programming, SIAM Review, vol.38, issue.1, pp.49-95, 1996.
DOI : 10.1137/1038003