Optimality Conditions in Semidefinite Programming

REFERENCES

• D. Alizadeh ( 1992 ). Optimization over the positive semidefinite cone: interior point methods and combinatorial applications . In: Advances in Optimization and Parallel Computing , (ed. P. Pardalos ). North-Holland , Amsterdam , 1 – 25 .
   Google Scholar
• M. F. Anjos and J.-B. Lasserre , eds. ( 2012 ). Handbook on Semidefinite, Conic and Polynomial Optimization . International Series in Operations Research & Management Science 166. Springer , New York .
   Google Scholar
• E. Bednarczuk , M. Pierre , E. Rouy , and J. Sokolowski ( 2000 ). Tangent sets in some functional spaces . Nonlinear Anal. 42 : 871 – 886 .
   Web of Science ®Google Scholar
• J. F. Bonnans , R. Cominetti , and A. Shapiro ( 1999 ). Second order optimality conditions based on parabolic second-order tangent sets . SIAM Journal on Optimization 9 ( 2 ): 466 – 492 .
   Web of Science ®Google Scholar
• J. F. Bonnans and A. Shapiro ( 2000 ). Perturbation Analysis of Optimization Problems . Springer Series in Operation Research. Springer , New York .
   Google Scholar
• S. Boyd and L. Vandenberghe ( 2004 ). Convex Optimization . Cambridge University Press , Cambridge , UK .
   Google Scholar
• S. Burer ( 2004 ). Semidefinite programming in the space of partial positive semidefinite matrices . SIAM J. Optim. 14 ( 1 ): 139 – 172 .
   Web of Science ®Google Scholar
• J.-S. Chen , X. Chen , and P. Tseng ( 2004 ). Analysis of nonsmooth vector-valued functions associated with second-order cones . Math. Program. 101 : 95 – 117 .
   Web of Science ®Google Scholar
• R. Cominetti ( 1990 ). Metric regularity, tangent sets and second-order optimality conditions . Appl. Math. Optim. 21 : 265 – 287 .
   Web of Science ®Google Scholar
• R. Cominetti and J.-P. Penot ( 1997 ). Tangent sets of order one and two to the positive cones of some functional spaces and applications . J. Applied Math. Optim. 36 ( 3 ): 291 – 312 .
   Web of Science ®Google Scholar
• J. Faraut and A. Korányi ( 1994 ). Analysis on Symmetric Cones . Oxford University Press , New York .
   Google Scholar
• B. Fares , D. Noll , and P. Apkarian ( 2002 ). Robust control via sequential semidefinite programming . SIAM J. Control Opt. 40 : 1791 – 1820 .
   Web of Science ®Google Scholar
• R. Fletcher ( 1985 ). Semidefinite matrix constraints in optimization . SIAM J. Control Optim. 23 : 493 – 513 .
   Web of Science ®Google Scholar
• A. Forsgren ( 2000 ). Optimality conditions for nonconvex semidefinite programming . Math. Program. 88A ( 1 ): 105 – 128 .
   Web of Science ®Google Scholar
• M. X. Goemans ( 1997 ). Semidefinite programming in combinatorial optimization . Math. Program. 79B ( 1–3 ): 143 – 161 .
   Google Scholar
• L. M. Graña Drummond and A. N. Iusem ( 2003 ). First order conditions for ideal minimization of matrix-valued problems . J. Convex Anal. 10 ( 1 ): 129 – 147 .
   Web of Science ®Google Scholar
• L. M. Graña Drummond , A. N. Iusem and B. F. Svaiter ( 2003 ). On first-order optimality conditions for vector optimization . Acta Math. Appl. Sin., Engl. Ser. 19 ( 3 ): 371 – 386 .
   Google Scholar
• J.-B. Hiriart-Urruty and J. Malick ( 2012 ). A fresh variational-analysis look at the positive semidefinite matrices world . J. Optim. Theory Appl. 153 ( 3 ): 551 – 577 .
   Web of Science ®Google Scholar
• A. Ioffe ( 1989 ). On some recent developments in the theory of second-order optimality conditions . Optimization, Proc. 5th French-German Conference, Varetz, Fr 1988 , Lecture Notes in Math. Springer , Berlin , 1405:55–68 .
   Google Scholar
• A. D. Ioffe ( 1994 ). On sensitivity analysis of nonlinear programs in Banach spaces: The approach via composite unconstrained optimization . SIAM J. Optim. 4 ( 1 ): 1 – 43 .
   Web of Science ®Google Scholar
• A. Iusem , R. Gárciga Otero ( 2002 ). Augmented Lagrangian methods for cone-constrained convex optimization in Banach spaces . J. Nonlinear Convex Anal. 3 ( 2 ): 155 – 176 .
   Google Scholar
• F. Jarre ( 2012 ). Elementary optimality conditions for nonlinear SDPs . In: Handbook on Semidefinite, Conic and Polynomial Optimization . (eds. M. F. Anjos and J.-B. Lasserre) . International Series in Operations Research & Management Science 166 . Springer , New York , 455 – 470 .
   Google Scholar
• V. Jeyakumar and M. J. Nealon ( 1999 ). Complete dual characterizations of optimality for convex semidefinite programming . In: Constructive, Experimental, and Nonlinear Analysis. Selected Papers of a Workshop, Limoges . (ed. Théra, Michel) . France , September 22–23 .
   Google Scholar
• H. Kawasaki ( 1992 ). Second-order necessary and sufficient optimality conditions for minimizing a sup-type function . Appl. Math. Optimization 26 ( 2 ): 195 – 220 .
   Web of Science ®Google Scholar
• L. Kong , L. Tunçel , and N. Xiu ( 2009 ). Clarke generalized Jacobian of the projection onto symmetric cones . Set-Valued Variational Anal. 17 ( 2 ): 135 – 151 .
   Web of Science ®Google Scholar
• S. J. Li , X. Q. Yang , and K. L. Teo (2003). Duality for semi-definite and semi-infinite programming. Optimization 52(4–5):507–528.
   Web of Science ®Google Scholar
• M. S. Lobo , L. Vandenberghe , S. Boyd , and H. Lebret ( 1998 ). Applications of second-order cone programming . Linear Algebra Appl. 284 ( 1–3 ): 193 – 228 .
   Web of Science ®Google Scholar
• R. D. C. Monteiro ( 2003 ). First- and second-order methods for semidefinite programming . Math. Program. B 97 : 209 – 244 .
   Web of Science ®Google Scholar
• Y. Nesterov and A. Nemirovskii ( 1994 ). Interior Point Polynomial Algorithms in Convex Programming . SIAM , Philadelphia .
   Google Scholar
• M. P. Pardalos and H. Wolkowicz , eds. ( 1998 ). Topics in Semidefinite and Interior Point Methods . Fields Institute Communications Series , Vol. 18 , Providence , RI , AMS .
   Google Scholar
• J.-P. Penot ( 1982 ). Regularity conditions in mathematical programming . Math. Programming 19 : 167 – 199 .
   Web of Science ®Google Scholar
• J.-P. Penot ( 1994 ). Optimality conditions in mathematical programming and composite optimization . Math. Programming Ser. A 67 ( 2 ): 225 – 245 .
   Web of Science ®Google Scholar
• J.-P. Penot ( 1999 ). Second-order conditions for optimization problems with constraints . SIAM J. Control Optim. 37 ( 1 ): 303 – 318 .
   Web of Science ®Google Scholar
• J.-P. Penot ( 2013 ). Calculus Without Derivatives . Graduate Texts in Maths. Vol. 266. Springer , New York .
   Google Scholar
• R. T. Rockafellar and R. JB Wets ( 1998 ). Variational Analysis . Springer , New York .
   Google Scholar
• R. Saigal , L. Vandenberghe , and H. Wolkowicz , eds. ( 2000 ). Handbook of Semidefinite Programming. Theory, Algorithms, and Applications . International Series in Operations Research & Management Science , Vol. 27 . Kluwer , Boston .
   Google Scholar
• A. Shapiro ( 1997 ). First and second-order analysis of nonlinear semidefinite programs . Math. Programming Series B 77 : 301 – 320 .
   Web of Science ®Google Scholar
• D. Sun ( 2006 ). The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications . Math. Oper. Res. 31 : 761 – 776 .
   Web of Science ®Google Scholar
• D. Sun and J. Sun ( 2002 ). Semismooth matrix-valued functions . Math. Oper. Res. 27 ( 1 ): 150 – 169 .
   Web of Science ®Google Scholar
• L. Vandenberghe and S. Boyd ( 1996 ). Semidefinite programming . SIAM Rev. 38 ( 1 ): 49 – 95 .
   Web of Science ®Google Scholar
• L. Vandenberghe and S. Boyd ( 1999 ). Applications of semidefinite programming . Appl. Numer. Math. 29 ( 3 ): 283 – 299 .
   Web of Science ®Google Scholar