Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method

References

• Baes M (2007) Convexity and differentiability properties of spectral functions and spectral mappings on Euclidean Jordan algebras. Linear Algebra Appl. 422(2):664–700.Crossref, Google Scholar
• Bertsekas DP (1982) Constrained Optimization and Lagrange Multipliers Methods (Academic Press, New York).Google Scholar
• Bertsekas DP (1999) Nonlinear Programming, 2nd ed. (Athena Scientific, Belmont, MA).Google Scholar
• Betts JT (1977) An accelerated multiplier method for nonlinear programming. J. Optim. Theory Appl. 21(2):137–174.Crossref, Google Scholar
• Birgin EG, Martínez JM (2014) Practical Augmented Lagrangian Methods for Constrained Optimization (Society for Industrial and Applied Mathematics, Philadelphia).Crossref, Google Scholar
• Bonnans JF, Ramírez CH (2005) Perturbation analysis of second-order cone programming problems. Math. Programming 104(2–3):205–227.Crossref, Google Scholar
• Bonnans JF, Shapiro A (2000) Perturbation Analysis of Optimization Problems (Springer-Verlag, New York).Crossref, Google Scholar
• Bonnans JF, Cominetti R, Shapiro A (1999) Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9(2):466–492.Crossref, Google Scholar
• Burer S, Monteiro RD (2003) A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Programming 95(2):329–357.Crossref, Google Scholar
• Burer S, Monteiro RD (2005) Local minima and convergence in low-rank semidefinite programming. Math. Programming 103(3):427–444.Crossref, Google Scholar
• Cominetti R (1990) Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(1):265–287.Crossref, Google Scholar
• Faraut J, Korányi A (1994) Analysis on Symmetric Cones, Oxford Mathematical Monographs (Clarendon Press, Oxford, UK).Google Scholar
• Faybusovich L (2006) Jordan-algebraic approach to convexity theorems for quadratic mappings. SIAM J. Optim. 17(2):558–576.Crossref, Google Scholar
• Faybusovich L (2008) Several Jordan-algebraic aspects of optimization. Optimization 57(3):379–393.Crossref, Google Scholar
• Fiala J, Kočvara M, Stingl M (2013) PENLAB: A MATLAB solver for nonlinear semidefinite optimization. Working paper, Numerical Algorithms Group, Oxford, UK, arXiv:1311.5240.Google Scholar
• Fukuda EH, Fukushima M (2016) The use of squared slack variables in nonlinear second-order cone programming. J. Optim. Theory Appl. 170(2):394–418.Crossref, Google Scholar
• Fukuda EH, Fukushima M (2017) A note on the squared slack variables technique for nonlinear optimization. J. Oper. Res. Soc. Japan 60(3):262–270.Crossref, Google Scholar
• Hiriart-Urruty JB, Lemaréchal C (1993) Convex Analysis and Minimization Algorithms I: Fundamentals, Grundlehren der mathematischen Wissenschaften (Springer-Verlag, Berlin).Crossref, Google Scholar
• Hock W, Schittkowski K (1980) Test examples for nonlinear programming codes. J. Optim. Theory Appl. 30(1):127–129.Crossref, Google Scholar
• Jongen HT, Stein O (2003) On the complexity of equalizing inequalities. J. Global Optim. 27(4):367–374.Crossref, Google Scholar
• Kanzow C, Ferenczi I, Fukushima M (2009) On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20(1):297–320.Crossref, Google Scholar
• Kawasaki H (1988) An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Programming 41(1–3):73–96.Crossref, Google Scholar
• Kleinmichel H, Schönefeld K (1988) Newton-type methods for nonlinearly constrained programming problems-algorithms and theory. Optimization 19(3):397–412.Crossref, Google Scholar
• Kong L, Tunçel L, Xiu N (2011) Equivalent conditions for Jacobian nonsingularity in linear symmetric cone programming. J. Optim. Theory Appl. 148(2):364–389.Crossref, Google Scholar
• Lourenço BF, Fukuda EH, Fukushima M (2018) Optimality conditions for nonlinear semidefinite programming via squared slack variables. Math. Programming 168(1–2):177–200.Crossref, Google Scholar
• Nocedal J, Wright SJ (1999) Numerical Optimization, 1st ed. (Springer-Verlag, New York).Crossref, Google Scholar
• Noll D (2007) Local convergence of an augmented Lagrangian method for matrix inequality constrained programming. Optim. Methods Software 22(5):777–802.Crossref, Google Scholar
• Pataki G (2000) The geometry of semidefinite programming. Wolkowicz H, Saigal R, Vandenberghe L, eds. Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (Kluwer Academic Publishers, Boston), 29–66.Crossref, Google Scholar
• Shapiro A (1997) First and second order analysis of nonlinear semidefinite programs. Math. Programming 77(1):301–320.Crossref, Google Scholar
• Sturm JF (2000) Similarity and other spectral relations for symmetric cones. Linear Algebra Appl. 312(1–3):135–154.Crossref, Google Scholar
• Sun D, Sun J (2008) Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33(2):421–445.Link, Google Scholar
• Sun D, Sun J, Zhang L (2008) The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Programming 114(2):349–391.Crossref, Google Scholar