## First and Second Order Analysis of Nonlinear Semidefinite Programs (1997)

Venue: | Mathematical Programming |

Citations: | 47 - 11 self |

### BibTeX

@ARTICLE{Shapiro97firstand,

author = {Alexander Shapiro},

title = {First and Second Order Analysis of Nonlinear Semidefinite Programs},

journal = {Mathematical Programming},

year = {1997},

volume = {77},

pages = {301--320}

}

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### Abstract

In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A secondorder analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed. Key words: Semidefinite programming, cone constraints, convex programming, duality, second-order optimality conditions, tangent cones, optimal value function, sensitivity analysis. AMS subject classification: 90C25, 90C30, 90C31 1 Introduction In this paper we consider the following optimization problem (P ) min x2IR m f(x) subject to G(x) 0: Here G : IR m ! S n is a mapping from IR m into the space S n of n \Theta n symmetric matrices and, for A; B 2 S n , the notation A B (the notation A B) means that the matrix A \Gamma B is positive semidefinite (negative semidefinite). Consider the cone K ae S n of positive semidefinite matrices. Then the co...