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

Venue: | Mathematical Programming |

Citations: | 48 - 10 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}

}

### Years of Citing Articles

### OpenURL

### 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...

### Citations

3679 | Variational Analysis
- Rockafellar, Wets
- 1998
(Show Context)
Citation Context ... convex, i.e. f(x) is convex and G(x) is psd-convex. The optimal value function /(A) can be analyzed then by a straightforward extension of a corresponding analysis used for nonlinear convex programs =-=[22]-=-,[23]. It is not difficult to verify that the function /(\Delta) is convex on S n . Suppose that the optimal value of (P ) is finite and that the Slater condition for the program (P ) holds. Because o... |

808 |
Inequalities: Theory of Majorization and Its Applications
- Marshall, Olkin
- 1979
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Citation Context ...T DF is the spectral decomposition of the matrix x and OE(D) applies componentwise to the diagonal entries of D, there exists a well developed theory relating convexity properties of OE to (2.1) (see =-=[11, 15]-=-). We deal here with a somewhat different situation when x can be an arbitrary vector. Proposition 2.1 The mapping G(x) is psd-convex if and only if for any v 2 IR n the function '(x) = v T G(x)v is c... |

493 | Interior point methods in semidefinite programming with applications to combinatorial optimization
- Alizadeh
- 1995
(Show Context)
Citation Context ...e at an extreme point of the feasible set and hence the upper bound (3.5) for the corresponding rank r holds at that point. In particular, if msn + 1, then rsn \Gamma 2 at an extreme optimal solution =-=[2, 19]-=-. Note that if msn + 1, then the feasible set G \Gamma1 (\GammaK) has a face of dimension greater than or equal to one provided the Slater condition holds. In that case there exists a linear function ... |

163 | Conjugate duality and optimization
- Rockafellar
- 1974
(Show Context)
Citation Context ...ex, i.e. f(x) is convex and G(x) is psd-convex. The optimal value function /(A) can be analyzed then by a straightforward extension of a corresponding analysis used for nonlinear convex programs [22],=-=[23]-=-. It is not difficult to verify that the function /(\Delta) is convex on S n . Suppose that the optimal value of (P ) is finite and that the Slater condition for the program (P ) holds. Because of the... |

161 |
Stable Mappings and their Singularities
- Golubitsky, Guillemin
- 1973
(Show Context)
Citation Context ...their implications on the semidefinite program (P ). We start with a discussion of the transversality condition. For a detailed study of the transversality concept and relevant references we refer to =-=[10]-=-. 6 Let X and Y be two finite dimensional vector spaces, W ae Y be a smooth manifold and g : X ! Y be a smooth (differentiable) mapping. Definition 3.1 It is said that g intersects W transversally at ... |

150 | Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method - Rubinstein, Shapiro - 1993 |

116 |
Introduction to Sensitivity and Stability Analysis in Nonlinear Programming
- Fiacco
- 1983
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Citation Context ... atsx(u) and G(x(u)) 2 W r for all u sufficiently close to u 0 . The following characterization of the differentiability properties ofsx(u) is then a consequence of the Implicit Function Theorem (cf. =-=[8, 29]-=-). For given x 0 2 IR m and\Omega 0 2 S n consider the quadratic function (y; d) = y T r 2 xx L(z 0 )y + 2y T r 2 xu L(z 0 )d + d T r 2 uu L(z 0 )d+ y T H xx (z 0 )y + 2y T H xu (z 0 )d + d T H uu (z ... |

101 | Complementarity and nondegeneracy in semidefinite programming
- Alizadeh, Haeberly, et al.
- 1997
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Citation Context ...every A 0 2 S n the vectors v ij , 1sisjsn \Gamma r, defined in proposition 3.2 (with G i = A i ), are linearly independent and the rank r of G(x) satisfies the inequality (3.4) for all x 2 IR m (cf. =-=[3, 26, 30]-=-). The transversality is an analogue of the condition of linear independence of the gradients of active constraints used in nonlinear programming (also called nondegeneracy condition in linear program... |

98 |
On matrices depending on parameters
- Arnold
- 1971
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Citation Context ...n + 1)=2 \Gamma (n \Gamma r)(n \Gamma r + 1)=2 in the linear space of n \Theta n symmetric matrices. Moreover, if A 2 W r , then the tangent space T (W r ; A) can be defined by linear equations (e.g. =-=[4, 30]-=-) T (W r ; A) = fZ 2 S n : e T i Ze j = 0; 1sisjsn \Gamma rg; (3.3) where e 1 ; :::; e n\Gammar , is a basis of the null space of the matrix A. By using this characterization of the tangent space T (W... |

63 |
Combinatorial optimization with interior point methods and semidefinite matrices
- ALIZADEH
- 1991
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Citation Context ...sd-convex) if it is convex with respect to the order relation imposed by the cone K. That is, the inequality tG(x) + (1 \Gamma t)G(y)sG(tx + (1 \Gamma t)y) (2.1) holds for any x; y 2 IR m and all t 2 =-=[0; 1]-=-. Condition (2.1) can be written in the following equivalent form tG(x) + (1 \Gamma t)G(y) \Gamma G(tx + (1 \Gamma t)y) 2 K: (2.2) In case the mapping G is defined on the set of symmetric matrices by ... |

41 |
Generalized Least-Squares Estimators in the Analysis of Covariance Structures
- Browne
- 1974
(Show Context)
Citation Context ... normally distributed population with the population covariance matrix A 0 , then \Gamma = 2M n (A 0\Omega A 0 )M n , where M n is a symmetric idempotent pattern matrix of rank n(n + 1)=2 (see, e.g., =-=[6]-=-). Suppose further that the program (P ) is convex, that the Slater condition for the program (PA 0 ) holds and that the optimal value /(A 0 ) is finite. It follows then from (5.1) that /(AN ) \Gamma ... |

39 |
Metric regularity, tangent sets, and second-order optimality conditions
- Cominetti
- 1990
(Show Context)
Citation Context ...der tangent set T 2 (\GammaK; G(x 0 ); [dG(x 0 )]y) (see equation (1.2) for a definition of the support function). Note that the right hand side 11 term in (4.11) is always less than or equal to zero =-=[7]-=-. Therefore the inequality (4.11) is weaker than the inequality y T r 2 xx L(x 0 ; \Omega\Gamma ys0. Let us calculate now the support function, given in the right hand side of (4.11), for a Lagrange m... |

39 |
On eigenvalue optimization
- Shapiro, Fan
- 1994
(Show Context)
Citation Context ...ty conditions of the problem (P ). The material of that section is mainly of a survey nature although it seems that general nonlinear semidefinite programs were not systematically studied before (cf. =-=[18, 30]-=-). In section 3 we study the geometry of the cone K. The material of that section is a summary and a simplification of results which appeared in various publications. In that section we make use of th... |

35 |
First order conditions for general nonlinear optimization
- Robinson
- 1976
(Show Context)
Citation Context ...ga\Gamma = 0; (2.8) \Omega G(x 0 ) = 0: (2.9) The MF-condition implies that the set of positive semidefinite matrices\Omega 2 S n , satisfying conditions (2.8) and (2.9), is nonempty and bounded (cf. =-=[13, 16, 21]-=-). Note that since G(x 0 )s0 and\Omegas0, condition (2.9) is equivalent to the condition\Omega ffl G(x 0 ) = 0, which is the standard complementarity condition under cone constraints. Note also that (... |

28 |
R.: Second derivatives for optimizing eigenvalues of symmetric matrices
- Overton, Womersley
- 1995
(Show Context)
Citation Context ...ty conditions of the problem (P ). The material of that section is mainly of a survey nature although it seems that general nonlinear semidefinite programs were not systematically studied before (cf. =-=[18, 30]-=-). In section 3 we study the geometry of the cone K. The material of that section is a summary and a simplification of results which appeared in various publications. In that section we make use of th... |

25 |
Sensitivity analysis of nonlinear programs and dierentiability properties of metric projections
- Shapiro
- 1988
(Show Context)
Citation Context ... atsx(u) and G(x(u)) 2 W r for all u sufficiently close to u 0 . The following characterization of the differentiability properties ofsx(u) is then a consequence of the Implicit Function Theorem (cf. =-=[8, 29]-=-). For given x 0 2 IR m and\Omega 0 2 S n consider the quadratic function (y; d) = y T r 2 xx L(z 0 )y + 2y T r 2 xu L(z 0 )d + d T r 2 uu L(z 0 )d+ y T H xx (z 0 )y + 2y T H xu (z 0 )d + d T H uu (z ... |

21 |
On eigenvalues of matrices dependent on a parameter
- Lancaster
- 1964
(Show Context)
Citation Context ...ional derivative ofsn (\Delta) at A in a direction Z is given by the smallest eigenvalue of the (n \Gamma r) \Theta (n \Gamma r) matrix E T ZE, provided the basis e 1 ; :::; e n\Gammar is orthonormal =-=[14]-=-. Finally, the cone T (K; A) is formed by those Z for which this directional derivative is nonnegative. Note that it follows from (3.3) and (3.6) that T (W r ; A) is the lineality space of T (K; A). T... |

20 |
First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems
- Maurer, Zowe
- 1979
(Show Context)
Citation Context ...ga\Gamma = 0; (2.8) \Omega G(x 0 ) = 0: (2.9) The MF-condition implies that the set of positive semidefinite matrices\Omega 2 S n , satisfying conditions (2.8) and (2.9), is nonempty and bounded (cf. =-=[13, 16, 21]-=-). Note that since G(x 0 )s0 and\Omegas0, condition (2.9) is equivalent to the condition\Omega ffl G(x 0 ) = 0, which is the standard complementarity condition under cone constraints. Note also that (... |

19 | Perturbation theory of nonlinear programs when the set of optimal solutions is not singleton - Shapiro - 1988 |

16 |
An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems, Mathematical Programming 41
- Kawasaki
- 1988
(Show Context)
Citation Context ... a matrix of Lagrange multipliers satisfying the first-order optimality conditions. We can write now second-order necessary optimality conditions for the program (P ) as follows (see [7, Theorem 4.2],=-=[12]-=-). Let x 0 be a locally optimal solution of the program (P ) and suppose that the MF-condition holds. Then to each y 2 C(x 0 ) corresponds a matrix \Omegas0 of Lagrange multipliers, satisfying the fir... |

13 |
On the Existence and Nonexistence of Lagrange Multipliers in Bachach Spaces
- Kurcyusz
- 1976
(Show Context)
Citation Context ...ga\Gamma = 0; (2.8) \Omega G(x 0 ) = 0: (2.9) The MF-condition implies that the set of positive semidefinite matrices\Omega 2 S n , satisfying conditions (2.8) and (2.9), is nonempty and bounded (cf. =-=[13, 16, 21]-=-). Note that since G(x 0 )s0 and\Omegas0, condition (2.9) is equivalent to the condition\Omega ffl G(x 0 ) = 0, which is the standard complementarity condition under cone constraints. Note also that (... |

13 |
Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis
- Shapiro
- 1982
(Show Context)
Citation Context ...2 = ! T 0 \Gamma! 0 and ! 0 = vec\Omega 0 . In particular, if \Gamma = 2M n (A 0\Omega A 0 )M n , then oe 2 = 2tr[(\Omega 0 A 0 ) 2 ]. For the minimum trace factor analysis this result was derived in =-=[25]-=-. Consider now the optimal solutionssxN =sx(AN ) and x 0 =sx(A 0 ). Suppose that the regularity conditions (specified in theorem 5.2) ensuring differentiability of the optimal solutionsx(A) at A = A 0... |

13 |
Extremal problems on the set of nonnegative definite matrices
- Shapiro
- 1985
(Show Context)
Citation Context ...i=1 x i A i . Then OE(\Omega\Gamma =\Omega ffl A 0 if c i +\Omega ffl A i = 0, i = 1; :::; m, and OE(\Omega\Gamma = \Gamma1 otherwise. Therefore in that case the dual problem takes the form (cf. [1], =-=[27]-=-) max\Omega 2Sn \Omega ffl A 0 subject to c i +\Omega ffl A i = 0; i = 1; :::; m; \Omegas0: (2.6) Consider now the following quadratic case, f(x) = c T x and the mapping G(x) is given in the form (2.4... |

12 | On regularity conditions in mathematical programming - Penot - 1982 |

10 | On uniqueness of Lagrange multipliers in optimization problems subject to cone constraints
- Shapiro
- 1997
(Show Context)
Citation Context ...cient condition for uniqueness of the Lagrange multipliers but in general is not necessary. For a derivation of sufficient and "almost" necessary conditions for uniqueness of Lagrange multip=-=liers see [32]-=-. In particular it is shown in [32] that if there exists a matrix\Omegas0 of Lagrange multipliers, satisfying optimality conditions (2.8) and (2.9), and the strict complementarity condition (2.13) hol... |

5 |
On the multiplicity of optimal eigenvalues
- Pataki
- 1994
(Show Context)
Citation Context ...affine mapping and x is an extreme point of the feasible set G \Gamma1 (\GammaK) of the program (P ), then the inequality r(r + 1)=2sn(n + 1)=2 \Gamma m (3.5) holds for the rank r of G(x) (see Pataki =-=[19]-=- for details). If, in addition, the objective function f(\Delta) is linear, then it attains its minimum value at an extreme point of the feasible set and hence the upper bound (3.5) for the correspond... |

4 | order necessary and sufficient optimality conditions under abstract constraints
- Bonnans, Cominetti, et al.
- 1996
(Show Context)
Citation Context ...4.18) and (4.19) vanishes. Nevertheless, even in the linear case the second term in those inequalities does not vanish and, in a sense, represents the curvature of the cone K. It is recently shown in =-=[5]-=- that the corresponding second-order conditions are also sufficient in the general setting of theorem 4.1. Let us finally remark the following. We have that the Lagrange multipliers matrix can be repr... |

4 |
On the unsolvability of inverse eigenvalues problems almost everywhere
- Shapiro
- 1983
(Show Context)
Citation Context ...every A 0 2 S n the vectors v ij , 1sisjsn \Gamma r, defined in proposition 3.2 (with G i = A i ), are linearly independent and the rank r of G(x) satisfies the inequality (3.4) for all x 2 IR m (cf. =-=[3, 26, 30]-=-). The transversality is an analogue of the condition of linear independence of the gradients of active constraints used in nonlinear programming (also called nondegeneracy condition in linear program... |

4 |
Directional differentiability of the optimal value function in convex semiinfinite programming
- Shapiro
- 1995
(Show Context)
Citation Context ...cription of the first-order differential behavior of the optimal value function /(u) in the convex case. This result can be viewed as an extension of (5.1) and is basically due to Gol'shtein [9] (see =-=[31]-=- for the required extension of Gol'shtein's result to the case of cone constraints). Theorem 5.1 Suppose that the unperturbed program (P ) is convex, that the set M of optimal solutions of (P ) is non... |

3 |
Theory of Convex
- Golâ€™shtein
- 1972
(Show Context)
Citation Context ...ive a description of the first-order differential behavior of the optimal value function /(u) in the convex case. This result can be viewed as an extension of (5.1) and is basically due to Gol'shtein =-=[9]-=- (see [31] for the required extension of Gol'shtein's result to the case of cone constraints). Theorem 5.1 Suppose that the unperturbed program (P ) is convex, that the set M of optimal solutions of (... |