AN EQUIVALENCY CONDITION OF NONSINGULARITY IN NONLINEAR SEMIDEFINITE PROGRAMMING

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1 J Syst Sci Complex (2010) 23: AN EQUVALENCY CONDTON OF NONSNGULARTY N NONLNEAR SEMDEFNTE PROGRAMMNG Chengjin L Wenyu SUN Raimundo J. B. de SAMPAO DO: /s Received: 2 February 2008 / Revised: 16 June 2009 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2010 Abstract n this paper, an equivalency condition of nonsingularity in nonlinear semidefinite programming, which can be viewed as a generalization of the equivalency condition of nonsingularity for linear semidefinite programming, is established under certain conditions of convexity. Key words Equivalency condition of nonsingularity, nonlinear semidefinite programming, semidefinite programming. 1 ntroduction n this paper, we consider the nonlinear semidefinite programming (nonlinear SDP): min X S n f(x) s.t. g(x) =0 (1) X 0 where f : S n R, g : S n R m are twice differentiable functions, S n denotes the subspace of all symmetric matrices in R n n,andx 0 a symmetric positive semidefinite matrix. f f and g are all linear (affine) functions, the nonlinear SDP problem (1) is reduced to a normal linear SDP problem, which has been extensively studied during the last decade, see, e.g., [1 4]. However, the research works on nonlinear SDP are much more recent, for the details, please refer to [5 9]. Chengjin L School of Mathematical Sciences, Nanjing Normal University, Nanjing , China; School of Mathematics and Computer Science, Fujian Normal University, Fuzhou , China. chengjin98298@sina.com. Wenyu SUN School of Mathematical Sciences, Nanjing Normal University, Nanjing , China. wysun@njnu.edu.cn. Raimundo J. B. de SAMPAO Pontifical Catholic University of Parana (PUCPR), Graduate Program in Production and Systems Engineering (PPGEPS), CEP Curitiba, Parana, Brazil. rsampaio@ppgia.pucpr.br. This research is supported by the National Natural Science Foundation of China under Grant No , the Natural Science Fund of Jiangsu Province under Grant No. BK , the nnovation Fund of Youth of Fujian Province under Grant No. 2009J05003 and CNPq Brazil.

2 AN EQUVALENCY CONDTON OF NONSNGULARTY 823 Using some results from the corresponding duality theory [4,10], it is not difficult to see that, under mild assumptions, the nonlinear SDP (1) has a solution if and only if the following optimality conditions (KKT conditions) hold: J X f(x) m λ i J X g i (X) S =0, i=1 g i (X) =0, i =1, 2,,m, (2) X 0, S 0, X, S =0, where λ =(λ 1,λ 2,,λ m ) R m, J X f(x) andj X g i (X) aref(réchet)-derivatives of f and g i at point X, respectively, and A, B := tr(a B) denotes the inner product of A, B R n n. t is well-known that for all X, S S n, the following equation is equivalent to the last three conditions in (2) (see [9, 11] for more detail), Π S n (X S) X =0, or Π S n (S X) S =0, where S n denotes the cone of all n n symmetric positive semidefinite matrices and Π S n ( ) the orthogonal projection on S n. Then the KKT conditions can be reformulated as J X f(x) m λ i J X g i (X) S i=1 F (X, λ, S) := g(x) =0, (3) Π S n (X S) X where g(x) =(g 1 (X),g 2 (X),,g m (X)). We denote the second F-derivative of function f at X by JXX 2 f(x). And by and we denote the identity matrix with appropriate size and the identity map, respectively. Let the Lagrange function be L(X, λ, S) =f(x) m i=1 λ ig i (X) X, S. For nonlinear SDP (1), we suppose that JXX 2 L(X,λ,S ) is self-adjoint positive semidefinite on S n in this paper. t follows that some convex SDP problems, for example, convex quadratic SDP in [12], are the special cases of problem (1). The aim of this paper is to deduce an equivalency condition of nonsingularity by analyzing nonsingularity of two special elements in the B-subdifferential of the function F (X, λ, S) atan optimal point. Notations Let Y := (X,λ,S ) be a KKT point of problem (1). We denote the linear operator J X g 1 (X ), J X g 2 (X ), A ( ) :=,. J X g m (X ), and let operator (A ) be the adjoint of A,so(A ) (λ) = m i=1 λ ij X g i (X ), where λ = (λ 1,λ 2,,λ m ).

3 824 CHENGJN L WENYU SUN RAMUNDO J. B. DE SAMPAO Assume that Q := X S has the following spectral decomposition: Q = P Λ (P ), where Λ is the diagonal matrix with eigenvalues λ 1 (Q ) λ 2 (Q ) λ n (Q )ofq as the diagonal entries, and P is the corresponding orthogonal matrix. Then X = Π S n (Q )=P Λ (P ), (4) where Λ is the diagonal matrix whose diagonal entries are the nonnegative parts of the respective diagonal entries of Λ. Define three index sets of positive, zero, and negative eigenvalues of Q, respectively, as α := {i : λ i (Q ) > 0}, β := {i : λ i (Q )=0}, γ := {i : λ i (Q ) < 0}, (5) and the numbers of elements in the above three index sets are denoted by α, β,and γ, respectively. Write Λ α 0 0 Λ = (6) 0 0 Λ γ and P =[P α P β P γ ]withp α R n α,p β R n β,andp γ R n γ. Define an α γ matrix U by U ij := λ i (Q ) λ i (Q ) λ α β j(q ), for all 1 i α, 1 j γ. The rest of the paper is organized as follows. n Section 2 we deduce an equivalency condition of nonsingularity in nonlinear SDP under the assumption of self-adjoint positive semidefiniteness of J 2 XX L(Y )ons n. Then, we conclude with some remarks in Section 3. 2 The Equivalency Condition of Nonsingularity n this section, an equivalency condition of nonsingularity, which can be viewed as a generalization of the equivalency condition for linear SDP in [13], will be established. First, some definitions and an important proposition, which will be used later on, are introduced. Then we deduce some lemmas for the proof of the equivalency condition. Finally, we establish the main result on the equivalency condition of nonsingularity for nonlinear SDP. The following two definitions can be found in [9, 14 17]. Definition 1 Let E and F be finite dimensional vector spaces, and suppose that the function Γ : E Fis locally Lipschitzian at E E. Then the set { } {E k } E, such that J B Γ (E E Γ (E k )exists, ):= V : (7) E k E and J E Γ (E k ) V. is said to be B-subdifferential of Γ at E. The generalized Jacobian C Γ (E )=conv( B Γ (E )), where conv denotes the convex hull.

4 AN EQUVALENCY CONDTON OF NONSNGULARTY 825 Some definitions in [13] are introduced as follows. The tangent cone of S n at X can be characterized as { T S n (X ):= H S n : ( ) ( ) } P β P γ H Pβ P γ 0. Moreover, the linearity spaces of T S n (X )andt S n (S ) are characterized as (X )) = {H S n : Pβ HP β =0,P β HP γ =0,P γ HP γ =0} (8) and (S )) = {H S n : Pα HP α =0, P α HP β =0,P β HP β =0}, (9) respectively. Moreover, an outer approximation set is defined to be app(λ,s ):={H S n : A H =0,Pβ HP γ =0,P γ HP γ =0}. (10) The following is the definition of constraint nondegeneracy introduced in [9, 13]. Definition 2 The constraint nondegeneracy holds at a feasible solution X S n to SDP (1) if A {0} S n (X = Rm )) S n, (11) or equivalently, A (X )) = R m. (12) For any Q S n, we define a linear-quadratic function Υ Q : S n S n R. Definition 3 [9] For any given H S n, define the linear-quadratic function Υ Q : S n S n R, which is linear in the first argument and quadratic in the second argument, by Υ Q (S, H) :=2 S, HQ H, (S, H) S n S n, where Q is the Moore-Penrose pseudo-inverse of Q. Denote by M(X )thesetofpoints(λ, S) R m S n such that (X,λ,S) is a KKT point of (1). Then we can state the strong second order sufficient condition for the SDP tailored from Sun [9] for the general nonlinear SDP. Definition 4 Let X S n be an optimal solution to the nonlinear SDP (1). We say that the strong second order sufficient condition holds at X if sup (λ,s) M(X ) { H, JXXL(X 2,λ,S)(H) Υ X ( S, H)} > 0 (13) { } for all 0 H app(λ, S). (λ,s) M(X ) Now, we need to introduce an important proposition in [9, 18]. Proposition 1 For any V B Π S n (Q )( C Π S n (Q ), respectively), thereexistsav β B Π β S (0) ( C Π β S (0), respectively) such that H α α Hα β U H α γ V (H) =P H α β V β ( H β β ) 0 H α γ (U (P ), H S n, (14) ) 0 0

5 826 CHENGJN L WENYU SUN RAMUNDO J. B. DE SAMPAO where H := (P ) HP. Conversely, for any V β B Π S β (0) ( C Π β S (0), respectively), there exists a V B Π S n (Q )( C Π S n (Q ), respectively) such that (14) holds, where denotes the Hadamard product. t is obvious that the zero mapping Vβ 0 0 and the identity mapping V β from S β to S β belong to B Π β S and V (0). Let V O and V be V in (14) with V β being replaced by V 0 β β respectively, and W O and W denote the elements in BF (Y )withv B Π S n (Q ) being VO and V, respectively. Next, we establish some lemmas, which characterize nonsingularity of the operators WO and W. Lemma 1 Suppose the operator WO is nonsingular, then the constraint nondegenerate condition (11) holds at X. Proof Assume on the contrary that (11) does not hold. t follows that A S n {0} (X )) 0 0 Rm S n, which implies that there exists 0 Δλ ΔS A S n {0} (X )). We obtain from Δλ ΔS {( A ) S n } that ( Δλ, ΔS), (A (H),H) =0, H S n = (A ) Δλ ΔS =0, (15) and from Δλ ΔS {0} (X )), we obtain that (P ) ΔSP, (P ) HP = ΔS, H =0, H (X )), (16) where (x, X), (y, Y ) := x y X, Y for all x, y R m and X, Y S n. Equation (16) together with (8) imply Pα ΔSP α =0, P α ΔSP β =0, and P α ΔSP γ =0. (17) From Proposition 1 and definition of VO,wehave Pα (ΔS)P α P α (ΔS)P β U (Pα (ΔS)P γ ) VO(ΔS) =P (Pα (ΔS)P β ) 0 0 (P ), (Pα (ΔS)P γ ) (U ) 0 0

6 AN EQUVALENCY CONDTON OF NONSNGULARTY 827 which, with the help of (17), implies VO (ΔS) =0 Sn. Therefore, by using (15), we have, for ΔX 0, that JXX 2 L(X,λ,S )(ΔX)(A ) (Δλ) ΔS 0 WO(ΔX, Δλ, ΔS) = A (ΔX) = 0 =0, V VO O (ΔS) (ΔX ΔS) ΔX which implies that WO is singular. The contradiction proves the conclusion. Lemma 2 Suppose that the operator W is nonsingular. Then the following condition holds for all ΔX S n : JXX 2 L(Y )(ΔX) =0, A (ΔX) =0, = ΔX =0. (18) Pα ΔXP γ =0, P β ΔXP γ =0, P γ ΔXP γ =0, Proof Assume on the contrary that there exists 0 ΔX S n such that the conditions in (18) hold. From the definition of V,wehave Pα (ΔX)P α P α (ΔX)P β U (Pα (ΔX)P γ ) V (P (ΔX) =P α (ΔX)P β ) Pβ (ΔX)P β 0 (P ), (Pα (ΔX)P γ ) (U ) 0 0 which, together with the last three conditions of (18), implies V (ΔX) =ΔX. Let(Δλ, ΔS) (0, 0) R m S n, from which together with the front two conditions of (18), we have JXX 2 L(Y )(ΔX)(A ) (Δλ) ΔS W (ΔX, Δλ, ΔS) = = V A (ΔX) V (ΔX ΔS) ΔX 0 0 =0, (ΔX) ΔX which implies that W is singular. The contradiction shows the conclusion. Lemma 3 Let X S n be an optimal solution to SDP (1). Assume that M(X ) = {(λ,s )} is a singleton. Then the strong second order sufficient condition (13) holds at X if condition (18) holds. Proof For all H S n,write H =(P ) HP. Since M(X )={(λ,s )}, the strong second order condition (13) becomes H, JXX 2 L(Y )(H) Υ X ( S,H) > 0 for all H app(λ,s )\{0}, which, by (4) (6) and the definition of Υ X ( S,H), is equivalent to H, JXXL(Y 2 λ j )(H) 2 (Q ) λ i α,j γ i (Q ) ( H ij ) 2 > 0 (19)

7 828 CHENGJN L WENYU SUN RAMUNDO J. B. DE SAMPAO for all H app(λ,s )\{0}. On the other hand, from condition (18), the self-adjoint positive semidefiniteness of J 2 XX L(Y ) and the fact that x Ax =0 Ax = 0 holds for all A S n and x R n, we have, for all H app(λ,s )\{0}, that either H, J 2 XXL(Y )(H) 0 or P α HP γ 0. f H, J 2 XX L(Y )(H) 0, from the self-adjoint positive semidefiniteness of J 2 XX L(Y ), we have H, J 2 XX L(Y )(H) > 0, so the inequality (19) holds. f P α HP γ 0, by using the self-adjoint positive semidefiniteness of J 2 XX L(Y ) again, we also have that (19) holds. Combining the above two cases, we complete the proof. Finally, with the help of the lemmas above, we are in a position to deduce the main result on the equivalency condition of nonsingularity for nonlinear semidefinite programming. Theorem 1 Let Y be a solution of optimality conditions (3). Then the following statements are equivalent: 1) The operators W O, W BF (Y ) are nonsingular. 2) The constraint nondegenerate condition (11) (or (12)) and the strong second order sufficient condition (13) hold at X. 3) All elements in C F (Y ) are nonsingular. 4) All elements in B F (Y ) are nonsingular. Proof 1) = 2): From Lemma 1, the constraint nondegenerate condition (10) holds at point X, which together with Proposition 3.1 in [9] conclude that M(X ) is a singleton. So we can get statement 2) by using Lemmas 2 and 3. 2) = 3): t follows directly from Proposition 3.2 of [9]. 3) = 4) = 1): These two relations hold obviously. 3 Conclusions n this paper, we present an equivalency condition of nonsingularity in nonlinear SDP under certain conditions of convexity, which is very important for studying nonlinear SDP. This equivalency condition of nonsingularity is suitable for some nonlinear SDP problems and can be viewed as a generalization of the equivalency condition of nonsingularity in linear SDP. Acknowledgements We thank the editors and two anonymous referees for their helpful comments and suggestions which greatly improve our manuscript. The authors also thank Professor Defeng Sun for helpful discussion. References [1] C. Helmberg, Semidefinite programming for combrinatorial optimization. URL: /helmberg/semidef.html. [2] M. J. Todd, Semidefinite optimization, Acta Numerica, 2001, 10: [3] L. Vandenberghe and S. Boyd, Semidefinite programming, SAM Review, 1996, 38: [4] H. Wolkowicz, R. Saigal, and L. Vandenberghe, Handbook of Semidefinite Programming, Kluwer Academic Publishers, Boston-Dordrecht-London, [5] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimizations, Springer, New York, 2000.

8 AN EQUVALENCY CONDTON OF NONSNGULARTY 829 [6] R. W. Freund, F. Jarre, and C. H. Vogelbusch, Nonlinear semidefinite programming: Sensitivity, convergence, and an application in passive reduced-order modeling, Mathematical Programming, 2007, 109: [7] C. Li and W. Sun, On filter-successive linearization methods for nonlinear semidefinite programs, Science in China Series A: Mathematics (Chinese Version), 2009, 39(8): , (English Version), 2009, 52(11). [8] C. Li and W. Sun, A nonsmooth Newton-type method for nonlinear semidefinite programming, Journal of Nanjing Normal University: Natural Science, 2008, 31: 1 7. [9] D. Sun, The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications, Mathematics of Operations Research, 2006, 31: [10] W. Sun and Y. Yuan, Optimization Theory and Methods: Nonlinear Programming, Springer, New York, [11] Z. S. Yu, Solving semidefinite programming problems via alternating direction methods, Journal of Computational and Applied Mathematics, 2006, 193: [12] R. H. Tütüncü, K. C. Toh, and M. J. Todd, Solving semidefinite quadratic-linear programs using SDPT3, Mathematical Programming, 2003, 95: [13] Z. Chan and D. Sun, Constraint nondegeneracy, strong regularity and nonsingularity in semidefinite programming, SAM Journal on Optimization, 2008, 19(1): [14] F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, [15] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 1993, 18: [16] L. Qi and J. Sun, A nonsmooth version of Newton s method, Mathematical Programming, 1993, 58: [17] D. Sun and J. Sun, Semismooth matrix value functions, Mathematics of Operations Research, 2002, 27: [18] J. S. Pang, D. F. Sun, and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, 2003, 28: