Finding a Strict Feasible Dual Initial Solution for Quadratic Semidefinite Programming

Transcription

1 Applied Mathematical Sciences, Vol 6, 22, no 4, Finding a Strict Feasible Dual Initial Solution for Quadratic Semidefinite Programming El Amir Djeffal Departement of Mathematics University of Batna, Algeria address: djeffal elamir@yahoofr Lakhdar Djeffal Departement of Mathematics University of Batna, Algeria address: lakdar djeffal@yahoofr Abstract In this paper, we propose an algorithm to comput the dual initial feasible solution for solving a primal-dual optimization problem (convex quadratic semidefinite programming) by a interior point methods Mathematics Subject Classification: 9C25, 9C3, 9C5, 9C2 Keywords: Quadratic programming, Convex nonlinear programming, Interior point methods Introduction The convex quadratic semidefinite programming (CQSDO) is a model which traduces many real applications In term of research, it is one of subject treated with fervour, in particular the problem of initialization in optimization problem [][2][3][5][7][6][4] Choice of starting primal and dual point is an important practical issue with a significant effect on the robustness of the algorithm A poor choice (X,y,S ) satisfying only the minimal conditions X, S In interior point methods, the successive iterates should be strictly feasible In consequence, a major concern is to find an initial primal feasible solution X The object of this paper is to find the initial dual feasible solution (y,s ) Some of the notations used throughout the paper are as follows: R n, R n + and R n ++ denote the set of vectors with n components, the set of nonnegative

2 23 El Amir Djeffal and Lakhdar Djeffal vectors and the set of positive vectors, respectively R nn denotes the set of n n real matrices F and 2 denote the Frobenius norm and the spectral norm for matrices, respectively S n, S n + and S n ++ denote the cone of symmetric, symmetric positive semi-definite and symmetric positive definite n n matrices, respectively We use the matrix inner product A B = Tr(A t B) In this paper, we consider the convex quadratic semidefinite optimization problem (CQSDO) in its standard form: min 2 X Ω(X)+C X st A i X = b i,,, m X (P ) and its dual problem is max b t y X Ω(X) 2 st y i A i + S = C +Ω(X), S, y R m (D) where Ω(X) :S n S n is a given self-adjoint positive semidefinite linear operation on S n, ie, for any A, B S n, then Ω(A) B = A Ω(B) and Ω(A) A To simplify matters we will restrict ourselves to the following special case Ω(X) = l H t i XH i where H i is a matrix in R nn and l is an integer not greater than n 2 The strict feasibility problem of (P ) and (D) is to find (X, y, S), where the strict feasible primal solution X is computed using the following problem: {X S + n,a i X = b i,,, m} To calculate the strict feasible dual solution, using the same help of the primal solution[4], but the main drawback is the vector y, because the algorithm we used is applicable only for positive elements We set y i = y + i y i Then, the problem of dual feasibility becomes as follows: { (y +,y,s) R m + R m + S n +, y i + A i } yi A i + S = C +Ω(X), (y i +,y i ) >,S S+ n (df )

3 Finding a strict feasible dual initial solution 23 On way to solve a strictly feasible problem consists in introducing an additional variabl λ as follows: min λ st y + i A i m y i A i + S + λ(c +Ω(X) m y + i >,y i >,S,λ y + i A i + m y i A i S )=C +Ω(X), (P a ) where (y + i,y i,s ) R m + Rm + S+ n is arbitrary The problem (P a ) can be written as a following linear semidefinite program: min C T st A i T = C +Ω(X), T (LSDP ) where C is the (2m + n +) (2m + n + ) symmetric matrix defined by: { if i = j =2m + n + C [i, j] =, otherwise and A i is the (2m + n +) (2m + n + ) symmetric matrix defined by: A i A i A i = I C +Ω(X) m y + i A i + m y i A i S and T is the (2m + n +) (2m + n + ) symmetric matrix defined by: Y + T = Y S λ where Y + is the (m m) matrix defined by: Y + = y + y + m

4 232 El Amir Djeffal and Lakhdar Djeffal Y is the (m m) matrix defined by: Y = [ ] T Lemma [5]: T is a solution of problem (df ) if and only if λ optimal solution of (LSDP ), with T, and λ sufficiently small (λ ε) y y m is an To comput the optimal solution of (LSDP ), we use only the second phase of the projective interior point method The corresponding algorithm is follows: 2 Algorithm for solving (LSDP ) Description of the algorithm (a) Initailization: ε = 8,λ =,T =, k = If max C +Ω(X) m y + k A i + m y k A i S k ε Stop, T k is an ε approximate solution of (df ) else go to (b) (b) If λ k ε, stop T k is an ε approximate solution of (df ) else go to (c) (c) step k set z = C T k

5 Finding a strict feasible dual initial solution determine L k, such that: T k = L kl t k comput (Cholesky decompozition) and we C k = L t k C L k A k i = L t k A il k,,, 2m + n + 3 Comput the matrix M and the vector d by: M ij = A k i Ak j + C +Ω(X), i,j=2m + n + d i = (C +Ω(X))z k C k A k i,2m + n + 4 Solve the linear system Mu = d 5 Comput V k = C k + 2m+n+ u i A k i v k = 2m+n+ (C +Ω(X))u i z k τ =( V k 2 + v 2 k ) 2 λ = (2m+n+)τ tr(v k) σ = V (2m+n+)τ 2 k 2 λ 2 β k = ζ(max( v k τ, λ + σ 2n + m)) 6 Comput T k+ = T k [ β k τ β k v k L k (V k v k I)L t k, with T k+ = Tk+ λ k+ 7 Take k = k + and go back to (b) 3 Conclusion In this paper, we presented an algorithm for comput the strictly feasible dual initial solution of a convex quadratic semidefinite problem We tried to take advantage of ideas of the projective method of Karmarkar applied to a linear semidefinite problem under special form The implementation of the algorithm remains which will require certain arrangements taking into account the best developments relative to the numerical performances ]

6 234 El Amir Djeffal and Lakhdar Djeffal References [] D Benterki, Résolution des problèmes de programmation semi-définie par la méthode de réduction du potentiel, Thèse de Doctorat, Université Ferhat Abbas, Sétif, 24 [2] D Benterki, JP Crouzeix, B Merikhi, A numerical implementation of an interior point method for semidefinite programming, Pesquisa Operacional 23 () (23) [3] D Benterki, B Merikhi, A modified algorithm for the strict feasibility, RAIRO Oper Res 35 (2) [4] D Benterki, A Keraghel, Finding a strict feasible solution of a linear semidefinite program, Applied Mathematics and Computation 27 (2) [5] IJ Lustig, Feasibility issues in a primal-dual interior point method for linear programming, Math Program 49 (99) [6] MR Oskoorouchi, JL Goffin, The analytic center cutting plane method with semidefinite cuts, SIAM J Optim 3 (23) [7] J Sun, KC Toh, JY Zhao, An analytic center cutting plane method for semidefinite feasibility problems, Math Oper Res 27 (22) Received: November, 2