2 The SDP problem and preliminary discussion

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1 Int. J. Contemp. Math. Sciences, Vol. 6, 2011, no. 25, Polynomial Convergence of Predictor-Corrector Algorithms for SDP Based on the KSH Family of Directions Feixiang Chen College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou, Chongqing, , P.R. China Abstract We establish the polynomiality of primal-dual interior-point algorithms for SDP based on any direction of the Kojima, Shindoh and Hara family of search directions. We show that the polynomial iterationcomplexity bounds of the well known algorithms for linear programming, namely, the predictor-corrector algorithm, carry over to the context of SDP. Mathematics Subject Classification: 90C33, 65G20, 65G50 Keywords: interior-point algorithm; polynomial complexity; path-following methods; semidefinite programming problems 1 Introduction Alizadeh [1] extends Yeprojective potential reduction algorithm for LP to SDP and argues that many known interior point algorithms for LP can also be transformed into algorithms for SDP in a mechanical way. Since then many authors have proposed interior-point algorithms for solving the SDP, including Monteiro [2,3], Zhang [4]. Most of these more recent works are concentrated on primal-dual algorithms. 2 The SDP problem and preliminary discussion Given C S n and (A i,b i ) S n R for i =1,...,m, a primal-dual pair of SDP problems is defined as (P ) min{c X : A i X = b i,,...,m,x 0}, (1)

2 1232 Feixiang Chen (D) max{b T y : y i A i + S = C, S 0}, (2) where b (b 1,...,b m ) T. The set of interior feasible solutions of (1) and (2) are F 0 (P ) {X S : X : A i X = b i,,...,m,x 0} F 0 (D) {(S, y) S R m : y i A i + S = C, S 0} respectively. The set of primal and dual optimal solutions consists of all the solutions (X, S, y) S+ n Sn + Rm to the following optimality system: XS =0, (3) y i A i + S = C, (4) A i X = b i,,...,m. (5) It is known that for every μ>0, the perturbed system XS = μi, (6) y i A i + S = C, (7) A i X = b i,,...,m. (8) has a unique solution, denoted by (X μ,s μ ), for every μ>0, and that the limit lim μ 0 exists and is a solution of (1). Using the square root X 1/2, (6) can also be alternatively expressed in the following symmetric form: X 1/2 SX 1/2 = μi. Path following algorithms for solving (1) are based on the idea of approximately tracing the central path. Application of Newton method for computing the solution of (2) with μ = ˆμ leads to the Newton search direction ( ΔX, ΔS) which solves the linear system X ΔS + ΔXS =ˆμI XS, (X + ΔX,S + ΔS) S n + S n +. (9) This system does not always have a solution. Kojima, Shindoh and Hara proposed the following modified Newton system of equation (6), (7), (8): X(ΔS+ ΔS)+(ΔX+ ΔX)S =ˆμI XS, (X+ΔX,S+ΔS) S n S n, ( ΔX, ΔS) F, (10)

3 Polynomial convergence of predictor-corrector algorithms 1233 A i ΔX =0,,...,m. (11) Δy i A i +ΔS =0, (12) where F is a linear subspace of R n n R n n satisfying the following assumptions: [A1]F S n Sn, dim F = n(n 1)/2 and F is monotone, that is U V 0 for every (U, V ) F. It was shown in [5] that system (10), (11), (12) has a unique solution. The symmetric component (ΔX, ΔS) of this solution is then used as a search direction to generate the next point. Theorem 2.1 System (10), (11), (12) has a unique solution. Lemma 2.2 Let X F 0 (P ) and (S, y) F 0 (D) be given and suppose that (ΔX, ΔX,ΔS, ΔS), is a solution of system (10), (11), (12) with ˆμ = σμ, then the following statements holds: (1) ΔS ΔX =0, (2) (X + αδx) (S + αδs) =(1 α + σα)(x S), α R. Lemma 2.3 For every α R, we have X(α)S(α) μ(α)i =(1 α)(xs μi) α(x ΔS + ΔXS)+α 2 ΔXΔS. (13) where X(α) =X + αδx, S(α) =S + αδs, and μ(α) =X(α) S(α)/n Proof. Follows immediately from lemma (2.2) and (10) with ˆμ = σμ. For a nonsingular matrix P R n n, consider the following operator H P : R n n S n defined as H P (M) 1 2 [PMP 1 +(PMP 1 ) T, M R n n. The operator H P has been used by Zhang [10] to characterize the central path of SDP problems. Lemma 2.4 For every θ R and α [0, 1], we have H X 1/2[X(α)S(α) μ(α)i] F (1 α) X 1/2 SX 1/2 μi F +α 2 δ x δ s +α δ x X 1/2 SX 1/2 θμi. (14) where δ x X 1/2 ΔXX 1/2 F, δ x X 1/2 ΔXX 1/2 F,δ s X 1/2 ΔSX 1/2 F (15) The proof of next lemma is straightforward and therefor we omit the details.

4 1234 Feixiang Chen Lemma 2.5 Let (X, S) N F (μ, γ) for some γ (0, 1), then X 1/2 S 1/2 2 (1 + γ)μ, (16) X 1/2 S 1/2 2 [()μ] 1, (17) X 1/2 SX 1/2 θμi F (γ +(1 θ) n)μ (18) ()nμ X S (1 + γ)nμ (19) Lemma 2.6 Let (X, S) N F (μ, γ) for some γ (0, 1), then max{δ x, δ x } γ +(1 σ) n, δ s γ +(1 σ) n μ, where δ x, δ x and δ s are defined in (15). Now we can ready to state the main results of the paper. Lemma 2.7 Suppose that (X, S) N F (μ, γ) for some γ (0, 1) and let (ΔX, ΔX,ΔS, ΔS) be the solution of (10), then the following inequality holds H X 1/2[X(α)S(α) μ(α)i] F {(1 α)γ+αγ γ +(1 σ) n +α 2 ( γ +(1 σ) n ) 2 }μ. Proof. It follows immediately from (14) with θ = 1, the assumption that (X, S) N F (μ, γ) and lemma 2.6. The following lemma is due to Monteiro (see Lemma 2.1 of [7])and plays a crucial role in our analysis. Lemma 2.8 Suppose that (X, S) S++ n S++ n and M R n n is a nonsingular matrix. Then, for every μ R, we have X 1/2 SX 1/2 μi F H M (XS μi) F, with equality holding if MXSM 1 S n. Lemma 2.9 Suppose V,Q R n n be given, and Q is nonsingular which satisfying H Q (V ) I < 1. (20) then, the matrix V is nonsingular. Lemma 2.10 Suppose γ (0, 1) and δ [0,n 1/2 ) be constants satisfying Γ 2( γ + δ )2 (1 δ n ) 1 < 1. (21) Suppose that (X, S) N F (μ, γ) for some μ>0, and that (ΔX, ΔX,ΔS, ΔS) is the solution of system (10), (11), (12) with ˆμ = σμ and σ =1 δ/ n. Then, (X +ΔX,S +ΔS) N F (σμ, Γ).

5 Polynomial convergence of predictor-corrector algorithms The predictor-corrector algorithm Algorithm-I Choose constants 0 <τ<1/2satisfying the conditions of Theorem 3.1 below. let ɛ (0, 1) and (X 0,S 0 ) F 0 (P ) F 0 (D) and μ 0 = X 0 S 0 /n be such that (X 0,S 0 ) N F (μ 0,τ) and set k =0. Repeat until μ k εμ 0, do (1) Choose s linear subspace F k satisfying [A1]; (2) Compute the solution (ΔXP k, ΔX k P, ΔSP k, ΔS k P ) of system (10), (11), (12) with (X, S) =(X k,s k ), F = F k and ˆμ =0; (3) Let α k max{α [0, 1] : (X k (α ),S k (α )) N F ((1 α )μ k, 2τ), α [0,α]}, where X k (α) =X k + αδxp k,sk (α) =S k + αδsp k ; (4) Let ( X k, Ŝk ) (X k,s k )+α k (ΔXP k, ΔSP k ) and μ k+1 =(1 α k )μ k ; (5) Choose s linear subspace F k satisfying [A1]; (6) Compute the solution (ΔXC k, ΔX k C, ΔSk C, ΔS k C ) of system (10), (11), (12) with (X, S) =( X k, Ŝk ), F = F k and ˆμ = μ k+1 ; (7) Set (X k+1,s k+1 ) ( X k, Ŝk )+(ΔXC k, ΔSk C ); (8) Increment k by 1. End Theorem 3.1 Assume that τ (0, 1/15]. Then algorithm-i satisfies the following statements: (1) for every k 0, (X k,s k ) N F (μ k,τ) and ( X k, Ŝk ) N F (2τ); (2) for every k 0, X k S k (1 α) k X 0 S 0, where α =1/O( n); (3) the algorithm terminates in at most O( nlogε 1 ) iterations. Proof. Statement (3) and the well-definedness of algorithm-i follow directly from (1) and (2). In turn, these two statements follow by a simple induction argument, the two lemmas below and relation (19). The following lemma analyzes the predictor step of Algorithm-II. Lemma 3.2 Suppose that (X, S) N F (μ, τ) for some τ (0, 1/2). For some subspace F satisfying [A1], let (ΔX P, ΔX P, ΔS P, ΔS P ) denote the solution of (10), (11), (12) with μ =0.Let α denote the unique positive root of the second-order polynomial p(α) defined as p(α) =( τ + n 1 τ )2 α 2 + τ[ τ + n +1]α τ. (22) 1 τ Then for any α [0, α], we have: (X(α),S(α)) (X + αδx P,S+ αδs P ) N F ((1 α)μ, 2τ). (23) Moveover, α =1/O( n).

6 1236 Feixiang Chen Proof. Using lemma with γ = τ and σ = 0, the fact that p(α) 0 for α [0, α], τ<1/2and (23), we conclude H X 1/2[X(α)S(α) μ(α)] F {(1 α)τ +( τ+ n 1 τ )2 α 2 + τ[ τ+ n ]α}μ 1 τ 2τμ(α)+p(α)μ 2τμ(α). An argument similar to the one used in lemma 2.10 together with lemma 2.2(2) and the fact that 2τ <1 and μ = 0 can be used to show that (24) holds. The assertion that α =1/O( n) follows by a straightforward verification. The following lemma analyzes the corrector step of Algorithm-II. Lemma 3.3 Suppose that ( X,Ŝ) N F (μ, 2τ) for some τ (0, 1/15). Let (ΔX C, ΔX C, ΔS C, ΔS C ) denote the solution of (10), (11), (12) with (X, S) = ( X,Ŝ), μ = μ and F satisfying [A1]. Then, we have: ( X +ΔX C, Ŝ +ΔS C) N F (μ, τ). Proof. Follows immediately from lemma 2.10 with σ = 1 and γ =2τ, and noting that Γ defined by (11) satisfies Γ τ when τ 1/15. References [1] F.Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim. 5(1995) [2] R.D.C. Monteiro, Primal-Dual Path-Following Algorithms for Semidefinite Programming, SIAM Journal on Optimization, Volume7, Issue3, pp: , [3] R.D.C. Monteiro and Y. Zhang, A unified analysis for a class of pathfollowing primal- dual interior-point algorithms for semidefinite programming, Math.program, pp: , [4] Yin Zhang, On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming, SIAM Journal on Optimization, pp: , [5] M. Shida, S. Shindoh and M. Kojima, Existence of Search Direction in Interior-Point Algorithms for the SDP and the Monotone SDLCP. SIAM Journal on Optimization, Vol.8, , Received: January, 2011

A SECOND ORDER MEHROTRA-TYPE PREDICTOR-CORRECTOR ALGORITHM FOR SEMIDEFINITE OPTIMIZATION

J Syst Sci Complex (01) 5: 1108 111 A SECOND ORDER MEHROTRA-TYPE PREDICTOR-CORRECTOR ALGORITHM FOR SEMIDEFINITE OPTIMIZATION Mingwang ZHANG DOI: 10.1007/s1144-01-0317-9 Received: 3 December 010 / Revised: