A Primal-Dual Interior-Point Filter Method for Nonlinear Semidefinite Programming

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1 The 7th Iteratioal Symposium o Operatios Research ad Its Applicatios (ISORA 8) Lijiag, Chia, October 31 Novemver 3, 28 Copyright 28 ORSC & APORC, pp A Primal-Dual Iterior-Poit Filter Method for Noliear Semidefiite Programmig Zhog-Yi Liu 1, We-Yu Su 2, 1 College of Sciece, Hohai Uiversity, Najig 2198, Chia 2 School of Mathematics ad Computer Scieces, Najig Normal Uiversity, Najig 2197, Chia Abstract This paper proposes a primal-dual iterior-poit filter method for oliear semidefiite programmig, which is a extesio of the work of Ulbrich et al. (Math. Prog., 1(2):379 41, 24). We use a mixed orm to tackle with trust regio costraits ad global covergece to first-order critical poits ca be proved. Keywords Noliear semidefiite programmig; Iterior-poit method; Primal-dual; Filter; Global covergece 1 Itroductio For the cocepts ad frame of filter methods, we refer to Fletcher et al. [1], Fletcher ad Leyffer [2], Fletcher et al. [3], Ulbrich et al. [5]. We cosider the followig oliear semidefiite programmig (SDP) problem: mi f (x), x R, s.t. h(x) =, X A x B, (1) where the fuctios f : R R ad h : R R m are sufficietly smooth. Here A is a liear operator defied by A x = i=1 x i A i for x R, ad A i S p, i = 1,...,, ad B S p are give matrices, where S p deotes the set of pth order real symmetric matrices. By X ad X, we mea that the matrix X is positive semidefiite ad positive defiite, respectively. Throughout this paper, we defie the ier product X,Z by X,Z = Tr(XZ) for two matrices X ad Z i S p, where Tr(M) deotes the trace of the matrix M. We also defie a adjoit operator A of A such that A Z is a dimesioal vector whose ith elemet is Tr(A i Z). The we have A x,z = x T (A Z) = (A Z) T x, where the superscript T deotes the traspose of a vector or a matrix. Correspodig author. zhyi@hhu.edu.c wysu@ju.edu.c

2 A Primal-Dual Iterior-Poit Filter Method Iterior-poit framework We retur to the oliear semidefiite programmig problem posed i (1). 2.1 Step computatio Let the Lagragia fuctio of problem (1) be defied by L(x,y,Z) = f (x) y T h(x) X,Z, where y R m ad Z S p are the Lagrage multiplier vector ad matrix which correspod to the equality ad positive semidefiiteess costraits, respectively. The Karush- Kuh-Tucker (KKT) coditios for optimality of problem (1) are give by xl(w) h(x) = X Z ad Here x L(x,y,Z) is give by ad the multiplicatio X Z is defied by X, Z. x L(x,y,Z) = f (x) h(x)y A Z, X Z = XZ + ZX. 2 It is kow that X Z = is equivalet to the relatio XZ = ZX =. We call (x,y,z) satisfyig X ad Z the iterior poit. To costruct a iteriorpoit algorithm, we itroduce a positive parameter ˆµ, ad we replace the complemetarity coditio X Z = by X Z = ˆµI, where I deotes the idetity matrix ad ˆµ >. The we try to fid a poit that satisfies the followig system of equatios: ad xl(x,y,z) h(x) X Z ˆµI X, Z. = (2) Throughout we will work with ˆµ = σ µ, where σ (,1) is a ceterig parameter ad To abbreviate otatio we set µ = Tr(XZ). w = (x,y,z) ad w = ( x, y, Z).

3 114 The 7th Iteratioal Symposium o Operatios Research ad Its Applicatios We apply a Newto-like method to the system of equatios (2). Let the Newto directios for the primal-dual variables by x R ad Z S p, respectively. We defie X = x i A i ad ote that X S p. i=1 Sice (X + X) (Z + Z) = σ µi ca be writte as (X + X)(Z + Z) + (Z + Z)(X + X) = 2σ µi, eglectig the oliear parts X Z ad Z X implies the Newto equatio for (2) G x h(x) y A Z = x L(x,y,Z) h(x) T x = h(x) XZ + Z X + X Z + ZX = 2σ µi XZ ZX, where G deotes the Hessia matrix of the Lagragia fuctio L(w) or its approximatio. To motivate our choice of the compoets i the filter ad the step decompositio, we rewrite the perturbed KKT-coditios i the form h(x) X Z µi + xl(x,y,z) (1 σ)µi =. (3) The first term i the right-had side of equality measures the proximity bo the quasicetral path, which is defied by P q µ = {(x,z) : h(x) =, X Z = µi}. Therefore it is atural to choose the measure of quasi-cetrality θ(x) = h(x) + X Z Tr(XZ) I F as the first compoet i the filter. The secod term o the left-had side of (3) measures complemetarity ad criticality. For the secod filter compoet we choose therefore the optimality measure Tr(XZ)/ + x L(w) 2. With this choice of the filter compoets it remais to defie correspodig tagetial ad ormal compoets of the trial step. We use the decompositio associated with the splittig (3). For the ormal step s = ( x, y, Z ), we thus choose G x h(x) y A Z h(x) T x X Z + Z X = h(x) X Z µi whereas our tagetial step s t = ( x t, y t, Z t ) is give by G x t h(x) y t A Z t h(x) T x t = xl(w) X Z t + Z X t (1 σ)µi, (4). (5)

4 A Primal-Dual Iterior-Poit Filter Method 115 Note that w = s + s t, ad the last expressio i (4) ad (5) are both for matrices, which are used to uify the form. However, it will be crucial that we exploit the flexibility of the step splittig to itroduce differet stepsizes for s ad s t i our trial step computatio. The tagetial step is the sum of a tagetial compoet s t 1, which is a solutio of the followig system of equatios G x t h(x) y t A Z t h(x) T x t X Z t + Z X t = xl(w) that attempts to reduce x L(w), with a predictor compoet s t 2, which is a solutio of the followig system of equatios G x t h(x) y t A Z t h(x) T x t X Z t + Z X t = (1 σ)µi that seeks the miimizatio of µ = Tr(XZ)/. Therefore, the tagetial step aims to reduce the optimality measure θ g (w) = Tr(XZ)/ + x L(w) 2. We itroduce as the positive scalar that primarily cotrols the legth of the step take alog w, forcig the damped compoets α ( )s ad α t ( )s t, to satisfy α ( )s, α t ( )s t. Havig these bouds i mid, ad requirig explicitly α t ( ) α ( ), we set { } α ( ) = mi 1, s, (6) { } { } α t ( ) = mi α ( ), s = mi 1, s, s t. (7) Hereby, we use for > the atural defiitio α ( ) = 1 for s = ad α t ( ) = α ( ) for s t = by usig the covetio mi{1, } = 1. Let also Thus, w( ) = (x( ),y( ),Z( )) = w + α ( )s + α t ( )s t, s( ) = (s x ( ),s y ( ),s Z ( )) = w( ) w = α ( ) + α t ( )s t. s( ) 2, ad here plays a role comparable to the trust-regio radius. We itroduce the otatio θ h (w) = h(x), θ c (w) = X Z Tr(XZ) I, θ L(w) = x L(w), which allows to rewrite the filter compoets as θ c (w), θ h (w), ad θ g (w) = Tr(XZ) + x L(w) 2.

5 116 The 7th Iteratioal Symposium o Operatios Research ad Its Applicatios Sice X Z might ot be zero matrix, a poit w that satisfies θ c (w) = θ h (w) = θ L (w) = ad X,Z, might ot be a KKT poit. The defiitio of θ g (w), however, guaratees that a poit w verifyig θ c (w) = θ h (w) = θ g (w) = ad X,Z, ideed a KKT poit. With the purpose of achievig a reductio o the fuctio θ g, we itroduce, at a give poit w, the quadratic model m(w( )) = Tr(XZ) Tr((X( ) X)Z) + Tr((Z( ) Z)X) + + x L(w) + 2 xwl(w)(w( ) w) 2 = Tr(X( )Z( )) Tr((X( ) X)(Z( ) Z)) + x L(w) + 2 xwl(w)(w( ) w) 2, by addig to the liearizatio of Tr(XZ)/ the squared orm of the liearizatio of x L(w). To shorte otatio we also set µ( ) = Tr(X( )Z( )). I order to prevet (X( ),Z( )) from approachig the boudary of the positive coe too rapidly we will keep the iteratio i the eighborhood { N (γ,m) = w : X, Z, X Z γ Tr(XZ), θ h (w) + θ L (w) M Tr(XZ) } with fixed γ (,1) ad M >. We will set i the ext subsectio that w N (γ,m) implies w( ) N (γ,m) wheever (, mi ] for a give costat mi >. 2.2 Step estimates The followig lemma measures the decrease o complemetarity obtaied by the ew iterate w( ). Lemma 1. Let A,B S, λ i (A) be its eigevalue, Λ = diag(λ i (A)), ad ρ(a) be its spectral radius, i.e., ρ(a) = max{ λ i (A), i = 1,...,}. The Proof. Sice A B ρ(a)ρ(b)i 2ρ(A)ρ(B)I (Q T AQQ T BQ + Q T BQQ T AQ) = ρ(a)ρ(b)i ΛQ T BQ + ρ(a)ρ(b)i Q T BQΛ Λρ(B)I ΛQ T BQ + ρ(b)λ Q T BQΛ = Λ(ρ(B)I Q T BQ) + (ρ(b)i Q T BQ)Λ holds for some orthogoal matrix Q, the the lemma follows easily.

6 A Primal-Dual Iterior-Poit Filter Method 117 Usig the above lemma ad Lemma 1 i Ulbrich et al. [5], we ca show if the curret poit w = (x,y,z) satisfies the cetrality requiremet X Z γi, so does the ext poit w( ) = (x( ),y( ),Z( )), provided is sufficietly small. 3 The iterior-poit filter method We choose θ ad θ g to form a filter etry, where θ measures feasibility ad θ g measures optimality. For the defiitios of domiace, filter, acceptable ad add, see Ulbrich et al. [5]. Algorithm 1. Primal-dual iterior-poit filter method Step. Choose σ (,1), ν (,1), γ 1,γ 2 >, < β,η,κ < 1, ad γ F (,1/3). Set F :=. Choose (X,Z ) ad y, ad determied γ (,1) such that X Z γµ with µ = Tr(X Z )/. Further, choose M > such that θ h (w )+θ L (w ) Mµ. Choose i > ad set k :=. Step 1. Set µ k := Tr(X k S k )/ ad compute s k ad st k by solvig the liear systems (4) ad (5), respectively, with (w, µ) = (w k, µ k ). Step 2. Compute k [, i k ] such that X k ( ), Z k ( ), X k ( ) Z k ( ) γµ k I for all [, k ]. Step 3. Compute the largest k (, k ] such that θ h (w k ( k )) + θ L(w k ( k )) Mµ k( k ). Set k := k. Step 4. If mi{θ(w k ),θ(w k ( ))} k mi{γ 1,γ 2 β k }, the go to Step 5. Otherwise, add w k to the filter ad call a restoratio algorithm that produces a poit w k+1 such that: w k+1 N (γ,m) with µ k+1 = Tr(X k+1 Z k+1 )/; w k+1 is acceptable to the filter; mi{θ(w k+1 ),θ(w k+1 ( ))} k+1 mi{γ 1,γ 2 β k } with k+1 = k. Set i k+1 := k, k := k + 1, ad go to Step 1. Step 5. If w k ( k ) is ot acceptable to the filter (ad m k (w k )-m k (w k ( k ))<κ mi{θ(w k ) 2, θ(w k ( )) 2 }), the go to Step 11. Step 6. If m k (w k ) m k (w k ( k )) =, the set ρ k :=. Otherwise, ρ k := θ g(w k ) θ g (w k ( k )) m k (w k ) m k (w k ( k )). Step 7. If ρ k < η ad m k (w k ) m k (w k ( k )) κ mi{θ(w k ) 2,θ(w k ( )) 2 }, the go to Step 11. Step 8. If m k (w k ) m k (w k ( k )) < κ mi{θ(w k ) 2,θ(w k ( )) 2 }, the add w k to the filter. Step 9. Choose i k k. Step 1. Set w k+1 := w k ( k ), k := k + 1, ad go to Step 1. Step 11. Set w k+1 := w k,s k+1 := s k,st k+1 := st k, k+1 := k/2. Set k := k + 1 ad go to Step 3.

7 118 The 7th Iteratioal Symposium o Operatios Research ad Its Applicatios 4 Global covergece to first-order critical poits The algorithm have two cases, the first is that ifiitely may iterates are added to the filter ad the secod is that the algorithm rus ifiitely but the filter is left with a fiite umber of etries. We summarize both situatios i the ext theorems. For the details of proof, see Liu ad Su [4]. Theorem 2. Suppose that ifiitely may iterates are added to the filter. The there exists a subsequece {k j } such that lim θ(w k j j ) =, lim θ g (w k j j ) =. It remais to cosider the case where the algorithm rus ifiitely but the filter is left with a fiite umber of etries. Theorem 3. Suppose that the algorithm rus ifiitely ad oly fiitely may iterates added to the filter. The lim θ(w k) =, k lim if k θ g(w k ) =. The mai result is obtaied by combiig Theorem 2 ad Theorem 3. Corollary 4. The sequece of iterates {w k } geerated by the primal-dual iterior-poit method (Algorithm 1) satisfies 5 Cocludig remarks lim if k θ(w k) + θ g (w k ) =. I this paper we exted the iterior-poit filter method to oliear semidefiite programmig. This work is trivial ad little difficulty happeed. From the paper, we ca see that it is atural that three-dimesioal filter methods ca be with thikig, where the first is for feasibility, the secod for cetrality, ad the third for optimizatio. I Liu ad Su [4], we preseted a three-dimesioal filter iterior-poit method for oliear semidefiite programmig. Refereces [1] R. Fletcher, S. Leyffer, ad P.L. Toit. O the Global Covergece of ad SLP-Filter Algorithm, Report 98/13, Dept of Mathematics, FUNDP, Namur (B), [2] R. Fletcher ad S. Leyffer. Noliear programmig without a pealty fuctio. Mathematical Programmig, 91(2): , 22. [3] R. Fletcher, N.I.M. Gould, S. Leyffer, P. L. Toit, ad A. Wächter. Global covergece of a trust-regio SQP-filter algorithm for geeral oliear programmig. SIAM Joural o Optimizatio, 13(3): , 22. [4] Z. Liu ad W. Su. A globally coverget primal-dual iterior-poit three-dimesioal filter method for oliear semidefiite programmig. Submitted to SIAM Joural o Optimizatio. [5] M. Ulbrich, S. Ulbrich, ad L.N. Vicete. A globally coverget primal-dual iterior-poit filter method for oliear programmig. Mathematical Programmig, 1(2):379 41, 24.