We introduce basic ideas of a nonsmooth Newton’s method and its application in solving semidefinite optimization (SDO) problems. In particular, the method can be used to solve both linear and nonlinear semidefinite complementarity problems. We also survey recent theoretical results in matrix functions and stability of SDO that are stemed from the research on the matrix form of the nonsmooth Newton’s method.Singapore-MIT Alliance (SMA

Sun, Jie

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1Nonsmooth Newton’s Method and SemidefiniteOptimizationJie SunAbstract— We introduce basic ideas of a nonsmooth Newton’smethod and its application in solving semidefinite optimization(SDO) problems. In particular, the method can be used to solveboth linear and nonlinear semidefinite complementarity problems.We also survey recent theoretical results in matrix functions andstability of SDO that are stemed from the research on the matrixform of the nonsmooth Newton’s method.Index Terms—semismooth functions, semidefinite optimiza-tion, Newton’s method, complementarity problems, stability, vari-ational inequality.I. MOTIVATIONA. Reduction of optimization problems to nonsmooth equationsThe Karush-Kuhn-Tucker system of a nonlinear optimizationproblem is often written as the complementarity problemFind x ≥ 0, such that F (x) ≥ 0 and x • F (x) = 0, (1)where x ∈ <n, F : <n → <n, F is continuously differentiable(smooth), and • stands for the inner product.Let <n+ be the nonnegative orthant of <n and let (·)+ be theprojection onto <n. It is easy to show that the complementarityproblem is equivalent to the following nonsmooth equationFind a root of G(x) = x− (x− F (x))+ = 0. (2)Thus, solving (1) is equivalent to solving (2).Now we generalize the above idea. Given a closed convex setK ⊆ <n, a mapping F : <n → <n, the variational inequalityproblem, denoted by VI (K,F ), is to find a vector x ∈ K suchthat( y − x )TF (x) ≥ 0, ∀ y ∈ K.The solution set of this problem is denoted SOL(K,F ). Offundamental importance to the VI is its normal map:FnorK (z) ≡ F (ΠK(z)) + z −ΠK(z), ∀ z ∈ <n, (3)where ΠK denotes the Euclidean projector onto K. It is wellknown that if x ∈ SOL(K,F ), then z ≡ x− F (x) is a zero ofFnorK ; conversely, if z is a zero of FnorK , then x ≡ ΠK(z) solvesthe VI (K,F ). When K is in addition a cone, the VI (K,F ) isequivalent to the CP (K,F ):K 3 x ⊥ F (x) ∈ K∗,Manuscript received October 31, 2002. This work was supported in part bySingapore-MIT Alliance and Grants RP314000-028/042-112 of National Uni-versity of Singapore.Singapore-MIT Alliance, National University of Singapore, Republic of Sin-gapore (e-mail: jsun@nus.edu.sg)(fax: 6779-2621).where K∗ is the dual cone of C; i.e., K∗ ≡ {y ∈ <n : yTx ≥0, ∀x ∈ K}.We note two points. 1. Problem (1) is a special case of CP(K,F ), where K = <n+. 2. The functions in (2) and (3) arecompositions of smooth functions and the projection function,hence are nonsmooth but Lipschitz continuous.B. Semidefinite optimization problemsLet Sn denote the space of n×n symmetric matrices; let Sn+and Sn++ denote the cone of n×n symmetric positive semidef-inite and positive definite matrices, respectively. We could con-sider problem VI (K,F ) and CP (Sn+, F ), respectively, whereK ⊂ Sn and F : Sn → Sn. These are, in a sense, the mostgeneral SDO problems. To fully understand them, more discus-sion ought to be made.We write A  0 to mean that A is a symmetric positivesemidefinite matrix. For any two matrices A and B in Sn, wewriteA • B ≡n∑i,j=1aijbij = tr(AB)for the Frobenius inner product between A and B, where “tr”denotes the trace of a matrix. The Frobenius norm on Sn is thenorm induced by the above inner product:‖A ‖ ≡√A • A =√√√√ n∑i,j=1a2ij .Under the Frobenius norm, the projection ΠSn+(A) of a matrixA ∈ Sn onto the cone Sn+ is the unique minimizer of the fol-lowing convex program in the matrix variable B:minimize ‖A−B ‖subject to B ∈ Sn+.Throughout the following discussion, we let A+ denote the(Frobenius) projection of A ∈ Sn onto Sn+. This projectionsatisfies the following complementarity condition:Sn+ 3 A+ ⊥ A+ −A ∈ Sn+, (4)where the ⊥ notation means “perpendicular under the abovematrix inner product”; i.e., C ⊥ D ⇔ C •D = 0 for any twomatrices C and D in Sn. The projection A+ has an explicitrepresentation. Namely, ifA = PΛPT , (5)where Λ is the diagonal matrix of eigenvalues of A and P is acorresponding orthogonal matrix of orthonormal eigenvectors,thenA+ = PΛ+PT ,where Λ+ is the diagonal matrix whose diagonal entries arethe nonnegative parts of the respective diagonal entries of Λ.Define three fundamental index sets associated with the matrixA:α ≡ { i : λi > 0 }, β ≡ { i : λi = 0 },γ ≡ { i : λi < 0 };these are the index sets of positive, zero, and negative eigenval-ues of A, respectively. WriteP = [ Wα Wγ Z ]with Wα ∈ <n×|α|, Wγ ∈ <n×|γ|, and Z ∈ <n×|β|. Thus thecolumns of Wα, Wγ , and Z are the orthonormal eigenvectorscorresponding to the positive, negative, and zero eigenvalues ofA, respectively.Any function from a topological vector space X to Sn iscalled a matrix function. In particular, a scalar matrix functionis a function F : Sn → Sn defined through a scalar functionand eigenvaluesF (X) = Pdiag (f(λ1(X)), ..., f(λn(X)))PT ,where f : < → < is a scalar function and PTXP =diag (λ1, ..., λn). For examples, X+ is a scalar matrix functionwith f(λ) = max(0, λ) = (λ)+ and |X| is defined as a scalarfunction with f(λ) = |λ|. However, it is obvious that not allmatrix functions from Sn to Sn are scalar matrix functions.II. THE SEMISMOOTH NEWTON’S METHODMost of the following results can be found in [7].Newton’s Methodxk+1 = xk − [F ′(xk)]−1F (xk) (6)is a classic method for solving the nonlinear equationF (x) = 0, (7)where F is a continuously differentiable function, i.e., a smoothfunction. Many other methods for solving (7) are related to thismethod.Suppose now that F is not a smooth function, but a locallyLipschitzian function. Then the formula (6) cannot be used.Let ∂F (xk) be the generalized Jacobian of F at xk, defined byClarke [3]. In this case, in stead of (6), one may usexk+1 = xk − V −1k F (xk) (8)where Vk ∈ ∂F (xk), to solve (7).What is the generalized Jacobian? Suppose F is a locallyLipschitzian function. According to Rademacher’s Theorem, Fis Freche´t differentiable almost everywhere. Denote the set ofpoints at which F is differentiable by DF . We write JF (x) forthe usual Freche´t derivative (the usual Jacobian) at x wheneverx is a point at which the Freche´t derivatives exist. Then thegeneralized Jacobian is the set∂F (x) = co{ limxi→xxi∈DFJF (xi)},where “co” is the convex hull [8].We say that F is semismooth at x if F is locally Lipschitzianat x andlimV∈∂F (x+th′)h′→h,t↓0{V h′}exists for any h.Theorem II.1: Suppose that F is a locally Lipschitzian func-tion. The following statements are equivalent.• F is semismooth at x;• F is directionally differentiable and for x + h ∈ DF onehasF (x+ h)− F (x)−∇F (x+ h)h = o(‖h‖) ∀h→ 0;• F is directionally differentiable andF ′(x+ h, h)− F ′(x, h) = o(‖h‖) ∀h→ 0;• for any V ∈ ∂F (x+ h), h→ 0,V h− F ′(x;h) = o(‖h‖).Semismoothness was originally introduced by Mifflin [5]for functionals. Convex functions, smooth functions and sub-smooth functions are examples of semismooth functions. Lin-ear combinations of semismooth functions are still semismoothfunctions.If any of the o(‖h‖) terms above is replaced by O(‖h‖1+p),where 0 < p ≤ 1, then we say that F is p-order semismoothat x. Note that p-order semismoothness (0 < p ≤ 1) im-plies semismoothness. In particular, 1-order semismoothnessis called strong semismoothness.Theorem II.2: (Global and Local Convergence) Suppose thatF is locally Lipschitzian and semismooth on S = {x : ‖x −x0‖ ≤ r}. Also suppose that for any V ∈ ∂F (x), x, y ∈ S, Vis nonsingular,‖V −1‖ ≤ β, ‖V (y − x)− F ′(x; y − x)‖ ≤ γ‖y − x‖,‖F (y)− F (x)− F ′(x; y − x)‖ ≤ δ‖y − x‖,where α = β(γ + δ) < 1 and β‖F (x0)‖ ≤ r(1 − α). Thenthe iterates (8) remain in S and converge to the unique solutionx∗ of (7) in S. Moreover, (for p-order semismooth F , resp.) theerror estimate‖xk−x∗‖ = o(‖xk−xk−1‖) (= O(‖xk − xk−1‖)1+p resp.)holds for large k. Thus, the semismooth Newton method issuperlinearly (quadratically) convergent.In the above theorem, the conclusion remains to be true if theset ∂F (x) is substituted by the set∂BF (x) ={limxi→xxi∈DFJF (xi)}.This allows to extend Newton’s method to solving problemslike |x| = 0.III. MATRIX VALUED FUNCTIONSThe following results can be found in [2], [9], [6], [10]. LetA,B ∈ Sn. Define a linear operator LA : Sn 7→ Sn asLA(B) = AB +BA.Lemma III.1: Let F : Sn 7→ Sn be a scalar matrix func-tion with respect to f. Then F is (continuously) differentiable(semismooth, strongly semismooth) at X if and only if fis (continuously) differentiable (semismooth, strongly semis-mooth) at λ1, ..., λn.Lemma III.2: The matrix function (X)+ is strongly semis-mooth and is in fact smooth at X if X is nonsingular with∇(X)+ = L−1|X| ◦ LX+ ,where L−1|X| is the inverse operator of L|X| and ◦ stands for thecomposition of operators.Theorem III.1: We may apply the semismooth Newton’smethod to the normal equation to solve VI and CP. The methodwill have a local quadratic convergence rate if all ∂BFnorK (z) arenonsingular at z∗.Associated with the projection problem (4) is the criticalcone of Sn+ at A ∈ Sn defined as:C(A;Sn+) ≡ T (A+;Sn+) ∩ (A+ −A )⊥,where T (A+;Sn+) is the tangent cone of Sn+ at A+ and (A+ −A)⊥ is the subset of matrices in Sn that are orthogonal to(A+−A) under the matrix inner product. The importance of thecritical cone in the local analysis of constrained optimization iswell known. In the present context, this cone can be completelydescribed [1]:C(A;Sn+) = {C ∈ Sn : WTγ CWγ = 0, WTγ CZ = 0,ZTCZ  0 }.The affine hull [8] of C(A;Sn+), which we denote L(A;Sn+), iseasily seen to be the linear subspace:{C ∈ Sn : WTγ CWγ = 0, WTγ CZ = 0 }.Theorem III.3: Consider CP (Sn+, F ). Suppose that for X ∈S(Sn+, F ) the Jacobian JF (X) of F at X is positive define onthe linear subspace L(X − F (X);S+). Then the conditions ofTheorem II.2 are satisfied.Theorems III.1 and III.3 provide a basis for a quadratically con-vergent Newton’s method for VI and CP(Sn+, F ), particularlyfor nonlinear cases.IV. STABILITY OF SEMIDEFINITE COMPLEMENTARITYPROBLEMSMost of the following results can be found in [6].How does the solution of a semidefinite variational inequalityproblem change if the given data has a small perturbation? Byreducing the semidefinite complementarity problem to a nons-mooth equation, we can study the stability of the semidefinitecomplementarity problem.Definition IV.1: A solution x∗ of the VI (K,F ) is said to bestrongly stable if for every open neighborhood N of x∗ satisfy-ingSOL(K,F ) ∩ clN = {x∗ }, (9)there exist two positive scalars c and ε such that for every con-tinuous function G satisfyingsupx∈K∩clN‖G(x)− F (x) ‖ ≤ ε,the set SOL(K,G) ∩ N is a singleton; moreover, for anothercontinuous function G˜ satisfying the same condition as G, itholds that‖x− x ′ ‖ ≤ c ‖ [F (x)−G(x) ]− [F (x ′)− G˜(x ′) ] ‖,where x and x ′ are the unique elements in the setsSOL(K,G) ∩N and SOL(K, G˜) ∩N , respectively.In essence, strong stability pertains to unique, continuous so-lution under small, continuous perturbations of F . Let us con-sider another concept.Definition IV.2: A function H : <n → <n is said to be alocally Lipschitz homeomorphism near a vector x if there ex-ists an open neighborhood N of x such that the restricted mapH|N : N → H(N ) is Lipchitz continuous and bijective, andits inverse is also Lipschitz continuous.We can now state the following result. The significance ofthis result is that the strong stability of a solution to a VI can bededuced from an inverse function theorem for the normal map.Theorem IV.3: Let F : K ⊆ <n → <n be locally Lipschitzcontinuous on the closed convex set K. Let x∗ ∈ SOL(K,F )be given. Let z∗ ≡ x∗ − F (x∗). The following statements areequivalent:(a) x∗ is a strongly stable solution of the VI (K,F );(b) z∗ is a strongly stable zero of FnorK ;(c) FnorK is a locally Lipschitz homeomorphism near z∗(d) There exist an open neighborhood Z of z∗ and a con-stant c > 0 such that‖FnorK (z)−FnorK (z′) ‖ ≥ c ‖ z−z ′ ‖, ∀ z, z ′ ∈ Z.The equivalence of statements in the above theorem remainsvalid for all locally Lipschitz continuous functions, of whichthe normal map FnorK is a special instance.Theorem IV.4: Let Φ : <n → <n be Lipschitz continuous inan open neighborhood D of a vector x∗ ∈ Φ−1(0). Considerthe following three statements:(a) every matrix in ∂Φ(x∗) is nonsingular;(b) Φ is a locally Lipschitz homeomorphism near x∗;(c) for every V ∈ ∂BΦ(x∗), sgn detV = ±1.It holds that (a) ⇒ (b) ⇒ (c). Assume in addition that Φ isdirectionally differentiable at x∗. Consider the following twoadditional statements:(d) Ψ ≡ Φ ′(x∗; ·) is a globally Lipschitz homeomor-phism;(e) for every V ∈ ∂BΨ(0), sgn detV = ±1.It holds that (b) ⇒ (d) ⇒ (e). Moreover, if (b) holds and Φis directionally differentiable at x∗, then the local inverse ofΦ near x∗, denoted Φ−1, is directionally differentiable at theorigin; and(Φ−1) ′(0;h) = Ψ−1(h), ∀h ∈ <n. (10)If Φ is semismooth on D then (b) ⇔ (c); in this case, the localinverse of Φ near x∗ is semismooth near the origin. Finally, ifΦ is semismooth on D and∂BΦ(x∗) ⊆ ∂BΨ(0), (11)then the four statements (b), (c), (d), and (e) are equivalent.The inclusion (11) plays an essential role for the statements(b) and (c) in Theorem IV.4, which pertain to the original func-tion Φ, to be equivalent to the corresponding statements (d) and(e), which pertain to the directional derivative Ψ.The next technical result establishes the equality (11) thatpaves the way for the application of Proposition IV.4 to SDOproblems.Lemma IV.1: Let A ∈ Sn be arbitrary. Let Ψ ≡ Π′Sn+(A; ·).It holds that∂BΠSn+(A) = ∂BΨ(0). (12)The following extends the semismooth inverse function the-orem to an semismooth implicit function theorem.Theorem IV.5: Assume that ∂BΦ(x∗) ⊆ ∂BΦ ′(x∗; ·)(0) andthat JwG(Φ(x∗), p∗)◦Φ ′(x∗; ·) is a globally Lipschitz homeo-morphism. There exist a neighborhood U of p∗, a neighborhoodV of x∗, and a Lipschitz continuous function x : U → V that issemismooth at p∗ such that for every p ∈ U , x(p) is the uniquevector in V satisfying G(Φ(x(p)), p) = 0. Moreover, for everyvector dp ∈ <m, x ′(p∗, dp) is the unique solution dx of thefollowing equation:JwG(Φ(x∗), p∗)Φ ′(x∗; dx) + JpG(Φ(x∗), p∗)dp = 0. (13)Applying this theorem to the normal map of CP (Sn+, F ),the condition of JwG(Φ(x∗), p∗) ◦ Φ ′(x∗; ·) being a globallyLipschitz homeomorphism can be interpreted as the uniquenessof the solution to the following linear complementarity problem(details omitted)C 3 S∗ ⊥ −Q+ ( JF (X∗) +AZ∗ )(S∗) ∈ C∗, (14)where AZ∗ is a certain computable operator at Z∗. We mayconclude thatTheorem IV.6: Let F : Sn → Sn be continuously differ-entiable in a neighborhood of a solution X∗ of the problemCP(Sn+, F ). The following three statements are equivalent.(a) X∗ is strongly stable for CP(Sn+, F );(b) for every Q ∈ Sn, the problem (14) has a unique so-lution that is Lipschitz continuous in Q;(c) for every V ∈ ∂BΠSn+(Z∗), sgn det((JF (X∗) +AZ∗) ◦ V + I − V ) = ±1.We may apply Theorem IV.5 to a parametric CP in SDO:Sn+ 3 X ⊥ F (X, p) ∈ Sn+, (15)where F : Sn × <m → Sn is a given mapping. In what fol-lows, we show how to calculate the directional derivative of animplicit solution function of the above problem at a base param-eter vector p∗ ∈ <m. For this purpose, let X∗ be a strongly sta-ble solution of the above problem at p∗. Assume that F is con-tinuously differentiable in a neighborhood of the pair (X∗, p∗).It follows that there exist open neighborhoods V ⊆ Sn+ of X∗and P ⊆ <m of p∗ and a locally Lipschitz continuous functionX : P → V such that for every p ∈ P , X(p) is the unique ma-trix in V that solves (15); moreover, the implicit solution func-tion X is semismooth at p∗. We wish to compute X ′(p∗; dp)for dp ∈ <m. For each p ∈ P , let Z(p) ≡ X(p)−F (X(p), p).We have X(p) = ΠSn+(Z(p)) andF (ΠSn+(Z(p)), p) + Z(p)−−ΠSn+(Z(p)) = 0.Taking the directional derivative of the above normal equationat p∗ along the direction dp and writing dZ ≡ Z ′(p∗; dp), weobtainJxF (X∗, p∗)Π ′Sn+(Z∗; dZ) + JpF (X∗, p∗)dp+dZ −Π ′Sn+(Z∗; dZ) = 0.Note that X ′(p∗; dp) = Π ′Sn+(Z∗; dZ). By the previous deriva-tion, we deduce that X ′(p∗; dp) is the unique solution S∗ of thelinear complementarity problem:C 3 S∗ ⊥ JpF (X∗, p∗)dp+( JxF (X∗, p∗)+AZ∗ )(S∗) ∈ C∗,where C ≡ T (X∗;Sn+)∩F (X∗, p∗)⊥ is the critical cone of theCP (Sn+, F (·, p∗)) at the solution X∗.REFERENCES[1] J.F. BONNANS AND A. SHAPIRO. Perturbation Analysis of Optimiza-tion Problems, Springer-Verlag (New York 2000).[2] X. CHEN, H.D. QI, AND P. TSENG. Analysis of nonsmooth symmet-ric matrix functions with applications to semidefinite complementarityproblems. SIAM Journal on Optimization, forthcoming.[3] F.H. CLARKE. Optimization and Nonsmooth Analysis, John Wiley(1983).[4] F. FACCHINEI AND J.S. PANG Finite-Dimensional Variational Inequal-ities and Complementarity Problems, Springer-Verlag (New York 2003),forthcoming.[5] R. MIFFLIN. Semismooth and semiconvex functions in constrained op-timization. SIAM Journal on Control and Optimization 15 (1977) 957–972.[6] J.-S. PANG, D. SUN, AND J. SUN, Semismooth Homeomorphismsand Strong Stability of Semidefinite and Lorentz Complementarity Prob-lems, Mathematics of Operations Research forthcoming.[7] L. QI AND J. SUN. A nonsmooth version of Newton’s method. Mathe-matical Programming 58 (1993) 353–367.[8] R.T. ROCKAFELLAR. Convex Analysis, Princeton Uniiversity Press(1970).[9] D. SUN AND J. SUN. Semismooth matrix valued functions. Mathemat-ics of Operations Research 27 (2002) 150–169.[10] J. SUN, D. SUN AND L. QI.Quadratic convergence of a squared smooth-ing Newton method for nonsmooth matrix equations and its applicationsin semidefinite optimization problems, Manuscript (2001).

Nonsmooth Newton’s Method and Semidefinite Optimization

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Nonsmooth Newton’s Method and Semidefinite Optimization

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