3 A merit function for SCCP with H-differentiable functions

Journal of Nonlinear and Convex Analysis, vol. 14, no. 2, pp. , 2013

On the H-differentiability of L¨ owner function with application in symmetric cone complementarity problem

Yu-Lin Chang

Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan

E-mail: [email protected]

Jein-Shan Chen ¹ Department of Mathematics National Taiwan Normal University

Taipei 11677, Taiwan E-mail: [email protected]

Weizhe Gu

Department of Mathematics School of Science Tianjin University Tianjin 300072, P.R. China E-mail: [email protected]

September 10, 2012

Abstract. Let K be the symmetric cone in a Jordan algebra V. For any function f from IR to IR, one can define the corresponding L¨owner function f^sc(x) on V by the spectral decomposition of x ∈ V with respect to K. In this paper, we study the relationship regarding H-differentiability between f^sc and f . The class of H-differentiable functions is known to be wider than the class of semismooth functions. Therefore, our result will contribute to solution analysis and solution methods for solving more general symmetric cone programs (SCP) and symmetric cone complementarity problems (SCCP). Besides, we also study a merit function approach for SCCP under H-differentiable condition.

In particular, for such class of complementarity problems, we provide conditions to guarantee every stationary point of the associated merit function to be a solution.

1Corresponding author. Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is supported by National Science Council of Taiwan.

Keywords. H-differentiable, symmetric cone, second-order cone, complementarity.

AMS subject classifications. 26B05, 26B35, 90C33, 65K05

1 Introduction and Preliminary

Let V be an n-dimensional vector space over the real field R, endowed with a bilinear mapping (x, y) 7→ x ◦ y from V × V into V. The pair (V, ◦) is called a Jordan algebra [9, 16] if the followings hold.

(i) x ◦ y = y ◦ x for all x, y ∈ V,

(ii) x ◦ (x²◦ y) = x²◦ (x ◦ y) for all x, y ∈ V.

Note that a Jordan algebra is not necessarily associative, i.e., x ◦ (y ◦ z) = (x ◦ y) ◦ z may not hold for all x, y, z ∈ V. We call an element e ∈ V the identity element if x ◦ e = e ◦ x = x for all x ∈ V. A Jordan algebra (V, ◦) with an identity element e is called a Euclidean Jordan algebra if there is an inner product h·, ·i

V such that (iii) hx ◦ y, zi

V = hy, x ◦ zi

V for all x, y, z ∈ V.

Given a Euclidean Jordan algebra A = (V, ◦, h·, ·i_V), we denote the set of squares as K :=x² | x ∈ V .

From [9, Theorem III.2.1], K is a symmetric cone which means that K is a self-dual closed convex cone with nonempty interior and for any two elements x, y ∈ intK, there exists an invertible linear transformation T : V → V such that T (K) = K and T (x) = y.

In the following, we present three examples of Euclidean Jordan algebras.

Example 1.1. Consider Rⁿ with the (usual) inner product and Jordan product defined respectively as

hx, yi =

n

X

i=1

xiyi and x ◦ y = x ∗ y ∀x, y ∈ Rⁿ

where x_i denotes the ith component of x, etc., and x ∗ y denotes the componentwise product of vectors x and y. Then, Rⁿis a Euclidean Jordan algebra with the nonnegative orthant Rⁿ+ as its cone of squares.

Example 1.2. The algebraSnof n × n real symmetric matrices. Let S^n×n be the space of all n × n real symmetric matrices with the trace inner product and Jordan product, respectively, defined by

hX, Y i_T := Tr(XY ) and X ◦ Y := 1

2(XY + Y X) ∀X, Y ∈ S^n×n.

Then, (S^n×n, ◦, h·, ·i_T) is a Euclidean Jordan algebra, and we write it as Sn. The cone of squares S^n×n+ inSn is the set of all positive semidefinite matrices in S^n×n.

Example 1.3. The Jordan spin algebra Ln. Consider Rⁿ (n > 1) with the inner product h·, ·i and Jordan product

x ◦ y :=

 hx, yi x0y + y¯ 0x¯



for any x = (x₀; ¯x), y = (y₀; ¯y) ∈ R × Rⁿ⁻¹. We denote the Euclidean Jordan algebra (Rⁿ, ◦, h·, ·i) byLn. The cone of squares, called the Lorentz cone (or second-order cone), is given by Ln⁺ := {(x₀; ¯x) ∈ R × Rⁿ⁻¹ | x₀ ≥ k¯xk}.

For any given x ∈ A, let ζ(x) be the degree of the minimal polynomial of x, i.e., ζ(x) := mink : {e, x, x², · · · , x^k} are linearly dependent .

Then, the rank of A is defined as max{ζ(x) : x ∈ V}. In this paper, we use r to denote the rank of the underlying Euclidean Jordan algebra. Recall that an element c ∈ V is idempotent if c² = c. Two idempotents ci and cj are said to be orthogonal if ci◦ cj = 0.

One says that {c₁, c₂, . . . , c_k} is a complete system of orthogonal idempotents if c²_j = c_j, c_j ◦ c_i = 0 if j 6= i for all j, i = 1, 2, · · · , k and Pk

j=1c_j = e.

An idempotent is primitive if it is nonzero and cannot be written as the sum of two other nonzero idempotents. We call a complete system of orthogonal primitive idempotents a Jordan frame. Now we state a version of the spectral decomposition theorem which is important for subsequent analysis.

Theorem 1.1 [9, Theorem III.1.2] Suppose that A is a Euclidean Jordan algebra with rank r. Then, for any x ∈ V, there exists a Jordan frame {c¹, · · · , cr} and real numbers λ₁(x), · · · , λ_r(x), arranged in the decreasing order λ₁(x) ≥ λ₂(x) ≥ · · · ≥ λ_r(x), such that

x = λ₁(x)c₁+ λ₂(x)c₂+ · · · + λ_r(x)c_r.

The numbers λj(x) (counting multiplicities), which are uniquely determined by x, are called the eigenvalues and tr(x) =Pr

j=1λ_j(x) the trace of x.

Since, by [9, Prop. III.1.5], a Jordan algebra (V, ◦) with an identity element e ∈ V is Euclidean if and only if the symmetric bilinear form tr(x ◦ y) is positive definite, we may define another inner product on V by hx, yi := tr(x ◦ y) for any x, y ∈ V. The inner product h·, ·i is associative by [9, Prop. II. 4.3], i.e., hx, y ◦ zi = hy, x ◦ zi for any x, y, z ∈ V. Every Euclidean Jordan algebra can be written as a direct sum of so-called simple ones. In finite dimensions, the simple Euclidean Jordan algebras come in four families with infinite cases, together with one exceptional case:

Theorem 1.2 [9, Theorem V.3.7] Suppose that A = (V, ◦, h·, ·iV) is a simple Euclidean Jordan algebra of rank r ≥ 3. Then, A is isomorphic to one of the following:

(i) The algebra Sn of n × n real symmetric matrices given by Example 1.2;

(ii) The algebra Hn of all n × n complex Hermitian matrices with trace inner product hx, yi_T := IRTr(xy^∗) and Jordan product x ◦ y := ¹₂(xy + yx) for any x, y ∈ H^n×n; (iii) The algebra Qn of all n × n quaternionic Hermitian matrices with trace inner product hx, yi_T := IRTr(xy^∗) and Jordan product x ◦ y := ¹₂(xy + yx) for any x, y ∈ Q^n×n;

(iv) The algebra O3 of all 3 × 3 octonionic Hermitian matrices with trace inner product hx, yi_T := IRTr(xy^∗) and Jordan product x ◦ y := ¹₂(xy + yx) for any x, y ∈ Ø^3×3; (v) The Jordan spin algebra Ln given by Example 1.3.

where the notation “∗” means the conjugate transpose, Tr(xy) denotes the trace of xy which is the multiplication of matrices x and y, and IRa means the real part of a.

Given an n-dimensional Euclidean Jordan algebra A = (V, h·, ·i, ◦) with K being its corresponding symmetric cone in V. For any scalar function f : IR → IR, we define a vector-valued function f^sc(x) (called L¨owner function) on V as

f^sc(x) = f (λ₁(x))c₁ + f (λ₂(x))c₂+ · · · + f (λ_r(x))c_r (1) where x ∈ V has the spectral decomposition

x = λ₁(x)c₁+ λ₂(x)c₂+ · · · + λ_r(x)c_r.

When V is the Jordan spin algebra Ln in which K corresponds the second-order cone (SOC), which is defined as

Kⁿ:= {(x₁, x₂) ∈ IR × IRⁿ⁻¹ | kx₂k ≤ x₁},

the function f^sc reduces to so-called SOC-function f^soc studied in [2, 3, 4, 5]. More specifically, under such case, the spectral decomposition for any x = (x₁, x₂) ∈ IR×IRⁿ⁻¹ becomes

x = λ₁(x)u⁽¹⁾_x + λ₂(x)u⁽²⁾_x , (2) where λ₁(x), λ₂(x), u⁽¹⁾x and u⁽²⁾x with respect to Kⁿ are given by

λ_i(x) = x₁+ (−1)ⁱkx₂k, u⁽ⁱ⁾_x =







1 2



1, (−1)ⁱ x₂ kx₂k



if x₂ 6= 0,

1 2



1, (−1)ⁱw



if x2 = 0,

for i = 1, 2, with w being any vector in IRⁿ⁻¹ satisfying kwk = 1. If x₂ 6= 0, the decomposition (2) is unique. With this spectral decomposition, for any function f : IR → IR, the L¨owner function f^sc associated with Kⁿ reduces to f^soc as below:

f^soc(x) = f (λ₁(x))u⁽¹⁾_x + f (λ₂(x))u⁽²⁾_x ∀x = (x₁, x₂) ∈ IR × IRⁿ⁻¹. (3) For SOC case, Chen, Chen and Tseng in [5] show that the L¨owner function f^soc inherits from f the properties of continuity, Lipschitz continuity, directional differentia- bility, Fr´echet differentiability, continuous differentiability, as well as semismoothness.

The H¨older continuity of f^soc and f is recently shown by the authors in [2]. Sun and Sun [23] extend some of the aforementioned results to more general symmetric cone case regarding f^sc. In addition, the H¨older continuity about f^sc and f for symmetric cone case is investigated by Lu and Huang in [17]. These results are useful in the design and analysis of smoothing and nonsmooth methods for solving symmetric cone programs (SCP) and symmetric cone complementarity problems (SCCP), see [4, 6, 19, 20] and references therein.

The concepts of H-differentiability and H-differential were introduced in [13] to study the injectivity on nonsmooth functions. As remarked in [13, 25, 26, 27, 28], the Fr´echet derivative of a Fr´echet differentiable function, the Clarke generalized Jacobian of a locally Lipschitz continuous function, the Bouligand subdifferential of a semismooth function, and the C-differential of C-differentiable function are all examples of H-differentials. It is known that any superset of an H-differential is an H-differential, H-differentiability implies continuity, and H-differentials satisfy simple sum, product and chain rules. Fur- thermore, an H-differentiable function need not to be locally Lipschitz continuous nor directionally differentiable. With the above facts, the class of H-differentiable functions is wider than the class of semismooth functions.

In this paper, we study whether the H-differentiability of the L¨owner function f^sc can be also inherited from f or not. Since the class of H-differentiable functions is known as wider than the class of semismooth functions, we believe that this result will contribute to solution analysis and solution methods towards more general SCP and SCCP. Besides, we also study a merit function approach for SCCP under H-differentiable condition. In particular, for such class of complementarity problems, we provide conditions to guar- antee every stationary point of the associated merit function to be a solution.

2 The relationship on H-differentiabilities between f

and f

In this section, we first review several concepts related to H-differentiability. Then, we present our first main result which says the H-differentiability of the vector-valued L¨owner function f^sc implies that of f .

The concepts of H-differentiability and H-differential of a function were first pro- posed by Gowda and Ravindran in [13]. Their motivation was to study a generalization (to nonsmooth case) of a result of Gale and Nikaido [11] which asserts that if the Ja- cobian matrix of a differentiable function f from a closed rectangle K ⊆ IRⁿ into IRⁿ is an P -matrix at each point of K, then f is one-to-one on K. More issues about H-differentiability have been studied in [12, 25, 26].

Definition 2.1 Given a function F : Ω ⊆ IRⁿ → IR^m, where Ω is an open set in IRⁿ and x^∗ ∈ Ω, we say that a nonempty subset T (x^∗), also denoted by T_F(x^∗), of IR^m×n is an H-differential of F at x^∗ if for every sequence x^k ∈ Ω converging to x^∗, there exist a subsequence x^kj and a matrix A ∈ T (x^∗) such that

F (x^kj) − F (x^∗) − A(x^kj − x^∗) = o(kx^kj− x^∗k).

We say that F is H-differentiable at x^∗ if F has an H-differential at x^∗.

A useful equivalent definition of an H-differential T_F(x^∗) is: for any sequence x^k :=

x^∗+ t_kd^k with t_k↓ 0 and kd^kk = 1 for all k, there exist convergent subsequences t_kj ↓ 0 and d^kj → d, and A ∈ T_F(x^∗) such that

j→∞lim

F (x^∗+ t_kjd^kj) − F (x^∗) tkj

= Ad.

Here are summaries of some well-known facts about H-differentiability, for more details please refer to [13, 25, 26, 27, 28].

Remark 2.1 (i) Any superset of an H-differential is an H-differential.

(ii) H-differentiability implies continuity.

(iii) If a function F : Ω ⊆ IRⁿ → IR^m is H-differentiable at a point ¯x, then there exist a constant L > 0 and a neighborhood B(¯x, δ) of ¯x with

kF (x) − F (¯x)k ≤ Lkx − ¯xk ∀x ∈ B(¯x, δ). (4) Conversely, if condition (4) holds, then T (¯x) := IR^m×n can be taken as an H- differential of F at ¯x.

(iv) Let f : Ω → IR be a real-valued function defined on an open set Ω ⊆ IRⁿ. Suppose that f is not H-differentiable at ¯x ∈ Ω. Then there exists a sequence {x^k} in Ω converging to ¯x and for all subsequence x^kj, there is no a ∈ IR such that

f (x^kj) − f (¯x) kx^kj− ¯xk → a.

Hence the set

nf (x^k)−f (¯x) kx^k−¯xk

o

is unbounded by Bolzano-Weierstrass Theorem. After taking subsequence, this is equivalent to saying that there exists a sequence {x^k} in Ω converging to ¯x such that

f (x^k) − f (¯x)

kx^k− ¯xk → ∞ or − ∞.

The following lemma gives a sufficient condition for H-differentiability. It is just a direct consequence of Bolzano-Weierstrass Theorem, we omit its proof.

Lemma 2.1 Let f : Ω → IR be a real-valued function defined on an open set Ω ⊆ IR.

Define subset A(¯x) with ¯x ∈ Ω as A(¯x) = f (x^k) − f (¯x)

kx^k− ¯xk : for all sequence {x^k} in Ω converging to ¯x



where we use the convention ⁰₀ = 1. Suppose A(¯x) is bounded. Then, the function f is H-differentiable at ¯x.

In the following, we present our first main result which says the H-differentiability of the L¨owner function f^sc implies that of f , and we also give a counter-example to show that the converse may not be true in general.

Theorem 2.1 Let f : IR → IR and f^sc be the corresponding L¨owner function defined in (1). Suppose f^sc is H-differentiable at x with x = λ₁(x)c₁+ λ₂(x)c₂+ · · · + λ_r(x)c_r. Then, f is H-differentiable at λi(x) for 1 ≤ i ≤ r.

Proof. We argue it by contradiction. Suppose that f is not H-differentiable at λ₁ = λ₁(x). From Remark 2.1(iv), there exists a sequence λ^k₁ converging to λ₁ such that

m_k = f (λ^k₁) − f (λ₁)

λ^k₁ − λ₁ → ∞ or − ∞. (5)

Define x^k = λ^k₁c₁+ λ₂(x)c₂ + · · · + λ_r(x)c_r. Then, we know x^k → x. By direct compu- tation, we also have

f^sc(x^k) − f^sc(x) = m_k(x^k− x)

where x^k−x = (λ^k₁−λ₁)c₁. Because f^sc is H-differentiable at x, there exist a subsequence x^kj and A ∈ IR^n×n such that

m_kj(x^kj − x) − A(x^kj − x)

kx^kj − xk → 0. (6)

For simplicity, we denote y_j = ^xkj−x

kx^kj−xk. In addition, by noting that the norm of ^xkj−x

kx^kj−xk

is 1, without lost of generality, we may assume that the sequence y_j converges to a y.

Now, with Ay_j → Ay and (6), we obtain

m_kjy_j → Ay

which contradicts (5). Similar arguments apply for the other λ_i(x). Thus, the proof is complete. 2

It is natural to ask whether the reverse implication holds or not. Unfortunately, the answer is uncertain. Here is a counter-example in SOC case. Consider a point x = (x₁, 0) ∈ IR × IRⁿ⁻¹ and h = (h₁, h₂) ∈ IR × IRⁿ⁻¹. First, we write down the difference

f^soc(x + h) − f^soc(x)

= h

f (x₁ + h₁− kh₂k) − f (x₁)i

v⁽¹⁾+h

f (x₁+ h₁+ kh₂k) − f (x₁)i

v⁽²⁾ (7) where v⁽ⁱ⁾ = ¹₂(1, (−1)ⁱh₂/kh₂k) for i = 1, 2. Suppose f is H-differentiable at x₁ = λ₁(x) = λ₂(x) and given an arbitrary sequence h^k = (h^k₁, h^k₂) converging to 0. By the definition of H-differentiability, there exist subsequence h^kj and real numbers a₁, a₂ ∈ T_f(x₁) such that

f

x₁+ h^k₁^j− kh^k₂^jk

− f (x₁) − a₁

h^k₁^j − kh^k₂^jk

= o

h^k₁^j − kh^k₂^jk , f

x₁+ h^k₁^j + kh^k₂^jk

− f (x₁) − a₂

h^k₁^j + kh^k₂^jk

= o

h^k₁^j + kh^k₂^jk . From the inequality√

a²+ b² ≥ ¹₂|a ± b|, it is not hard to verify that o(h^k₁^j ± kh^k₂^jk) are also small “o” function o(kh^kjk) of kh^kjk. Plugging these into equation (7), we have

f^soc(x + h^kj) − f^soc(x)

= h

f (x₁+ h^k₁^j− kh^k₂^jk) − f (x₁)i

v⁽¹⁾+h

f (x₁+ h^k₁^j+ kh^k₂^jk) − f (x₁)i v⁽²⁾

= a₁+ a₂ 2



h^k₁^j, h^k₂^j



+ a₂− a₁

2 kh^k₂^jk, h₁ h^k₂^j kh^k₂^jk

!

+ o(kh^kjk).

The first term is linear with respect to h^kj = (h^k₁^j, h^k₂^j), but the second term is the trouble one which is nonlinear in general when a1 6= a2. This is the trouble place that the reverse implication cannot be guaranteed at present.

3 A merit function for SCCP with H-differentiable functions

Recently, applications to nonlinear complementarity problems (NCP) and variational inequalities under H-differentiability have been considered in [25, 27, 28]. In this sec- tion, similar applications are extended to SCCP under H-differentiable condition which is a wider class of SCCPs than traditional SCCPs.

The formulations of SCCPs is to find x, y ∈ V and ζ ∈ V such that

hx, yi = 0, x ∈ K, y ∈ K, (8)

x = F (ζ), y = G(ζ), (9)

where h·, ·i is the Euclidean inner product, F : V → V and G : V → V are H- differentiable mappings, V is the Cartesian product of simple Jordan algebras, and K is the Cartesian product of corresponding symmetric cones, i.e.,

V = V1× · · · × VN and K = K₁× · · · × K_N.

Here each n_i-dimensional space Vi is a simple Jordan algebra with n₁, · · · , n_N ≥ 1, n₁ + · · · + n_N = n, and

K_i :=x²_i | x_i ∈ Vi .

For any x, y ∈ V, we write x = (x1, . . . , x_N), y = (y₁, . . . , y_N) with x_i, y_i ∈ Vi. Then, x ◦ y = (x₁◦ y₁, · · · , x_N◦ y_N) and hx, yi = hx₁, y₁i + · · · + hx_N, y_Ni. Therefore, the SCCP is equivalent to finding an ζ ∈ V such that

Fi(ζ) ∈ Ki, Gi(ζ) ∈ Ki, hFi(ζ), Gi(ζ)i = 0, i = 1, 2, · · · , N. (10) An important special case of SCCP corresponds to G(ζ) = ζ for all ζ ∈ V, namely, (8)-(9) reduces to

hF (ζ), ζi = 0, F (ζ) ∈ K, ζ ∈ K. (11)

Next, we turn into the merit function approach for SCCP under H-differentiable condition. To this end, we recall that a smooth function ψ : V × V → IR+ is called a merit function if

ψ(x, y) = 0 ⇐⇒ (x, y) satisfies (8).

A popular merit function is

ψ_FB(x, y) = 1

2kφ_FB(x, y)k² (12)

where φ_FB : V × V → V is the well-known Fisher-Burmeister (FB) complementarity function defined by

φ_FB(x, y) = (x²+ y²)¹² − x − y. (13) It is known that φ_FB(x, y) = 0 if and only if (x, y) satisfies (8) by [14, Proposition 6].

With this fact, the SCCP can be expressed as an unconstrained (global) minimization problem associated with the merit function ψ_FB:

min

ζ∈V f (ζ) := ψ_FB(F (ζ), G(ζ)). (14) The following proposition describes what we just mentioned.

Proposition 3.1 [18, Lemma 2.2] Let φ_FB and ψ_FB be defined as in (13) and (12), re- spectively. Then, ψ_FBis continuously differentiable everywhere. Furthermore, ∇_xψ_FB(0, 0) =

∇_yψ_FB(0, 0) = 0; and if (x, y) 6= (0, 0),

∇_xψ_FB(x, y) =

L_xL⁻¹_(x2+y2)1/2 − I

φ_FB(x, y),

∇_yψ_FB(x, y) =

L_yL⁻¹_(x2+y2)1/2 − I

φ_FB(x, y), where I denotes the identity operator from V to V.

Lemma 3.1 [18, Proposition 3.3] Let φ_FB and ψ_FB be defined as in (13) and (12), respectively. Then, for any (x, y) ∈ V, the following hold.

(a) h∇_xψ_FB(x, y), ∇_yψ_FB(x, y)i ≥ 0, with equality holding if and only if φ_FB(x, y) = 0.

(b) ψ_FB(x, y) = 0 ⇐⇒ ∇_xψ_FB(x, y) = 0 ⇐⇒ ∇_yψ_FB(x, y) = 0.

Let T_F(ζ) and T_G(ζ) denote the H-differentials of F and G, respectively. Since a Fr´echet differentiable function is H-differentiable, by Proposition 3.1 and using the chain rule for H-differentiable functions, the H-differential of f defined as in (14) can be written as

T_f(ζ) (15)

= {M ∇xψ_FB(F (ζ), G(ζ)) + N ∇yψ_FB(F (ζ), G(ζ)) | M ∈ TF(ζ), N ∈ TG(ζ)} . Now, we present the main result for merit function approach which indicates un- der what condition every stationary point of (14) is a solution of the SCCP with H- differentiable condition. This is answered in Proposition 3.2 whereas Proposition 3.3 provides a descent direction for non-stationary point. To establish it, we need the defi- nition of the Cartesian P₀-property for a linear transformation from V to V. Specifically, a linear transformation Υ : V → V is said to have the Cartesian P0-property if for any 0 6= ζ = (ζ₁, · · · , ζ_N) ∈ V, there exists an index ν ∈ {1, 2, . . . , N } such that ζν 6= 0 and hζ_ν, (Υζ)_νi ≥ 0.

Proposition 3.2 Let φ_FB and ψ_FB be defined as in (13) and (12), respectively, and f be given by (14). Suppose F and G are H-differentiable and the H-differentials of F and G satisfy one of the following conditions:

(i) for every ζ ∈ V, ∀M ∈ TF(ζ), N ∈ T_G(ζ), M, −N are column monotone, i.e., for any u, v ∈ V,

M u + (−N )v = 0 =⇒ hu, vi ≥ 0. (16)

(ii) for every ζ ∈ V, ∀M ∈ TF(ζ), N ∈ T_G(ζ), N is invertible and N⁻¹M has the Cartesian P0-property.

Then, there hold

0 ∈ T_f(ζ) ⇐⇒ ψ_FB(F (ζ), G(ζ)) = 0

Proof. “⇐=” This direction is easy to verify. To see this, from Proposition 3.1(b), it is clear that ψ_FB(F (ζ), G(ζ)) = 0 implies

(∇_xψ_FB(F (ζ), G(ζ)), ∇_yψ_FB(F (ζ), G(ζ))) = 0, which yields Tf(ζ) = {0} by applying (15).

“=⇒” For this direction, suppose that 0 ∈ T_f(ζ). From (15), there exist M ∈ T_F(ζ) and N ∈ T_G(ζ) such that

M ∇_xψ_FB(F (ζ), G(ζ)) + N ∇_yψ_FB(F (ζ), G(ζ)) = 0. (17) (a) If condition (i) is satisfied, from the column monotonicity of M and −N , we know

h∇_xψ_FB(F (ζ), G(ζ)), ∇_yψ_FB(F (ζ), G(ζ))i ≤ 0.

This together with Lemma 3.1(b) implies ψ_FB(F (ζ), G(ζ)) = 0.

(b) If condition (ii) is satisfied. From (17), we have

N⁻¹M ∇_xψ_FB(F (ζ), G(ζ)) + ∇_yψ_FB(F (ζ), G(ζ)) = 0. (18) For any u = (u₁, . . . , u_N), v = (v₁, . . . , v_N) ∈ V with ui, v_i ∈ Vi, we write

∇xψ_FB(u, v) = (∇x1ψ_FB(u1, v1), · · · , ∇xNψ_FB(uN, vN)) ,

∇_yψ_FB(u, v) = (∇_y1ψ_FB(u₁, v₁), · · · , ∇_yNψ_FB(u_N, v_N)) .

Assume that ζ is not a solution of ψ_FB(F (ζ), G(ζ)) = 0, applying Lemma 3.1(b) gives

∇_xψ_FB(F (ζ), G(ζ)) 6= 0 and ∇_yψ_FB(F (ζ), G(ζ)) 6= 0. (19) By the Cartesian P₀-property of N⁻¹M , there exists an index ν ∈ {1, 2, . . . , N } such that ∇_xνψ_FB(F_ν(ζ), G_ν(ζ)) 6= 0 and

∇_xνψ_FB(F_ν(ζ), G_ν(ζ)),N⁻¹M ∇_xψ_FB(F_ν(ζ), G_ν(ζ))

ν ≥ 0.

On the other hand, it follows from (18) that

∇xνψ_FB(F_ν(ζ), G_ν(ζ)),N⁻¹M ∇_xψ_FB(F_ν(ζ), G_ν(ζ))

ν

= − h∇xνψ_FB(Fν(ζ), Gν(ζ)), ∇yνψ_FB(Fν(ζ), Gν(ζ))i . Combining last two equations and using Lemma 3.1(a) yield

h∇_xνψ_FB(F_ν(ζ), G_ν(ζ)), ∇_yνψ_FB(F_ν(ζ), G_ν(ζ))i = 0.

Hence, φ_FB(F_ν(ζ), G_ν(ζ)) = 0 which contradicts (19). Therefore, under condition (ii), we prove that 0 ∈ T_f(ζ) implies ψ_FB(F (ζ), G(ζ)) = 0. 2

Remark 3.1 A different merit function for the the SCCP with F and G being H- differentiable is considered in [24]. In fact, [24, Theorem 4.1] also provides a condition, under which any stationary point of merit function is a solution of the SCCP. However, that condition is stricter than condition (ii) in Proposition 3.2.

Proposition 3.3 Let φ_FB and ψ_FB be defined as in (13) and (12), respectively, and f be given by (14). Suppose F and G are H-differentiable and the H-differentials of F and G satisfy assumption (16). In the case of 0 /∈ T_f(ζ), if there exists ¯N ∈ T_G(ζ) which is invertible, then

d_FB(ζ) := −( ¯N⁻¹)^T∇_xψ_FB(F (ζ), G(ζ)) is a descent direction of f at ζ.

Proof. From the definition of a descent direction for an H-differentiable function at a point, it is sufficient to prove that for some M ∈ T_F(ζ) and N ∈ T_G(ζ), there holds

hd_FB(ζ), M ∇_xψ_FB(F (ζ), G(ζ)) + N ∇_yψ_FB(F (ζ), G(ζ))i < 0. (20) In fact, for any M ∈ TF(ζ) (dropping the argument “(F (ζ), G(ζ))” for simplicity), we have

d_FB(ζ), M ∇_xψ_FB + ¯N ∇_yψ_FB

= −( ¯N⁻¹)^T∇_xψ_FB, M ∇_xψ_FB + ¯N ∇_yψ_FB

= −∇xψ_FB, ¯N⁻¹M ∇xψ_FB − h∇xψ_FB, ∇yψ_FBi

≤ − h∇_xψ_FB, ∇_yψ_FBi ,

where the inequality follows from the fact ¯N⁻¹M is a positive semi-definite matrix (this is guaranteed from the invertibility of ¯N and assumption (16)). By Lemma 3.1(b), the right-hand side is non-positive and equals to zero if and only if ψ_FB(F (ζ), G(ζ)) = 0. On the other hand, applying Proposition 3.2 gives

ψ_FB(F (ζ), G(ζ)) = 0 ⇐⇒ 0 ∈ T_f(ζ).

Therefore, in the case of 0 /∈ T_f(ζ), the right-hand cannot equal zero, so it must be negative. Thus, (20) is satisfied which says d_FB(ζ) is a descent direction. 2

It is known that the SCCP is closely related to the Karush-Kuhn-Tucker (KKT) optimality conditions for the convex symmetric cone program (CSCP):

min g(x)

s.t. Ax = b, x ∈ K (21)

where A : V → IR^m is a linear operator, b ∈ IR^m and g : V → IR is a convex and smooth (continuously differentiable) function with its gradient mapping ∇g : V → V being H-differentiable. Especially, when K is a second order cone, the assumption of g being a convex and smooth function is equivalent to the condition ∇g : IRⁿ→ IRⁿbeing a monotone function. Furthermore, if ∇g satisfies either

(a) ∇g is Fr´echet differentiable on IRⁿ, or (b) ∇g is locally Lipschitzian on IRⁿ,

then ∇g is H-differentiable at any x ∈ IRⁿ. For each case, the H-differential of ∇g is the set {∇²g(x)} and the generalized Jacobian

∂(∇g)(x) = convn

k→∞lim ∇²g(x^k) | x^k ∈ D∇g, x^k → xo ,

respectively, where D_∇gdenotes the set Fr´echet differentiable points of ∇g in IRⁿ. In par- ticular, under each of the aforementioned two cases, it is well known that the convexity of g (or the monotonicity of ∇g) is equivalent to the conclusion that the H-differentials of ∇g consist of positive semi-definite (p.s.d.) matrices, see [15, Proposition 2.3]. In summary, under the smoothness of g,

g is convex

⇐⇒ ∇g is monotone (22)

⇐⇒ the H-differential of ∇g consists of p.s.d. matrices for case (a) or (b).

The above discussions in SOC case raise the motivation of investigating such equiv- alences in general case, i.e., is (22) true in general? In fact, for the general case without convexity of g, one direction is known true, i.e., if the H-differential of ∇g consists of positive semi-definite matrices then ∇g is a monotone function, see the following theorem.

Theorem 3.1 [13, Theorem 4] If h : IRⁿ→ IRⁿis H-differentiable at each point x ∈ IRⁿ with T_h(x) consisting of positive semi-definite (positive definite) matrices, then h is a monotone (strictly monotone) function.

Theorem 3.1 indicates that the H-differential of an H-differentiable function consist- ing of positive semi-definite matrices provides a sufficient condition for monotonicity of this function. However, the H-differential consisting of positive semi-definite matrices is not a necessary condition, namely, we don’t know whether the opposite side of Theorem 3.1 is true or not. What can we achieve for necessary case? Below are results describing the opposite direction of Theorem 3.1.

Proposition 3.4 Suppose h : IRⁿ→ IRⁿ is a monotone and H-differentiable mapping.

(a) For n = 1, if a ∈ T_h(x) for a point x ∈ IR, then a ≥ 0.

(b) For n ≥ 2, if A ∈ T_h(x) for a point x ∈ IRⁿ, then A is positive semi-definite in the subspace E ⊆ IRⁿ where

E = {d ∈ IRⁿ| d satisfies (24)}. (23) In particular, from the equivalent definition of Definition 2.1, for A ∈ T_h(x), there exists some d ∈ IRⁿ satisfying

j→∞lim

h(x + t_kjd^kj) − h(x) tkj

= Ad (24)

for some sequence x^k := x + t_kd^k with t_k ↓ 0 and kd^kk = 1 for all k, and for any convergent subsequence t_kj ↓ 0 and d^kj → d.

Proof. (a) From Definition 2.1, we have

h(x + t_kjd^kj) − h(x) = at_kjd^kj+ o t_kj

for some sequence x^k := x + t_kd^k with t_k ↓ 0 and kd^kk = 1 for all k, and for any convergent subsequence tkj ↓ 0 and d^kj → d, where d = 1 or −1 in this case. By the monotonicity of h,

0 ≤ h(x + t_kjd^kj) − h(x)
t_kjd^kj = at²_kj d^kj
2

+ o t²_kj

= at²_kj + o t²_kj which implies a ≥ 0.

(b) From equation (24), we know

h(x + t_kjd^kj) − h(x) = At_kjd^kj+ o t_kj
. By the monotonicity of h again,

0 ≤ hh(x + t_kjd^kj) − h(x), t_kjd^kji = t²_kj(d^kj)^TAd^kj + o t²_kj

, and hence

d^TAd = lim

j→∞

t²_k

j(d^kj)^TAd^kj+ o t²_k

j

 t²_k

j

≥ 0

which says A is positive semi-definite in the subspace E defined in (23). 2

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