Differences of Polyhedra in Matrix Space and Their Applications to ...

C 2006) JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 130, No. 3, pp. 429–440, September 2006 ( DOI: 10.1007/s10957-006-9120-x

Differences of Polyhedra in Matrix Space and Their Applications to Nonsmooth Analysis1 Y. GAO2 Communicated by F. Giannessi Published Online: 3 November 2006

Abstract. Formulas of the differences of polyhedra in matrix space are proposed. Based on these formulas, the differences of polyhedra can be calculated by solving systems of linear inequalities. A modified algorithm for calculating one element of the differences is presented also. The motivation for this work is to compute the Clarke generalized Jacobian, the B-differential, and one of their elements via the quasidifferential. Applications to Newton methods for solving nonsmooth equations are discussed. Key Words. Nonsmooth analysis, quasidifferential calculus, difference of convex compact sets, Clarke generalized Jacobian, nonsmooth equations.

1. Introduction The Demyanov difference and the Rubinov difference for convex compact sets in n play important roles in quasidifferential calculus. They serve mainly for the description of the relation between the Clarke generalized gradient and the quasidifferential; see for instance Refs. 1–7. In Ref. 8, the notions of these differences were extended to the matrix space m×n . Based on these differences, representations of the Clarke generalized Jacobian and the B-differential via the quasidifferential were developed. Nevertheless, the definitions of these differences do not allow one to express them explicitly for general sets. In the present paper, we consider the polyhedral case. We give formulas of these differences for polyhedra. Furthermore, methods for calculating these differences and one of their elements are developed. Thus, the Clarke generalized 1 This

project was sponsored by the Shanghai Education Committee, Grant 04EA01, by the Education Ministry of China, and by the Shanghai Government, Grant T0502. The author thanks two anonymous referees and Professor F. Giannessi for valuable suggestions and comments. 2 Professor, School of Management, University of Shanghai for Science and Technology, Shanghai, China. 429 C 2006 Springer Science+Business Media, Inc. 0022-3239/06/0900-0429/0

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JOTA: VOL. 130, NO. 3, SEPTEMBER 2006

Jacobian and the B-differential can be calculated via the quasidifferential for a certain class of nonsmooth functions. Throughout this paper, m×n denotes the m × n real matrix space, cl denotes the closure, co denotes the convex hull, meas is the Lebesgue measure, ∇ is the gradient, B(x, δ) is the the ball in n , with x and δ as its center and radius respectively, J is the Jacobian, DF is the set where F is differentiable; in the vector case, max and min denote componentwise maximum and componentwise minimum, respectively. ˜ x (W ) The maxface and the minface of W ⊂ m×n , denoted by Gx (W ) and G respectively, are defined by Gx (W ) = w ∈ W |wx = max wx , w∈W ˜ x (W ) = w ∈ W |wx = min wx . G w∈W

The support function PW of W ⊂ m×n is defined by PW (x) = max wx, w∈W

x ∈ n .

By Ref. 9, when W ∈ n , PW (x) is a convex function on n with ∂PW (x) = Gx (W ), where ∂ denotes the subdifferential in the sense of convex analysis; PW is differentiable at x if and only if Gx (W ) is a singleton. A set T ⊂ n is called of full measure with respect to n if n \T is a set of measure zero. Let U, V ⊂ m×n be convex compect. By virtue of Ref. 8, given a full measure set T such that the Jacobians of PU and PV exist at each point x ∈ T , · ·· and the Rubinov difference U − V of U and V the Demyanov difference U −V are defined by · = cl{J PU (x) − J PV (x) | x ∈ T }, U −V ·· U−V = [Gx (U ) − Gx (V )]. x=0

2. Formulas of the Differences · ·· In this section, we give the formulas of U −V and U − V for the polyhedra U and V . Let

[U, V ] = [U1 , V1 ] × · · · × [Um , Vm ],

(1)

where Ui , Vi ∈ for i = 1, . . . , m. n

Theorem 2.1. Let U and V be given in (1), let Ui = co{uij |j ∈ Ji }, and let Vi = co{vik |k ∈ Ki }, for i = 1, . . . , m, uij , vik ∈ n , where Ji , Ki are finite index sets. Without loss of generality, suppose that uis = vit , ∀s, t ∈ Ji , s = t,


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and vis = vit , ∀s, t ∈ Ki , s = t, for i = 1, . . . , m. Given a pair of index sets j1 ∈ J1 , . . . , jm ∈ Jm and k1 ∈ K1 , . . . , km ∈ Km , let us construct the following system of linear inequalities, denoted by (L(j1 , . . . , jm ; k1 . . . km )), with y ∈ n as the variable: (L(j1 , . . . , jm ; k1 , . . . , km ))

Then,

· U −V =

T u1s − u1j1 y < 0, ... ... ... ... T u − umjm y < 0, T ms v1t − v1k1 y < 0, ... ... ... ... T vmt − vmkm y < 0,

u1j1 − v1k1 , . . . , umjm − vmkm

∀s ∈ J1 \{j1 }, ... ... ... ..., ∀s ∈ Jm \{jm }, ∀t ∈ K1 \{k1 }, ... ... ... ..., ∀t ∈ Km \{km }.

T (L(j , . . . , j ; k , . . . , k )) is 1 m 1 m . consistent jp ∈ Jp , kp ∈ Kp , 1 ≤ p ≤ m (2)

Proof.

Let

y solves (L(j1 , . . . , jm ; k1 , . . . , km ))}, (j1 , . . . jm ; k1 , . . . , km ) = {y ∈ jp ∈ Jp , kp ∈ Kp , p = 1, . . . , m, n

and Uij s = {y ∈ n |(uij − uis )T y = 0}, Vikt = {y ∈ n |(vik − vit )T y = 0}, Ji (y) = j ∈ Ji |uTij y = max uTis y , s∈Ji T y = max vitT y , Ki (y) = k ∈ Ki |vik t∈Ki

j, s ∈ Ji , i = 1, . . . , m, k, t ∈ Ki , i = 1, . . . , m, y ∈ n , i = 1, . . . , m, y ∈ n , i = 1, . . . , m.

We prove next the following assertion: ⎛ ⎞ meas ⎝n \ (j1 , . . . , jm ; k1 , . . . , km )⎠ = 0. jp ∈Jp ,kp ∈Kp

Let y∈ / n \

(j1 , . . . , jm ; k1 , . . . , km ).

jp ∈Jp ,kp ∈Kp

This means that y∈ /\ jp ∈Jp ,kp ∈Kp

(j1 , . . . , jm ; k1 , . . . , km ).

(3)

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Moreover, y∈ / (j1 , . . . , jm ; k1 , . . . , km ),

∀jp ∈ Jp , kp ∈ Kp , p = 1, . . . , m.

Hence, there exists an index 1 ≤ i ≤ m such that at least one index set between Ji (y) and Ki (y) is not a singleton. Otherwise, both Ji (y) and Ki (y) are singletons for i = 1, . . . , m, say, J1 (y) = {j1 }, . . . , Jm (y) = {jm }, K1 (y) = {k1 }, . . . , Km (y) = {km }. Thus, y = y solves (L(j1 , . . . , jm ; k1 , . . . , km )). This contradicts y ∈ / (j1 , . . . , jm ; k1 , . . . , km ). We might as well assume that there exists an index 1 ≤ i0 ≤ m such that Ji0 (y) is not a singleton. This entails that there exist j0 , s0 ∈ Ji0 (y), j0 = s0 , such that y ∈ Ui0 j0 s0 . As a consequence, ⎛ ⎞ ⎝ y∈ Uij s ⎠ . 1≤i≤m

j,s∈ji

We conclude that n \

(j1 , . . . , jm ; k1 , . . . , km )

jp ∈Jp ,kp ∈Kp

⊂

1≤i≤m

Uij s

j,s∈Ji

Vikt

.

(4)

k,t∈Ki

The assumptions that uis = uit , ∀s, t ∈ Ji , s = t and vis = vit , ∀s, t ∈ Ki , s = t, for i = 1, . . . , m, yield that Uij s and Vikt are (n − 1)-dimensional subspaces in n , so that their measures are zero. Since both Ji and Ki are finite index sets, one has that

·Uij s Vikt meas = 0. (5) 1≤i≤m

j,s∈Ji

k,t∈Ki

Combining (4) with (5) leads to (3). It is not hard to see that y ∈ (j1 , . . . , jm ; k1 , . . . , km ) implies that Gy (Ui ) = {uiji },

Gy (Vi ) = {viki },

i = 1, . . . , m.

Therefore, PUi and PVi are differentiable at y with ∇PUi (y) = uiji and ∇PVi (y) = viji for i = 1, . . . , m; morcover, PU and PV are differentiable at y with J PU (y) = (u1j1 , . . . , umjm )T and J PV (y) = (v1k1 , . . . , vmkm )T . Hence, PU and PV are differentiable in (j1 , . . . , jm ; k1 , . . . , km ) for j1 ∈ J1 , . . . , jm ∈ Jm and k1 ∈ K1 , . . . , km ∈ Km . Taking


T =

433

(j1 , . . . , jm ; k1 , . . . , km )

jp ∈Jp ,kp ∈KP · in the definition of U −V , we obtain (2). This completes the proof of the theorem.

Remark 2.1. Each system (L(j1 , . . . , jm ; k1 , . . . , km )) has cardKi − 2) strictly linear inequalities with n variables.

m i=1

(cardJi +

· Theorem 2.1 enables us to calculate the set U −V by determining the consistency of the system (L(j1 , . . . , jm ; k1 , . . . , km )) for each pair of index sets j1 ∈ J1 . . . , jm ∈ Jm and k1 ∈ K1 , . . . , km ∈ Km . If the system (L(j1 , . . . , jm ; k1 , . . . , km )) is consistent, then T · ; u1j1 − v1k1 , . . . , umjm − vmkm ∈ U −V

otherwise, T · u1j1 − v1k1 , . . . , umjm − vmkm ∈ U −V . In this way, the consistency of m i=1 (cardJi × cardKi ) systems of strictly linear inequalities needs to be determined. Determining the consistency of a system of strictly linear inequalities can be transformed into solving a linear programming problem (see Ref. 4). Theorem 2.2. Suppose that both U and V are given in Theorem 2.1. For a pair of index sets j1 ∈ J1 , . . . , jm ∈ Jm and k1 ∈ K1 , . . . km ∈ Km , let us construct the system of linear inequalities (M(j1 , . . . jm ; k1 . . . , km )) as follows, with the y ∈ n as the variable: (M(j1 , . . . jm ; k1 , . . . km ))

(u1s − u1j1 )T ≤ 0, ... ... ... (ums − umjm )T y ≤ 0, (v1t − v1k1 )T y ≤ 0, ... ... ... (vmt − vmkm )T y ≤ 0,

∀s ∈ J1 \{j1 }, ... ..., ∀s ∈ Jm \{jm }, ∀t ∈ K1 \{k1 }, ... ..., ∀t ∈ Km \{km }.

Then, one has that

T (u1j1 − v1k1 , . . . , umjm − vmkm )| (M(j1 , . . . , jm ; k1 , . . . , km )) has nonzero ¨ = U −V . solutions, jp ∈ Jp , kp ∈ Kp , 1 ≤ p ≤ m (6)

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Proof. Given a pair of index sets j1 ∈ J1 , . . . , jm ∈ Jm and k1 , . . . , km ∈ Km , let us construct the two systems of linear incqualities as follows: (Uj1 ,...,jm )

(u1s − u1j1 )T y ≤ 0, ... ... ... (ums − umjm )T y ≤ 0,

∀s ∈ J1 \{j1 }, ... ..., ∀s ∈ Jm \{jm },

(Vk1 ,...,km )

(v1t − u1k1 )T y ≤ 0, ... ... ... (vmt − vmkm )T y ≤ 0,

∀t ∈ K1 \{k1 }, ... ..., ∀t ∈ Km \{km },

where y ∈ n . According to the definition of the maxface, we have that Gx (U ) = {(u1j1 , . . . , umjm )T | uTiji x = max uTis x, ji ∈ Ji , i = 1, . . . , m} s∈Ji ⎧ ⎫ (u1s − u1j1 )T x ≤ 0, ∀s ∈ J1 \{j1 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ... ... ... ... ..., T ... = (u1j1 , . . . , umjm ) T (ums − umjm ) x ≤ 0, ∀s ∈ Jm \{jm } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ji ∈ Ji , i = 1, . . . , m = {(u1j1 , . . . , umjm )T | x solves (Uj1 ,...,jm ), jp ∈ Jp , p = 1, . . . m}. Likewise, it is obtained that Gx (V ) = {(v1k1 , . . . , vmkm )T

| x solves(Vk1 ,...,km ), kp ∈ Kp , p = 1, . . . , m}.

Since the system (M(j1 , . . . , jm ; k1 , . . . , km )) is the combination of the systems (Uj1 , . . . ,j m ) and (Vk1 ,...,km ), it follows that Gx(U ) − Gx (V )

x solves (Uj1 ,...,jm ), (Vk1 ,...,km ), = (u1j1 , . . . , umjm )T − (v1k1 , . . . , vmkm )T jp ∈ Jp , kp ∈ Kp , p = 1, . . . , m T x solves (M(j1 , . . . , jm ; k1 , . . . , km )), . = (u1j1 − v1k1 , . . . , umjm − vmkm ) jp ∈ Jp , kp ∈ Kp , 1 ≤ p ≤ m Furthermore, ¨ = U −V [Gx (U ) − Gx (V )]

x=0

x = 0 solves (M(j1 , . . . , jm ; k1 , . . . , km )), = (u1j1 − v1k1 , . . . , umjm − vmkm )T jp ∈ Jp , kp ∈ Kp , 1 ≤ p ≤ m ⎫ ⎧ ⎨ (M(j1 , . . . , jm ; k1 , . . . , km )) ⎬ T ; = (u1j1 − v1k1 , . . . , umjm − vmkm ) has nonzero solutions, ⎭ ⎩ jp ∈ Jp , kp ∈ Kp , 1 ≤ p ≤ m

i.e., (6) holds. This completes the proof of the theorem.


Evidently, each system (M(j1 , . . . , jm k1 , . . . , km )) has cardKi − 2) linear inequalities with n variables.

435

m

i=1 (cardJi

+

3. Calculating One Element of the Demyanov Difference ˙ is developed. Nevertheless, In Section 2, a method of calculating the set U −V ˙ is this calculation is quite time consuming. Actually, only one element of U −V required in some applications. In this section, we develop a robust approach to ˙ when both U and V are given as in Theorem 2.1. calculating an element of U −V For simplicity, from here on we denote ui+m k = vik ,

Ji+m = Ki ,

Ui+m = Vi ,

i = 1, . . . , m,

and p = 2m. Thus, Ui+m = Vi = co{ui+m |j ∈ Ji+m },

i = 1, . . . , m,

and denote jm+1 = k1 , . . . , jp = km ; thus, k 1 ∈ K1 , . . . , k m ∈ Km . Then, the system (L(j1 , . . . jm ; k1 , . . . km )) is rewritten as (Lj1 ,...,jp )

(u1s − u1j1 )T y < 0, ... ... ...

∀s ∈ J1 \{j1 }, ... ...,

(ups − upjp )T y < 0,

∀s ∈ Jp \{jp }.

Lemma 3.1. Suppose that ai ∈ n , i ∈ I and aj = ak , ∀j, k ∈ I, j = k, where I is a finite index set. Assume that i0 ∈ I satisfies ai0 = maxi∈I ai . Then, aiT ai0 < aiT0 ai0 ,

∀i ∈ I \{i0 }.

Given a set of vectors yi ∈ n for i = 1, . . . , p, define some index sets as follows: J1 (y1 ) = j ∈ J1 uT1j y1 > uT1s y1 , ∀s ∈ J1 \{j } , (7a) T T Ji (y1 ) = j ∈ Ji uij y1 = max uis y1 , i = 2, . . . , p, (7b) s∈Ji (7c) Ji (y1 , y2 ) = j ∈ Ji (y1 ) uTij y2 = max uTis y2 , i = 2, . . . , p, s∈Ji (y1 )

436


Ji (y1 , . . . , yi−1 ) = j ∈ Ji (y1 , . . . , yi−2 ) uTij yi−1 =

max

s∈Ji (y1 ,...,yi−2 )

i = 3, . . . , p, Ji (y1 , . . . , yi ) = j ∈ Ji (y1 , . . . , yi−1 ) uTij yi > uTis yi , ∀s ∈ Ji (y1 , . . . , yi−1 )\{j } , i = 2, . . . , p.

uTis yi−1 , (7d) (7e)

Obviously, for any set of vectors yi ∈ n , i = 1, . . . , m, each index set Ji (y1 , . . . , yj ), 1 ≤ j < i, is nonempty, whereas the index sets Ji (y1 , . . . , yi ), i = 1, . . . , p, are not guaranteed to be nonempty. However, by definition, it is easy to verify that Ji (y1 , . . . , yj ) ⊂ Ji (y1 , . . . , yj −1 ), 2 ≤ j ≤ i, (uis − uij )T yt ≤ 0, ∀s ∈ Ji , j ∈ Ji (y1 , . . . , yi ),

t ≤ i.

(8a) (8b)

Theorem 3.1. Let y1 , . . . , yp ∈ n , let Ji (y1 , . . . , yi ) for i = 1, . . . , p, be nonempty, and let ji ∈ Ji (y1 , . . . , yi ) for i = 1, . . . , p. Then, the system (Lj1 , . . . ,jp ) is consistent. Proof. We will prove that there exist scalars i > 0 for i = 2, . . . , p such that y1 + 2 y2 + · · · + p yp solves the system (Lj1 , . . . ,jp ). We use mathematical induction on p. In the case p = 1, the definition of J1 (y1 ) yields (u1s − u1j1 )T y1 < 0,

∀s ∈ J1 \{j1 },

that is to say y1 solves (Lj 1 ). Now, we suppose that there exist scalars i > 0 for i = 2, . . . , p − 1 such that y1 + 2 y2 + · · · + p−1 yp−1 solves (Lj1 , . . . , j p−1 ), i.e., (uis − uiji )T (y1 + 2 y2 + · · · + p−1 yp−1 ) < 0, ∀s ∈ Ji \{ji },

i = 1, . . . , p − 1.

Notice that the solution set of the system of linear inequalities Ay < 0, y ∈ n is an open convex cone in n (Ref. 9), where A ∈ m×n . Therefore, the solution set of (Lj1 ,...,jp−1 ) is an open convex cone. Then, there exists a small number (1) > 0 such that 0 < p < (1) implies that (uis − uiji )T (y1 + 2 y2 + · · · + p yp ) < 0, ∀s ∈ Ji \{ji }, i = 1, . . . , p − 1.

(9)

From the definition of Jp (y1 , . . . , yp ), it follows that (ups − upjp )T yp < 0,

∀s ∈ Jp (y1 , . . . , yp−1 )\{jp }.

(10)


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The inequalities (8) and (10) yield (ups − upjp )T (y1 + 2 y2 + · · · + p yp ) < 0, ∀s ∈ Jp (y1 , . . . , yp−1 )\{jp }.

(11)

Notice that Jp (y1 , . . . , yp ) ⊂ Jp (y1 , . . . , yp−1 ) ⊂ · · · ⊂ Jp (y1 ) ⊂ Jp . Therefore, an index s ∈ (Jp \Jp (y1 , . . . , yp−1 ))\{jp } leads to two cases. In the first case, there exists an index t with 2 ≤ t ≤ p − 1 (where t depends on s) such that s ∈ Jp (y1 , . . . , yt−1 )\{jp },

s ∈ Jp (y1 , . . . , yt )\{jp }.

The definition of Jp (y1 , . . . , yt ) implies that (ups − upjp )T yt < 0. In the second case, s ∈ Jp (y1 )\{jp }. Thus, (ups − upjp )T y1 < 0. To sum up, for any s ∈ (Jp \Jp (y1 , . . . , yp−1 ))\{jp }, there is an index t ≤ p − 1(t depends on s) such that (ups − upjp )T yt < 0. Therefore, there exists a small number (2) > 0 such that 0 < p < (2) implies that (ups − upjp )T (y1 + 2 y2 + · · · + p yp ) < 0, ∀s ∈ (Jp \Jp (y1 , . . . , yp−1 ))\{jp }.

(12)

Combining (11) with (12) and choosing p such that 0 < p < min (1) , (2) , one has that (ups − upjp )T (y1 + 2 y2 + · · · + p yp ) < 0,

∀s ∈ Jp \{jp }.

(13)

Furthermore, (9) and (13) lead to that y1 + 2 y2 + · · · + p yp solves the system (Lj1 ,...,jp ). By virtue of mathematical induction, the theorem holds. ˙ can be transBased on Theorems 2.1 and 3.1, calculating an element of U −V formed into finding a set of vectors yi ∈ n for i = 1, . . . , p such that all index

438


sets Ji (y1 , . . . , yi ) for i = 1, . . . , p are nonempty. This work can be implemented in the two steps listed below: (i) Choose an index j1 ∈ J1 with u1j1 = max{ u1s (x) | s ∈ J1 } and set y1 = u1j1 . (ii) Choose the indices ji ∈ Ji (y1 , . . . , yi−1 ) with uiji = max{ uis |j ∈ Ji (y1 , . . . , yi−1 ),

i = 2, . . . p,

and set yi = uiji for i = 2, . . . , p. According to Lemma 3.1, the set of indices ji ∈ Ji , for i = 1, . . . , p, chosen according to the above principle guarantees that ji ∈ Ji (y1 , . . . , yi ), for i = 1, . . . , p. Now, we present the detailed algorithmic scheme. Algorithm 3.1. Step 0. Given the integer p, the index sets Ji , i = 1, . . . , p, and the vectors uij ∈ n , j ∈ Ji , i = 1, . . . , p, set l = 1. Step 1. Determine the index set Jl = j ∈ Jl | ulj = max uls s∈Jl

and choose an index jl ∈ Jl . Step 2. If l = p, calculate (u1j1 − um+1jm+1 , . . . , umjm − upjp )T , an element ˙ , and stop; otherwise, continue. of U −V Step 3. Determine the index sets Ji = j ∈ Ji uTij uljl = max uTis uljl , i = l + 1, . . . , p. s∈Ji

Step 4. Set Ji = Ji , i = l + 1, . . . , p. Replace l by l + 1 and loop to Step 1. ˙ is obtained by executing Algorithm Remark 3.1. An element of U −V 3.1. pThe number of multiplications performed in Algorithm 3.1 is less than n i=1 i cardJi . 4. Applications It is well known that many that existing numerical methods for the solution of nonsmooth equation, for instance Newton methods, are proposed under the condtion that at least one element of the Clarke gneralized Jacobian or the Bdifferential for the corresponding funciton at each iteration point can be computed (Ref. 10). However, this computation cannot be performed for some more or less complicated functions. On the other hand, the quasidifferentials can be computed easily for a large class of functions. So, it is probably the investigation of the


439

Clarke generlized Jacobian and the B-differential via the quasidifferential for some functions. As in Ref. 3, F : n → m is called quasidifferentiable at x ∈ n , in the sense of Demyanov and Rubinov, if it is directionally differentiable at x and there exists a pair of convex compact sets ∂F (x), ∂F (x) ⊂ m×n such that F (x; d) = max vd + min wd, v∈∂F (x)

w∈∂F (x)

∀d ∈ n .

The pair of sets DF (x) = [∂F (x), [∂F (x)] is called the quansidifferential of F at x. Suppose that F : n → m is locally Lipschitzain. By the definition in Ref. 10, ∂B F (x) = lim J F (xn ) | xn → x, xn ∈ DF xn →x

is called the B-differential of F at x. According to the definition in Ref. 3, ∂Cl F (x) = co∂B F (x) is called the Clarke generalized Jacobian of F at x. In light of Ref. 8, one has that ˙ (x)) ⊂ ∂B F (x), ∀F ∈ M(x), ∂F (x)−(−∂F ¨ ∂B F (x) ⊂ ∂F (x)−(−∂F (x)), ∀F ∈ M(x),

(14) (15)

are families of functions F from an open set X ⊂ n , with where M(x) and M(x) m x ∈ X to ; for the definitions, see Ref. 8. Thus, (14) and (15) can be used to estimate the B-differential and compute one of their elements when both ∂F (x) and ∂F (x) are polyhedra Consider the following nonsmooth equation: G(x) = 0,

(16)

where G : n → n is Lipschitzian. The Newton method of solving (16) is given by xk+1 = xk − Vk−1 G(xk ), where Vk is an element of ∂B G(x) (Ref. 10). So, the method given in this paper can be applied to the Newton methods for solving system of nonsmooth equations, in which their quasidifferentials can be calculated easily. References 1. DEMYANOV, V. F., On a Relation between the Clarke Subdifferential and Quasidifferential, Vestnik Leningrad University, Vol. 13, pp. 183–189, 1981.

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2. DEMYANOV, V. F., and SUTTI, C., Representation of the Clarke Subdiferential for a Regular Quasidifferentiable Function, Journal of Optimization Theory and Applications, Vol. 87, pp. 553–561, 1995. 3. DEMYANOV, V. F., and RUBINOV, A. M., Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main, Germany, 1995. 4. GAO, Y., Demyanov Difference of Two Sets and Optimality Conditions of Lagrange Multipliers Type for Constrained Quasidifferentiable Optimization, Journal of Optimization Theory and Applications, Vol. 104, pp. 377–394, 2000. 5. RUBINOV, A. M., Differences of Convex Compact Sets and Their Applications in Nonsmooth Analysis, Nonsmooth Optimization Methods and Applications, Edited by F. Giannessi, Gordon and Breach Science Publishers, Singapore, Republic of Singapore, pp. 366–378, 1992. 6. RUBINOV, A. M., and AKHUNDOV, I. S., Difference of Convex Compact Sets in the Sense of Demyanov and Its Application in Nonsmooth Analysis, Optimization, Vol. 23, pp. 179–188, 1992. 7. RUBINOV, A. M., and VLADIMIROV, A. A., Differences of Convex Compacta and Metric Spaces of Convex Compacta with Applications: A Survey, Quasidifferentiability and Related Topics, Edited by V. F. Demyanov and A. M. Rubinov, Kluwer Academic Publishers, Dordrecht, Holland, pp. 263–296, 2000. 8. GAO, Y., Representation of the Clarke Generalized Jacobian via the Quasidifferential, Journal of Optimization Theory and Applications, Vol. 123, pp. 519–532, 2004. 9. HIRIART-URRUTY, J. B., and LEMARECHAL, C., Convex Analysis and Minimization Algorithms, Springer Verlag, Berlin, Germany, 1993. 10. QI, L., Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations, Mathematics of Operations Research, Vol. 18, pp. 227–253, 1993.