For Peer Review Only
On optimality conditions for nonsmooth vector problems in normed spaces via generalized Hadamard directional
derivatives
Journal: Optimization Manuscript ID GOPT-2022-0196 Manuscript Type: Original Article Date Submitted by the Author: 02-May-2022
Complete List of Authors: Phan, Tinh; Hue University, Faculty of sciences, Department of Mathematics
Keywords:
Second order tangent sets, second order approximations, second order Hadamard directional derivative, second order optimality condition
<a href="http://www.ams.org/mathscinet/msc/msc2020.html"
target="_blank">2020 Mathematics Subject Classification</a>:
49K27, 90C30, 90C34, 90C48
URL: http:/mc.manuscriptcentral.com/gopt
For Peer Review Only
On optimality conditions for nonsmooth vector problems in normed spaces via generalized
Hadamard directional derivatives
∗Phan Nhat Tinh†
Abstract. By introducing the concepts of generalized Hadamard di- rectional derivatives, we establish first and second order optimality con- ditions for nonsmooth vector problems with set constraint in normed spaces. Our results generalize, sharpen and strengthen some recent known ones. Illustrative numerical examples are also given.
Key Words. Second order tangent sets; second order approximations;
second order Hadamard directional derivative; second order optimality condition.
AMS subject classifications. 49K27, 90C30, 90C34, 90C48
1 Introduction
The study on first and second order optimality conditions for scalar/vector nonsmooth problems still attract the attention of researchers in the last years through numerous publications, see [1-3, 6-17, 20-27, 29-30] and references therein. The main tools of the study are generalized deriva- tives [ 14 , 16 , 21 , 1 , 22 , 10 ] and generalized directional derivatives [ 2 , 3 , 13 , 20 , 29 ]. In this paper we consider the following general nonsmooth problem
minf(x) s.t. x∈S, (P)
where f : X → Y with X, Y are normed spaces, Y is ordered by a pointed, closed and convex cones C with nonempty interior and S is a nonempty subset ofY.
The behaviors of objective functions on first and second order tan- gent directions to the feasible sets at points under consideration reveal important information on optimality. Using this approach, Penot [ 28 ] investigates the optimality of the problem (P) when Y = R and f is twice differentiable. Bigi [ 4 , 5 ] presents results when X, Y are finite di- mensional spaces andf is twice differentiable. Jimenes and Novo [ 18 , 19 ] generalize results in [ 4 , 5 ] to the case of infinite dimensional spaces. To treat problems in the nonsmooth case, Luc and Jeyakumar [16] introduce the concepts of pseudo Jacobian and pseudo Hessian, Tuan and Khanh [
∗This research was partially supported by the Vietnam Institute for Advanced Study in Mathe- matics
†University of Hue, Faculty of Sciences, VietnamURL: http:/mc.manuscriptcentral.com/gopt 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
22] use the concepts of approximations introduced by Allali and Amahroq [ 1 ] while Jimenez and Novo [ 13 , 20 ] choose Dini and Hadamard sec- ond order directional derivatives to establish optimality conditions. This article is a continuity of above works. By introducing the concepts of generalized Hadamard second order directional derivatives in a natural way, we obtain results on first and second order optimality conditions of local optimal solutions for the problem (P) which generalize, sharpen and strengthen some known results of the authors mentioned above.
The paper is organized as follows. In the next section we recall con- cepts of first and second order tangent sets and concepts of approxima- tions. After that we introduce the concepts of generalized second order Hadamard directional derivatives together with some basic properties.
Section 3 is devoted to first order optimality conditions and the last section provides results on second order conditions.
2 Preliminaries
LetX be a normed space, S ⊂X, x∈X and r >0. We denote by clS, intS, coneS, SX and B(x, r) the closure of S, the interior of S, the cone generated by S, the unit sphere in X and the open ball centered at x with radiusr, respectively. Let ¯x∈clS and u∈X.
The feasible direction cone of S at ¯xis defined by
T0(S; ¯x) :={v ∈X :∃ δ >0 such that ¯x+tv∈S,∀t∈(0, δ]}
and thefirst order tangent cone toS at ¯x is defined by
T(S; ¯x) := {v ∈X :∃tn↓0, vn →v such that ¯x+tnvn ∈S,∀n}.
It is clear that T0(S; ¯x) ⊂ T(S; ¯x), the converse is true when S is a polyhedral convex set.
The second order tangent cone of S at (¯x, u) in the sense of Penot [ 28 ] is the set
T00(S; ¯x;u) :={z ∈X :∃sn, tn↓0, zn→z such that ¯x+tnu+tnsnzn ∈S,∀n}.
Particularly, T00(S; ¯x; 0) = T(S; ¯x) because xn = ¯x +αnzn ∈ S with αn = tnsn → 0 and zn → z. We also see that if u /∈ T(S; ¯x), then T00(S; ¯x;u) =∅.
The parabolic second order tangent set [ 28 ] is defined as
T2(S; ¯x;u) :={z ∈X :∃tn↓0, zn →z such that ¯x+tnu+t2nzn∈S,∀n}.
The theasymptotic second order tangent cone given by Cambini et al.[ 7 ] is
T02(S; ¯x;u) :={z ∈X :∃ sn, tn↓0, tn/sn →0, zn→z such that
¯
x+tnu+tnsnzn∈S,∀n}.
The interior tangent cone of S at ¯x is defined by
IT(S; ¯x) :={v ∈URL: http:/mc.manuscriptcentral.com/goptX :∃δ >0 such that ¯x+tw ∈S, ∀t∈(0, δ], ∀w∈B(v, δ)}.
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Remark 2.1 (i) −IT(S; ¯x) = IT(−S;−¯x).
(ii) ([ 18 , Proposition 2.3(ii)]) If S is convex and intS6=∅, then IT(intS,x) =¯ IT(S; ¯x) = int cone(S−x).¯
The following concept of tangentiability is similar to the concept of asymptotability of [ 26 ] and plays an important role in establishing suf- ficient conditions of optimal solutions.
Definition 2.1 ([ 10 ]) Let ¯x∈ S. We say thatS is first order tangen- tiable at ¯xif for everyε >0, there is a neighborhood U of the origin such that
(S−x)¯ ∩U ⊂[T(S; ¯x)]ε
where [T(S; ¯x)]ε :={x∈X : d(x/kxk, T(S; ¯x)≤ε} ∪ {0}.
S is said to besecond order tangentiable at ¯x if for everyu∈T(S; ¯x) and for every ε >0, there is a neighborhood U of the origin such that
cone[(S−x)¯ ∩U]∩(u+U)⊂u+ [T00(S; ¯x;u)]ε
Remark 2.2 IfS is second order tangentiable at ¯x, then it is also first order tangentiable at that point. To see this, takeu= 0 and apply equal- ity T”(S,x,¯ 0) = T(S; ¯x). Moreover, in finite dimensional spaces, every nonempty subset is first and second order tangentiable at any element of it ( [ 10 , Proposition 2.2]).
Next we present the concepts of approximations [ 10 ] which is a gener- alized version of concepts of approximations introduced by Jourani and Thibault in [ 21 ] and by Allali and Amahroq in [ 1 ]. Approximations can be used as generalized derivatives to investigate the behavior of functions in a neighborhood of points under consideration.
Let ¯x, u, v ∈ X. A sequence {xn} ⊂ X is said to converge to ¯x in the direction u, denoted xn →u x, if¯ ∃ tn ↓ 0, un → u such that xn= ¯x+tnun, ∀ n, and it is said to converge to ¯x in the direction (u, v), denoted xn→(u,v)x, if¯
∃ tn, sn ↓0, vn→v such that xn= ¯x+tnu+tnsnvn, ∀ n.
It is implied immediately from above definitions that xn →u x¯⇔xn→(u,0) x.¯
Let Y be another normed space and r : X → Y. The function r is said to have the limit 0 as x converges to 0 in the direction u, denoted
x→limu0r(x) = 0,if
∀ {xn} ⊂X, xn→u 0 ⇒ r(xn)→0.
The function r is said to have the limit 0 as x converges to 0 in the direction (u, v), denoted lim
x→(u,v)0r(x) = 0, if
∀ {xURL: http:/mc.manuscriptcentral.com/goptn} ⊂X, xn→(u,v)0 ⇒ r(xn)→0.
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Let L(X, Y) (B(X, Y)) denote the space of continuous linear map- pings from X to Y (continuous bilinear mappings from X ×X to Y).
For A ⊂ L(X, Y) and x ∈ X, A(x) := {a(x) : a ∈ A} ⊂ Y. Par- ticularly, when A is a singleton, A = {a} for some a, A(x) is iden- tified with a(x) for convenience. For B ⊂ B(X, Y) and x, x0 ∈ X, B(x, x0) := {b(x, x0) : b ∈ B}. The following definition in [ 22 ] will be needed in the sequel.
Definition 2.2 (i) Let an, a ∈ L(X, Y). The sequence {an} pointwise converges toa and denoted as an→p a or a= p-liman if
n→∞liman(x) = a(x),∀x∈X.
A similar definition is adopted forbn, b∈B(X, Y).
(ii) A subset A ⊂ L(X, Y) is said to be relatively p-compact if each sequence{an} ⊂Ahas a subsequence which pointwise converges to some a∈L(X, Y). A similar definition is adopted for B ⊂B(X, Y).
Let f :X →Y be a vector function.
Definition 2.3 ([10]) A nonempty subsetAf ⊂L(X, Y) is called afirst order approximation of f at ¯x ∈ X if for every direction u ∈ X \ {0}, there exists a function ru : X → Y with lim
x→u0ru(x) = 0, such that for every sequence {xn} ⊂X converging to ¯xin the direction u,
f(xn)∈f(¯x) +Af(xn−x) +¯ kxn−xkr¯ u(xn−x).¯
Definition 2.4 ([ 10 ]) Let Af ⊂L(X, Y),Bf ⊂B(X, Y) be nonempty sets. We say thatf admits (Af, Bf) as a second order approximation at
¯
x∈X if
(i) Af is a first order approximation of f at ¯x.
(ii) For every direction (u, z) 6= (0,0), there exists a function r(u,z) : X → Y with lim
x→(u,z)0r(u,z)(x) = 0 such that for every sequence
{xn} ⊂X converging to ¯x in the direction (u, z), f(xn)∈
f(¯x) +Af(xn−x) +¯ 12Bf(xn−x, x¯ n−x) +¯ kxn−xk¯ 2r(u,z)(xn−x).
From Definition 2.4, if f is twice Fr´echet differentiable at ¯x, then (Df(¯x), D2f(¯x)) is a second order approximation of f at ¯x. Moreover, first and second order approximations in the sense of Allali and Amahroq [ 1 ] are also approximations in the sense of Definition 2.4 (see, [ 10 ]).
Consequently, Clarke generalized Jacobian of locally Lipschitz functions fromRn to Rm are first order approximations.
Definition 2.5 ( [ 14 ]) Letg :Rn→Rbe a mapping of classC1,1. The Clarke generalized Hessian of g at ¯x∈Rn is defined as
∂C2g(¯x) := co{lim
k→∞D2g(xk) : xk →x, D¯ 2g(xk) exists}.
URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
It is known [ 10 ] that if g :Rn→R isC1,1 in a neighborhood U of ¯x, then (Dg(¯x), ∂C2g(¯x)) is a second order approximation off at ¯x.
Finally, we introduce the concepts of generalized Hadamard direc- tional derivatives in a natural way. They are used as main tools to establish optimality conditions.
Let S ⊂ X, h : S → Y. We recall that h is called directional dif- ferentiable at ¯x ∈ S in the direction u ∈ T0(S; ¯x) if the following limit exists.
h0(¯x;u) := lim
t↓0
h(¯x+tu)−h(¯x)
t .
h is said to be Hadamard directional differentiable at ¯x ∈ S in the direction u∈T(S; ¯x) if the following limit exists.
dh(¯x;u) := lim
t↓0;v→u x+tv∈S¯
h(¯x+tv)−h(¯x)
t .
Definition 2.6 The generalized Hadamard directional derivative of h at ¯x∈S in the direction u∈T(S; ¯x) is defined as the following set
dGh(¯x;u) := {lim
n→∞
h(¯x+tnun)−h(¯x)
tn | tn ↓0, un →u,x¯+tnun∈S,∀n}.
If dGh(¯x;u) 6= ∅, then we say h is generalized Hadamard directional differentiable at ¯x in the directionu.
It is immediate from the above definition that if h is Hadamard direc- tional differentiable at ¯x in the direction u, thendGh(¯x;u) = {dh(¯x;u)}.
Iff :X →Y is differentiable at ¯x∈S, then f|S is Hadamard directional differentiable at ¯x in every direction u∈T(S; ¯x) and
df|S(¯x;u) =Df(¯x)(u).
(Where, f|S denotes the restriction of f to S and Df(¯x) denotes the derivative of f at ¯x ). The relation between first order approxima- tions and Hadamard directional differentiability are given in the following proposition.
Proposition 2.1 Let f : X → Y. Assume that f admits a first order approximation Af at x¯∈S.
i) If Af is relatively p-compact and bounded, then f|S is generalized Hadamard directional differentiable at x¯ in every direction u∈T(S; ¯x)\ {0}.
ii) Particularly, if Af is a singleton, thenf|S is Hadamard directional differentiable at x¯ in every direction u ∈ T(S; ¯x)\ {0} and df|S(¯x;u) = Af(u).
Proof. Let u ∈ T(S; ¯x)\ {0} be arbitrary. By Definition 2.3, there is a function ru : X → Y with lim
x→u0r(x) = 0 such that for every sequence {xn} ⊂X, xn→u x¯ we have
f(xn)URL: http:/mc.manuscriptcentral.com/gopt∈f(¯x) +Af(xn−x) +¯ kxn−xkr¯ u(xn−x).¯ (1) 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
i) Since u ∈ T(S; ¯x), there are sequences tn ↓ 0, un → u so that xn:= ¯x+tnun∈S. By (1), for every n, there is an∈Af so that
f|S(¯x+tnun)−f|S(¯x) tn
=an(un) +kunkru(tnun).
Since Af is relative p-compact we may assume an →p a, for some a ∈ L(X, Y). By the boundedness of Af, an(un)→a(u). Hence
n→∞lim
f|S(¯x+tnun)−f|S(¯x)
tn =a(u)
which impliesdGf|S(¯x;u)6=∅.
ii) Consider the case Af is a singleton. Let tn ↓ 0, un → u be ar- bitrary so that ¯x+tnun ∈ S,∀n. Using arguments as in i), we obtain
n→∞lim
f|S(¯x+tnun)−f|S(¯x)
tn =Af(u). Hence df|S(¯x;u) = Af(u).
Definition 2.7 Let ¯x∈S, u∈T(S; ¯x). Assume thatdh(¯x;u) exists.
i) Let z ∈ T00(S; ¯x, u). The Hadamard generalized second order direc- tional derivative of h at ¯x ∈ S in the direction (u, z) is defined as the following set
d00Gh(¯x;u, z) :={lim
n→∞
h(¯x+tnu+tnsnzn)−h(¯x)−tndh(¯x;u)
tnsn | tn, sn↓0;
zn→z; ¯x+tnu+tnsnzn∈S,∀n}.
We say that h is Hadamard second order directional differentiable at
¯
xin the direction (u, z) if the following limit exists d00h(¯x;u, z) := lim
t,s↓0,w→z x+tu+tsw∈S¯
h(¯x+tu+tsw)−h(¯x)−tdh(¯x;u)
ts .
ii) Letz ∈T2(S; ¯x, u). TheHadamard generalized parabolic second order directional derivative of h at ¯x ∈ S in the direction (u, z) is defined as the following set
d2Gh(¯x;u, z) :={lim
n→∞
h(¯x+tnu+t2nzn)−h(¯x)−tndh(¯x;u)
t2n | tn ↓0;zn→z;
¯
x+tnu+t2nzn ∈S,∀n}.
We say that h is Hadamard parabolic second order directional differ- entiable at ¯x in the direction (u, z) if the following limit exists
d2h(¯x;u, z) := lim
t↓0,w→z
¯
x+tu+t2w∈S
h(¯x+tu+t2w)−h(¯x)−tdh(¯x;u)
t2 .
iii) Let z ∈ T02(S; ¯x, u). The Hadamard generalized asymptotic second order directional derivative ofhat ¯x∈S in the direction (u, z) is defined as the following set
URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
d2G,0h(¯x;u, z) :={lim
n→∞
h(¯x+tnu+tnsnzn)−h(¯x)−tndh(¯x;u) tnsn
| tn, sn↓0;
tn
sn →0; zn →z; ¯x+tnu+tnsnzn∈S,∀n}.
We say that h is Hadamard asymptotic second order directional dif- ferentiable at ¯x in the direction (u, z) if the following limit exists
d20h(¯x;u, z) := lim
t,s↓0,w→z
¯
x+tsu+ts2w∈S
h(¯x+tsu+ts2w)−h(¯x)−tsdh(¯x;u)
ts2 .
Obviously ,d2Gh(¯x;u, z), d2G,0h(¯x;u, z)⊂d00Gh(¯x;u, z). Ifd2h(¯x;u, z), d20h(¯x;u, z) exist, then
d2Gh(¯x;u, z) = {d2h(¯x;u, z)}
d2G,0h(¯x;u, z) = {d20h(¯x;u, z)}.
The concept of Hadamard parabolic second order directional derivative were first introduced by Ben-Tal and Zowe [ 2 , 3 ] and the concept of Hadamard asymptotic second order directional derivative were used by Gutirrez et al. [ 13 ]. Here we propose generalized versions of these concepts.
Proposition 2.2 Let f : X → Y be given. Assume that f admits (Af, Bf)as a second order approximation at x¯∈S withAf is a singleton andBf is relatively p-compact and bounded. For every u∈T(S; ¯x)\ {0}, we have
(i)
d2Gf|S(¯x;u, z)6=∅,∀z ∈T2(S; ¯x;u).
(ii)
d2G,0f|S(¯x;u, z)6=∅,∀z ∈T02(S; ¯x;u).
Proof. (i) Let u ∈ T(S; ¯x) \ {0}, z ∈ T2(S; ¯x;u) be arbitrary. Then there are sequencestn →0+, zn→z so thatxn := ¯x+tnu+t2nzn∈S,∀n.
From the definitions of approximations, there exists functionr(u,z) :X→ Y with lim
x→(u,v)0r(u,z)(x) = 0 such that for every n there exist bn ∈ Bf
satisfying
f(xn)−f(¯x) =Af(xn−x) +¯ 1
2bn(xn−x, x¯ n−x) +¯ kxn−xk¯ 2r(u,z)(xn−x).¯ (2) By Proposition 2.1, df|S(¯x;u) = Af(u). Then it follows from (2) that
f(¯x+tnu+t2nzn)−f(¯x)−tndf|S(¯x;u)
t2n =
=Af(zn) + 1
2bn(u+tnzn, u+tnzn) +ku+tnznkr(u,z)(xn−x).¯ URL: http:/mc.manuscriptcentral.com/gopt
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
We may assume that bn →p b, for some b ∈ B(X, Y). Taking the above equalities to the limit as n→ ∞ , we get
n→∞lim
f(¯x+tnu+t2nzn)−f(¯x)−tndf|S(¯x;u) t2n
=Af(z) + 1 2b(u, u) which impliesd2Gf|S(¯x;u, z)6=∅.
(ii) Similarly.
Proposition 2.3 If f : X → Y is twice differentiable at ¯x ∈ S, then for every u∈T(S; ¯x), we have
d2f|S(¯x;u, z) =Df(¯x)(z) + 1
2D2f(¯x)(u, u), ∀z ∈T2(S; ¯x;u) d20f|S(¯x;u, z) =Df(¯x)(z), ∀z ∈T02(S; ¯x;u).
Proof. Applying Taylor’s formula, one has f(x) =f(¯x) +Df(¯x)(x−x) +¯ 1
2D2f(¯x)(x−x, x¯ −x) +¯ ◦(kx−xk¯ 2). (3) Letu∈T(S; ¯x) be arbitrary. From the note after Definition 2.6, one has
df|S(¯x;u) =Df(¯x)(u).
Let z ∈ T02(S; ¯x;u), be arbitrary. From (3), for t, s > 0, w ∈ X with
¯
x+tsu+ts2w∈S, we have
f|S(¯x+tsu+ts2w)−f|S(¯x)−tsdf|S(¯x;u) ts2
=Df(¯x)(w) + 1
2tD2f(¯x)(u+sw, u+sw) + ◦(t2s2ku+swk2) t2s2 t.
Hence,
d20f|S(x;u, z) = lim
t,s↓0,w→z
¯
x+tsu+ts2w∈S
f|S(¯x+tsu+ts2w)−f|S(¯x)−tsdf|S(¯x;u) ts2
=Df(¯x)(z).
Similarly, we can also prove that d2f|S(x;u, z) =Df(¯x)(z) + 1
2D2f(¯x)(u, u),∀z ∈T2(S; ¯x;u).
3 First-order Optimality conditions
In this section, we establish first order optimality necessary/sufficient conditions in terms of generalized Hadamard directional derivatives for the problem (P) stated in Section 1.
minf(x) s.t. x∈S. (P)
URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
We recall that an element ¯x ∈ S is called a local efficient solution (re- spectively, local ideal efficient solution, local strict efficient solution, local weak efficient solution) of (P) if there exists a neighborhoodU of ¯x such that for every x∈(S\ {¯x})∩U,
f(x)∈/f(¯x)−(C\ {0}) (respectively,
f(x)∈f(¯x) +C f(x)∈/f(¯x)−C f(x)∈/ f(¯x)−intC.)
We say that ¯xis a local strict efficient solution of order k ( with k≥1 is an integer) of (P) if there exist a neighborhood U of ¯x and α > 0 such that
f(¯x) +B(0, αkx−xk¯ k)
∩(f(x) +C) =∅, ∀x∈(S\ {¯x})∩U.
From the above definitions, we have implications
ideal efficient solution ⇒ efficient solution ⇒ weak efficient solution strict efficient solution of orderk⇒strict efficient solution⇒efficient solution
Theorem 3.1 (First order necessary condition) Let x¯∈S.
i) If x¯ is a local weak efficient solution of (P), then dGf|S(¯x;u)∩ −intC =∅, ∀u∈T(S; ¯x).
ii) If x¯ is a local ideal solution of (P), then dGf|S(¯x;u)⊂C, ∀u∈T(S; ¯x).
iii) If x¯ is a local strict efficient solution of order 1 of (P), then dGf|S(¯x;u)∩ −C =∅, ∀u∈T(S; ¯x)\ {0}.
Proof. i) Let u ∈ T(S; ¯x), l ∈ dGf|S(¯x;u) be arbitrary. Then there exist sequences tn ↓ 0, un → u with ¯x+tnun ∈ S,∀n, such that l =
n→∞lim
f(¯x+tnun)−f(¯x)
tn . Since ¯x is a local weak efficient solution, we have f(¯x+tnun)−f(¯x)
tn ∈ −intC,/ forn sufficiently large, which impliesl /∈ −intC.
ii) Similarly.
iii) Suppose to the contrary that there exists u ∈ T(S; ¯x)\ {0} such that
dGf|S(¯x;u)∩ −C 6=∅ (4) Then there are sequences un →u, tn →0+ with xn := ¯x+tnun ∈S,∀n, satisfying
n→∞lim
f(¯x+tnun)−f(¯x)
tn ∈ −C.
URL: http:/mc.manuscriptcentral.com/gopt 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
By assumption on ¯x, there exist a neighborhood U of ¯x and a positive number α such that
(f(¯x) +B(0, αkx−xk))¯ ∩(f(x) +C) =∅, ∀x∈(S\ {¯x})∩U. (5) Set l:= lim
n→∞
f(¯x+tnun)−f(¯x)
tn .Then for n sufficiently large, we have f(¯x+tnun)−f(¯x)
tn ∈l+B(0,αkuk 2 ).
Hence, for n sufficiently large,
f(¯x+tnun)−tnl ∈f(¯x) +B(0, tnαkuk
2 )⊂f(¯x) +B(0, αkxn−xk)¯ which contradicts (5).
Theorem 3.1 i) strengthens Theorem 4.2 (i) [ 19 ] in whichf is differen- tiable at ¯x and Theorem 5.2 (i) [ 13 ] in whichf is Hadamard directional differentiable at ¯x . Theorem 3.1 iii) is more general and stronger than the ’only if’ part of Theorem 4.1 [ 20 ] in which X is finite dimensional and f is Hadamard directional differentiable.
Corollary 3.1 Assume that f|S is Hadamard directional differentiable at x¯∈S in every direction u∈T(S; ¯x).
i) If x¯ is a local weak efficient solution of (P), then df|S(¯x;u)∈ −intC/ =∅, ∀u∈T(S; ¯x).
ii) If x¯ is a local ideal solution of (P), then df|S(¯x;u)∈C, ∀u∈T(S; ¯x).
iii) If x¯ is a local strict efficient solution of order 1 of (P), then df|S(¯x;u)∈ −C,/ ∀u∈T(S; ¯x)\ {0}.
Proof. It is immediate from Theorem 3.1 sincedGf|S(¯x;u) = {df|S(¯x;u)},
∀u∈T(S; ¯x).
Corollary 3.2 Assume that f is directional differentiable at x¯ ∈ S in every directionu∈T0(S; ¯x).
i) If x¯ is a local weak efficient solution of (P), then f0(¯x;u)∈ −intC,/ ∀u∈T0(S; ¯x).
ii) If x¯ is a local ideal solution of (P), then f0(¯x;u)∈C, ∀u∈T0(S; ¯x).
iii) If x¯ is a local strict efficient solution of order 1 of (P), then f0(¯x;u)∈ −C,/ ∀u∈T0(S; ¯x)\ {0}.
Proof. It is immediate from Theorem 3.1 since f0(¯x;u) ∈ dGf|S(¯x;u),
∀u∈T0(S; ¯x).
Specially, we have immediately from Corollary 3.2URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Corollary 3.3 Assume that x¯∈ intS and f is differentiable at x. If¯ x¯ is a local ideal solution of (P), then
Df(¯x) = 0.
Obviously, the conditions in Theorem 3.1 are not sufficient for a fea- sible point to be a local weak/ideal/strict efficient solution, even if S is convex. For instant, let us consider the following example.
Example 3.1 Assume that R2 is ordered by the following cone C:={(x, y) : 0≤y≤2x}.
LetS :={(x, y)∈R2 : y≥x2} and letf :R2 →R2 defined by f(x, y) =
((y, y), y > x2 (−x2,−x2), y≤x2. Consider the following problem
minf(x), s.t. x∈S. (P1) Set ¯x= (0,0).We have
T(S; ¯x) ={(u1, u2) : u2 ≥0}.
It can be verified thatf|S is Hadamard directional differentiable at ¯x in every directionu∈T(S; ¯x) and
df|S(¯x;u) = (u2, u2)∈C,∀u= (u1, u2)∈T(S; ¯x).
However, ¯x is not a local weak efficient solution of (P1) since for every n we have xn:= (1n,n12)∈S and
f|S(xn) = (− 1 n2,− 1
n2)∈f|S(¯x)−intC.
For establishing first order sufficient conditions, we need the following lemma.
Lemma 3.1 Let ¯x∈S and{xn} ⊂S\ {¯x}, xn→x. Assume that¯ S is tangentiable at ¯x and T(S,x) is locally compact at the origin. Then the¯ sequencen
xn−¯x kxn−¯xk
o
has a convergent subsequence.
Proof. Since S is tangentiable at ¯x, by passing to a subsequence if necessary, we may assume that
d
xn−x¯
kxn−xk¯ , T(S; ¯x)
< 1 n, ∀n.
Then for every n, there exists un ∈T(S,x) such that¯
xn−x¯ kxn−xk¯ −un
< 1 n.
Since T(S,x) is a cone and locally compact at 0, there exists a subse-¯ quence{ukn} ⊂ {un}which converges to some point u. Then n x
kn−¯x kxkn−¯xk
o is also converges toURL: http:/mc.manuscriptcentral.com/goptu.
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Theorem 3.2 (First order sufficient condition) Suppose that S is first order tangentiable at x¯∈S and the tangent cone T(S; ¯x) is locally com- pact at the origin. Assume that f admits a relatively p-compact and bounded first order approximation Af at x. If¯
dGf|S(¯x;u)∩ −C =∅, ∀ u∈T(S; ¯x)∩SX, (6) then x¯ is a local strict efficient solution of order 1 of (P).
Proof. Suppose to the contrary that ¯x is not a strictly local efficient solution of order 1. Then for every n ∈ N \ {0}, there exists xn ∈ B(¯x,n1)∩(S\ {¯x}) such that
[f(¯x) +B(0,1
nkxn−xk)]¯ ∩[f(xn) +C]6=∅.
Hence, for every n, there are an ∈B(0,n1kxn−xk)], c¯ n ∈C satisfying f(xn)−f(¯x) =an−cn.
By Lemma 3.1, we may assume that xn → x¯ in some direction u ∈ T(S; ¯x)∩SX. Then there exist tn ↓0, un →u such thatxn = ¯x+tnun. By Definition 2.4, there are functionru :X →Y with lim
x→u0ru(x) = 0 and an∈Af, n= 1,2, ..,satisfying
f(xn)−f(¯x) =an(xn−x) +¯ kxn−xkr¯ u(xn−x).¯ (7) Therefore,
an−cn
tn = f(¯x+tnun)−f(¯x)
tn =an(un) +kunkru(xn−x).¯ (8) Since Af is relatively p-compact, we may assume that an→p a, for some a ∈ L(X, Y), which together the boundedness of Af imply an(un) → a(u). Then (8) yields
n→∞lim
an−cn
tn = lim
n→∞
f(¯x+tnun)−f(¯x)
tn =a(u).
Since
n→∞lim
an−cn tn
= lim
n→∞
−cn tn
∈ −C, we have
dGf|S(¯x;u)∩ −C 6=∅
which contradicts (6).
When X, Y are finite dimensional we have
Corollary 3.4 Assume thatX =Rn, Y =Rm andf is locally Lipschitz at x¯∈S. If
dGf|S(¯x;u)∩ −C =∅, ∀ u∈T(S; ¯x)∩SX, then x¯ is a local strict efficient solution of order 1 ofURL: http:/mc.manuscriptcentral.com/gopt (P).
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Proof. It is immediate from Theorem 3.2 since every set in a finite dimensional space is first order tangentiable at any point of it ([ 10 , Proposition 2.2]) and the Clarke generalized Jacobian is an compact ap-
proximation.
Next we present an alternate theorem to establish first order sufficient optimality conditions.
Theorem 3.3 Assume that S is first order tangentiable at x¯ ∈ S and the tangent coneT(S; ¯x)is locally compact at the origin. Iff|S is Hadamard directional differentiable at x¯ in every direction u∈T(S; ¯x)∩SX and
df|S(¯x;u)∈ −C,/ ∀ u∈T(S; ¯x)∩SX, then x¯ is a local strict efficient solution of order 1 of (P).
Proof. Suppose to the contrary that ¯x is not a strictly local efficient solution of order 1. Then for every n ∈ N \ {0}, there exists xn ∈ B(¯x,n1)\ {¯x}such that
[f(¯x) +B(0,1
nkxn−xk)]¯ ∩[f(xn) +C]6=∅.
Hence, for every n, there are an ∈B(0,n1kxn−xk)], c¯ n ∈C satisfying f(xn)−f(¯x) =an−cn.
By Lemma 3.1, we may assume that xn → x¯ in some direction u ∈ T(S; ¯x)∩SX. Then there exist tn ↓0, un →u such thatxn = ¯x+tnun. We have
df|S(¯x;u) = lim
n→∞
f(¯x+tnun)−f(¯x) tn
= lim
n→∞
an−cn
tn
= lim
n→∞
−cn tn
∈ −C
which contradicts assumptions.
Theorem 3.3 is more general and stronger than the ’only if’ part of Theorem 4.1 [ 20 ] in which X is finite dimensional and f is required Hadamard directional differentiable.
For local ideal efficient solutions, we have following results whose proofs are similar to the ones of above results.
Theorem 3.4 Suppose that S is first order tangentiable at x¯ ∈ S and the tangent coneT(S; ¯x)is locally compact at the origin. Assume that f admits a relatively p-compact and bounded first order approximation Af
at x. If¯
dGf|S(¯x;u)⊂intC, ∀ u∈T(S; ¯x)∩SX, then x¯ is a local ideal efficient solution of (P).
URL: http:/mc.manuscriptcentral.com/gopt 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Corollary 3.5 Assume thatX =Rn, Y =Rm andf is locally Lipschitz at x¯∈S. If
dGf|S(¯x;u)⊂intC, ∀ u∈T(S; ¯x)∩SX, then x¯ is a local ideal efficient solution of (P).
Theorem 3.5 Assume that S is first order tangentiable at x¯ ∈ S and the tangent coneT(S; ¯x)is locally compact at the origin. Iff|S is Hadamard directional differentiable at x¯ in every direction u∈T(S; ¯x)∩SX and
df|S(¯x;u)∈intC, ∀ u∈T(S; ¯x)∩SX, then x¯ is a local ideal efficient solution of (P).
Example 3.2 Let H be an infinite dimensional Hilbert space with an orthonormal countable basis {e1, e2, . . . , en, . . .}. Denote S2 := {t1e1 + t2e2 : t2 ≥t21}, Sn :={ten: t≥n}, n = 3,4, . . . and
S :=
∞
S
n=2
Sn.
It can see that S is first order tangentiable at 0 and, moreover, there is no finite dimensional subspace of H that contains it.
Assume that R2 is ordered by the following cone C :={(y1, y2) : 1
2y1 ≤y2 ≤2y1}.
Define a functionf :H →R2 as follows. For everyx=
∞
P
n=1
tnen∈H,
f(x) =
(|t1|+t2,|t1|+t2), x∈S2 (−
∞
P
n=1
t2n,−
∞
P
n=1
t2n), x /∈S2. Consider the following problem
minf(x), s.t. x∈S. (P2) Set ¯x:= 0. We have
T(S; ¯x) = {t1e1+t2e2 : t2 ≥0}.
Clearly,T(S; ¯x) is locally compact at the orgirin. It also can be verified that f|S is Hadamard directional differentiable at ¯x in every direction u=t1e1+t2e2 ∈T(S; ¯x)∩SX and
df|S(¯x;u) = (|t1|+t2,|t1|+t2)∈intC.
Then by Theorem 3.5, ¯xis a local ideal solution of (P2). We should note that the spaceH is infinite dimensional. So we can not apply results in [ 13 , 20 ] to investigate the problem (P2).
URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
4 Second-order Optimality conditions
Consider the problem (P) as in preceding section minf(x) s.t. x∈S . Denote
C0(f|S,x) :=¯ {u∈T(S; ¯x) : df|S(¯x;u) exists, df|S(¯x;u)∈/ intC}
C(f|S,x) :=¯ {u∈T(S; ¯x) : df|S(¯x;u) exists, df|S(¯x;u)∈ −C}.
Theorem 4.1 (Second order necessary condition) If x¯ ∈ S is a local weakly efficient solution of (P), then for every u∈C(f|S,x), we have¯
d00Gf|S(¯x;u, z)∩ −int cone(C+df|S(¯x;u)) =∅,∀z ∈T00(S; ¯x;u).
Proof. Suppose to the contrary that there exist u ∈ C(f|S,x),¯ z ∈ T00(S; ¯x, u) such that
d00Gf|S(¯x;u, z)∩ −int cone(C+df|S(¯x;u))6=∅. (9) Let l ∈ d00Gf|S(¯x;u, z)∩ −int cone(C +df|S(¯x;u)) be arbitrary. Then there are tn ↓0, sn ↓0, zn→z with xn := ¯x+tnu+tnsnzn∈ S,∀n such that
l= lim
n→∞
f(xn)−f(¯x)−tndf|S(u)
tnsn ∈ −int cone(C+df|S(u)). (10) Since df|S(¯x;u)∈ −C, by Remark 2.1, we have
−int cone(C+df|S(¯x;u)) =IT(−intC, df|S(¯x;u)) (11) which together with (10) imply
l ∈IT(−intC, df|S(u)). (12) By the definition of the cone IT, there is δ >0 so that
df|S(¯x;u) +s(l+B(0, δ))⊂ −intC, ∀s∈(0, δ]. (13) Therefore,
tdf|S(¯x;u) +ts(l+B(0, δ))⊂ −intC, ∀t >0, s∈(0, δ].
By (10), for n sufficiently large,
f(xn)−f(¯x)−tndf|S(u)
tnsn ∈l+B(0, δ).
Hence,
f(xn)−f(¯x)∈tndf|S(u) +tnsn(l+B(0, δ))⊂ −intC
which contradicts the definition of ¯x. The proof is complete.
Since d2Gh(x;u, z), d2G,0h(x;u, z) ⊂ d00Gh(x;u, z), we have immediately from Theorem 4.1URL: http:/mc.manuscriptcentral.com/gopt
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
For Peer Review Only
Corollary 4.1 If x¯∈S is a local weakly efficient solution of (P), then for every u∈C(f|S,x), we have¯
d2Gf|S(¯x;u, z)∩ −int cone(C+df|S(¯x;u)) =∅,∀z ∈T2(S; ¯x;u) d2G,0f|S(¯x;u, z)∩ −int cone(C+df|S(¯x;u)) =∅,∀z ∈T02(S; ¯x;u).
Corollary 4.1 immediately implies Theorem 5.2 (ii) [ 13 ] since in which f is assumed asymptotic and parabolic Hadamard second order direc- tional differentiable.
For local ideal solution, we have
Theorem 4.2 (Second order necessary condition) If x¯ ∈ S is a local ideal solution of (P), then for every u∈C(f|S,x), we have¯
d00Gf|S(¯x;u, z)⊂C, ∀z ∈T00(S; ¯x;u).
Proof. Let u ∈ C(f|S,x), z¯ ∈ T00(S; ¯x;u) and l ∈ d00Gf|S(¯x;u, z) be arbitrary. Then by Definition 2.7, there exist sequencestn, sn↓0,zn →z with ¯x+tnu+tnsnzn ∈S such that
l= lim
n→∞
f(¯x+tnu+tnsnzn)−f(¯x)−tndf|S(¯x;u) tnsn
. By assumption, for n large enough,
f(¯x+tnu+tnsnzn)−f(¯x)−tndf|S(¯x;u)
tnsn ∈C
which impliesl ∈C.
For the casef is twice differentiable, sinced2Gf|S(¯x;u, z), d2G,0f|S(¯x;u, z)⊂ d00Gf|S(¯x;u, z), we have immediately from Corollary 4.1, Theorem 4.2 and Proposition 2.3
Corollary 4.2 Assume that f is twice differentiable at x¯∈S.
(i) If x¯ is a local weak efficient solution of (P), then for every u ∈ C(f|S,x), we have¯
Df(¯x)(z) + 1
2D2f(¯x)(u, u)∈ −int cone(C/ +df|S(¯x;u)), ∀z ∈T2(S; ¯x;u) Df(¯x)(z)∈ −int cone(C/ +df|S(¯x;u)), ∀z ∈T02(S; ¯x;u).
(ii) If x¯ is a local ideal solution of (P), then for every u ∈C(f|S,x),¯ we have
Df(¯x)(z) + 1
2D2f(¯x)(u, u)∈C, ∀z ∈T2(S; ¯x;u) Df(¯x)(z)∈C, ∀z ∈T02(S; ¯x;u).
Corollary 4.2 coincides with Theorem 4.2 (ii) [ 19 ] and generalizes Corol- lary 3.2 [ 28 ] in whichY =R.
Corollary 4.3 Assume that x¯∈intS and f is twice differentiable at x.¯ If x¯ is a local ideal solution of (P), then
D2f(¯x)(u, u)∈C,∀u∈X.
URL: http:/mc.manuscriptcentral.com/gopt 3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
