Karush-Kuhn-Tucker necessary conditions for local Pareto minima of constrained multiobjective programming problems

Karush-Kuhn-Tucker necessary conditions for local Pareto minima of constrained multiobjective programming problems DO VAN LUU Department of Mathematics and Informatics Thang Long Univers

Karush-Kuhn-Tucker necessary conditions for local Pareto minima of constrained multiobjective

programming problems

DO VAN LUU Department of Mathematics and Informatics

Thang Long University

Abstract Under the constraint qualification of Abadie type we establish necessary efficiency conditions described by inconsistent inequalities, Karush-Kuhn-Tucker necessary conditions and strong Karush-Kuhn-Tucker necessary conditions for lo-cal Pareto minima of nonsmooth multiobjective programming problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators Note that all the constraint functions involving in the considering problem are not necessarily continuous, except inactive constraints

The Karush-Kuhn-Tucker conditions of nonsmooth multiobjective programming problems is a significal topic in optimization If Lagrange multipliers corresspond-ing to all the components of the objective function are positive, they are called strong Karush-Kuhn-Tucker conditions The Lagrange multiplier rules in terms of different subdifferentials for nonsmooth optimization problems have been studied

by many authors (see, e g., [4], [6 - 16], and references therein).The notion of convex and compact convexificator was first introduced by Demyanov [2] This is a generalization of the notions of upper convex and lower concave approximations in [3] The notions of nonconvex closed convexificator and approximate Jacobian were introduced by Jeyakumar and Luc in [6] and [7], respectively They have provided good calculus rules for establishing necessary optimality conditions in nonsmooth optimization The notion of convexificator is a generalization of some notions of known subdifferentials such as the subdifferentials of Clarke [1], Michel-Penot [15], Mordukhovich [16]

In this paper, under the constraint qualifications of Abadie type we establish necessary efficiency conditions described by inconsistent inequalities, Karush-Kuhn-Tucker necessary conditions and strong Karush-Kuhn-Karush-Kuhn-Tucker necessary conditions for local Pareto minima of nonsmooth multiobjective programming problems involv-ing inequality, equality and set constraints in Banach spaces in terms of convexifi-cators

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2 Notions and definitions

Let X be a Banach space, X∗topological dual of X and f a extended-real-valued function defined on X The lower (resp upper) Dini directional derivatives of f at

x ∈ X in a direction v ∈ X are defined, respectively, by

f−(x; v) = lim inf

t↓0

f (x + tv) − f (x)

 resp f+(x; v) = lim sup

t↓0

f (x + tv) − f (x)

t



In case f+(x; v) = f−(x; v), we denote their common value by f0(x; v), which is called Dini derivative of f at x in the direction v The function f is Dini differentiable at

x if its Dini derivative at x exists in all directions Recall [6] that the function f

is said to have an upper (resp lower) convexificator ∂∗f (x) (resp ∂∗f (x)) at x if

∂∗f (x) (resp ∂∗f (x)) ⊆ X∗ is weak∗ closed, and

f−(x; v) 6 sup

ξ∈∂ ∗ f (x)

hξ, vi (∀v ∈ X),

 resp f+(x; v) > inf

ξ∈∂ ∗ f (x)

hξ, vi (∀v ∈ X)



A weak∗ closed set ∂∗f (x) ⊆ X∗ is said to be a convexificator of f at x if it is both upper and lower convexificator of f at x Note that upper and lower convexificators are not unique For a locally Lipschitz function, the Clarke subdifferential and the Michel-Penot subdifferential are convexificators of f at x (see [6]) The function f

is said to have an upper semi-regular convexificator ∂∗f (x) at x if ∂∗f (x) ⊆ X∗ is weak∗ closed, and

f+(x; v) 6 sup

ξ∈∂ ∗ f (x)

Following [6], if equality holds in (1) then ∂∗f (x) is called an upper regular convex-ificator For a locally Lipschitz function and regular in the sense of Clarke [1], the Clarke subdifferential is an upper regular convexificator (see [4])

Example 2.1 Let f : R → R be defined as

f (x) :=











x4− 4x3+ 4x2, if x ∈ Q∩] − ∞, 0],

where Q is the set of rationals Then

f+(0; v) =

(

v, if v > 0,

0, if v < 0,

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f−(0; v) = 0 (∀v ∈ R).

The set {0; 1} is an upper semi-regular convexificator of f at x = 0, and so it is upper convexificator of f at x The set {0} is lower convexificator of f at x Recall [1] that the contingent cone to a set C ⊆ X at a point x ∈ C is defined as K(C; x) =nv ∈ X : ∃ vn→ v, ∃ tn↓ 0 such that x + tnvn∈ C, ∀ no

The cone of sequential linear directions (or sequential radial cone) to C at x ∈ C is

Z(C; x) =nv ∈ X : ∃ tn↓ 0 such that x + tnv ∈ C, ∀ no Note that both these cones are nonempty and Z(C; x) ⊆ K(C; x)

for local Pareto minima

Let f, g, h be mappings from a Banach space X into Rm, Rn, R`, respectively, and let C be a subset of X Then f, g, h are of the forms: f = (f1, , fm), g = (g1, , gn), h = (h1, , h`), where f1, , fm, g1, , gn, h1, , h` are extended-real-valued functions defined on X For the sake of simplicity, we set: I = {1, , n},

J = {1, , m} and L = {1, , `} We shall be concerned with the following multiobjective programming problem (MP):

min f (x) s.t x belongs to M :=nx ∈ C : gi(x) 6 0, i ∈ I, hj(x) = 0, j ∈ Lo

We set I(x) = {i ∈ I : gi(x) = 0} Recall that a point x ∈ M is said to be a local Pareto minimizer of Problem (MP) if there exists a number δ > 0 such that there is

no x ∈ M ∩ B(x; δ) satisfying

fk(x) 6 fk(x) (∀ k ∈ J ),

fs(x) < fs(x) at least one s ∈ J, where B(x; δ) stands for the open ball of radius δ around x

For x ∈ X and s ∈ J , we set

Qs(x) =nx ∈ C : fk(x) 6 fk(x) (∀k ∈ J, k 6= s), gi(x) 6 0

(∀i ∈ I(x)), hj(x) = 0 (∀j ∈ L)

o

If hj is Dini differentiable at x for all j ∈ L, we put

C(Qs(x); x) =

n

v ∈Z(C; x) : fk−(x; v) 6 0 (∀k ∈ J, k 6= s),

gi−(x; v) 6 0 (∀i ∈ I(x)), h0j(x; v) = 0 (∀j ∈ L)o

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The relationship between Z(Qs(x); x) and C(Qs(x); x) is shown in the following result

Theorem 3.1 Let x ∈ M and hj is Dini differentiable at x for all j ∈ L Then, for

s ∈ J ,

Z(Qs(x); x) ⊆ C(Qs(x); x)

Proof For v ∈ Z(Qs(x); x), there exists tn ↓ 0 such that x + tnv ∈ Qs(x) Hence,

x + tnv ∈ C, and

fk(x + tnv) 6 fk(x) (∀k ∈ J, k 6= s),

gi(x + tnv) 6 0 = gi(x) (∀i ∈ I(x)),

hj(x + tnv) = 0 (∀j ∈ L)

These reduce to that v ∈ Z(C; x), and

fk−(x; v) 6 lim inf

n→+∞

fk(x + tnv) − fk(x)

tn 6 0 (∀k ∈ J, k 6= s),

gi−(x; v) 6 lim inf

n→+∞

gi(x + tnv) − gi(x)

tn 6 0 (∀i ∈ I(x)),

h0j(x; v) = lim

n→+∞

hj(x + tnv) − hj(x)

Consequently, v ∈ C(Qs(x); x), and so the result follows

To derive Kuhn-Tucker necessary conditions for local Pareto minima of Problem (MP), we introduce the following constraint qualification of Abadie type (CQs):

C(Qs(x); x) ⊆ Z(Qs(x); x)

A necessary condition for local Pareto minima of (MP) can be stated as follows Theorem 3.2 Let x be a local Pareto minimum of (MP), and let T be an arbitrary nonempty closed convex subcone of Z(C; x) with vertex at the origin Assume that the constraint qualification (CQs) holds for some s ∈ J , and the function hj is Dini differentiable at x for all j ∈ L, the function gi is continuous at x for all i /∈ I(x) Then, the following system has no solution v ∈ T :

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Proof Contrary to the conclusion, suppose that there exists v0 ∈ T satisfying (2) -(5) Then v0∈ C(Qs(x); x), and hence, v0∈ Z(Qs(x); x) as (CQs) holds Therefore, there exists a sequence tn↓ 0 such that x + tnv0∈ Qs(x), and so x + tnv0 ∈ C and

fk(x + tnv0) 6 fk(x) (∀k ∈ J, k 6= s),

gi(x + tnv0) 6 0 (∀i ∈ I(x)),

hj(x + tnv0) = 0 (∀j ∈ L)

For i /∈ I(x), one has that gi(x) < 0 In view of the continuity of gi (i /∈ I(x)), there

is a natural number N such that for all n > N, gi(x + tnv0) 6 0 (∀i /∈ I(x)) Hence, for all n > N , gi(x + tnv0) 6 0 (i ∈ I) Since x is a local Pareto minimum of (MP),

it results that for all n > N , fs(x + tnv0) > fs(x), which leads to the following

fs+(x; v0) > lim sup

n→+∞

fs(x + tnv0) − fs(x)

But this conflicts with (2) This completes the proof

To derive necessary conditions for efficiency of Problem (MP) we introduce the following assumption

Assumption 3.1 There exists an index s ∈ J such that the function fs admits

a nonempty bounded upper semi-regular convexificator ∂∗fs(x) at x; for every k ∈

J, k 6= s, and i ∈ I(x), the functions fk and gi admit upper convexificators ∂∗fk(x) and ∂∗gi(x) at x, respectively; all the functions gi(i /∈ I(x)) are continuous at x; all the functions hj(j ∈ L) are Gˆateaux differentiable at x with Gˆateaux derivatives

∇Ghj(x)

For s ∈ J and a nonempty closed convex subcone T of Z(C; x), we set

HTs(x) =[ nco∂∗fs(x) + X

k∈J,k6=s

λkco∂∗fk(x) + X

i∈I(x)

µico∂∗gi(x) +X

j∈L

γj∇Ghj(x)

+ T0: λk> 0 (∀k ∈ J, k 6= s), µi> 0 (∀i ∈ I(x)), γj ∈ R (∀j ∈ L)o, where co indicates the convex hull

We are now in a position to formulate a Karush-Kuhn-Tucker necessary condition for local Pareto minimum of Problem (MP)

Theorem 3.3 Let x be a local Pareto minimum of Problem (MP) Assume that Assumption 3.1 is fulfilled and the constraint qualification (CQs) holds for some

s ∈ J Suppose, in addition, that the set HTs(x) is weak∗ closed for some nonempty closed convex subcone T ⊆ Z(C; x) with vertex at the origin Then, there exist

λk > 0 (∀k ∈ J, k 6= s), µi > 0 (∀i ∈ I(x)), γj ∈ R (∀j ∈ L) such that

0 ∈ co∂∗fs(x) + X

k∈J,k6=s

λkco∂∗fk(x)

i∈I(x)

µico∂∗gi(x) +X

j∈L

γj∇Ghj(x) + T0

(6)

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Proof We invoke Theorem 3.2 to deduce that the system (2) - (5) has no solution

v ∈ T , where (9) is of the form: h∇Ghj(x), vi = 0 (∀j ∈ L) Hence, according to Assumption 3.1, the following system is also impossible v ∈ T :

sup

ξ s ∈co∂ ∗ f s (x)

sup

ξ k ∈co∂ ∗ f k (x)

hξk, vi 6 0 (∀k ∈ J, k 6= s), (8)

sup

ζ i ∈co∂ ∗ g i (x)

Let us show that

Assume the contrary, that 0 /∈ Hs

T(x) Observing that HTs(x) is convex and weak∗ closed, a separation theorem for a weak∗ closed convex set and a point outside that set (see, [5], Theorem 3.4) can be applied, and yields the existence of 0 6= v0 ∈ X such that

hζ, v0i < 0 (∀ζ ∈ Hs

This implies that

hξs, v0i + X

k∈J,k6=s

λkhξk, v0i + X

i∈I(x)

µihζi, v0i

j∈L

γjh∇Ghj(x), v0i + hη, v0i < 0,

(13)

for all ξs∈ co∂∗fs(x), λk> 0, ξk∈ co∂∗fk(x)(k ∈ J, k 6= s), µi > 0, ζi∈ co∂∗gi(x)(i ∈ I(x)), γj ∈ R(j ∈ L), η ∈ T0

For λk = 0(∀k ∈ J, k 6= s), µi = 0(∀i ∈ I(x)), γj = 0(∀j ∈ L), η = 0, in view of the boundness of ∂∗fs(x), it follows from (13) that

sup

ξ s ∈co∂ ∗ f s (x)

Let us show that

sup

ξk∈co∂ ∗ fk(x)

hξk, v0i 6 0 (∀k ∈ J, k 6= s) (15)

If this were not so, there would exist k0 ∈ J, k06= s such that

sup

ξk0∈co∂ ∗ fk0(x)

hξk0, v0i > 0

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Then, for λk = 0(∀k ∈ J, k 6= k0, s), µi = 0(∀i ∈ I(x)), γj = 0(∀j ∈ L), η = 0,

ξs∈ ∂∗fs(x), by taking λk0 be large enough, we get that

hξs, v0i + λk0 sup

ξk0∈co∂ ∗ fk0(x)

hξk0, v0i > 0,

as | hξs, v0i |< +∞ But it follows from (13) that

hξs, v0i + λk0 sup

ξk0∈co∂ ∗ fk0(x)

hξk0, v0i 6 0

Thus we arrive at a contradiction, and so (15) follows Analogously, we obtain

sup

ζ i ∈co∂ ∗ g i (x)

Moreover, we also can see that

Indeed, suppose that this were false, that is h∇Ghj0(x), v0i 6= 0 for some j0 ∈ L Then for λk = 0(∀k ∈ J, k 6= s), µi = 0(∀i ∈ I(x)), η = 0 and ξs ∈ ∂∗fs(x), in view

of the boundness of ξs, by letting γj0 be sufficiently large if h∇Ghj(x), v0i > 0, while

γj0 < 0 with its absolute value be large enough if h∇Ghj(x), v0i < 0, we shall arrive

at a contradiction with (13), and so (17) holds

It can see that v0 ∈ T In fact, if it were not so, there would exist η0 ∈ T0 such that hη0, v0i > 0 By letting λk = 0(∀k ∈ J, k 6= s), µi = 0(∀i ∈ I(x)), γj = 0(∀j ∈ L), for α sufficiently large, αη0 ∈ T0, and so we arrive at a contradiction with (13) Therefore, hη, v0i 6 0(∀η ∈ T0) Since T is closed convex, it is also weakly closed, and hence, v0 ∈ T00 = T It follows from (14) - (17) that the system (7) - (10) has a solution v0 ∈ T : a contradiction Consequently, (11) holds, and so there exist

λk > 0 (∀k ∈ J, k 6= s), µi > 0 (∀i ∈ I(x)), γj ∈ R (∀j ∈ L) such that the inclusion (6) holds The proof is complete

Remark 3.1 In Theorem 3.3, ∂∗fk(x)(k ∈ J, k 6= s) and ∂∗gi(x)(i ∈ I(x)) may be unbounded The functions fk(k ∈ J, k 6= s), gi(i ∈ I(x)), hj(j ∈ L) may be not continuous

To derive strong Karush-Kuhn-Tucker necessary conditions for efficiency of Prob-lem (MP) with positive Lagrange multipliers corresponding to all the components

of the objective, we introduce the following assumption

Assumption 4.1 For every k ∈ J , the function fk admits a nonempty bounded upper semi-regular convexificator ∂∗fk(x) at x; for every i ∈ I(x), the function gi

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admits upper convexificator ∂∗gi(x) at x; all the functions gi(i /∈ I(x)) are continuous

at x; all the functions hj(j ∈ L) are Gˆateaux differentiable at x

In what follows we shall give a strong Karush-Kuhn-Tucker necessary condition for local Pareto minimum in which Lagrange multipliers corresponding to all the components of the objective function are positive

Theorem 4.1 Let x be a local Pareto minimum of Problem (MP) Assume that Assumption 4.1 is fulfilled, the constraint qualification (CQs) holds for all s ∈ J , and the set HTs(x) weak∗ closed for some nonempty closed convex subcone T of Z(C; x) with vertex at the origin and all s ∈ J Then, there exist λk > 0 (∀k ∈

J ), µi> 0 (∀i ∈ I(x)), γj ∈ R (∀j ∈ L) such that

k∈J

λkco∂∗fk(x) + X

i∈I(x)

µico∂∗gi(x) +X

j∈L

γj∇Ghj(x) + T0

Proof It is easy to see that Assumption 4.1 implies Assumption 3.1 for all s ∈ J

We invoke Theorem 3.3 to deduce that for every s ∈ J , there exist λ(s)k > 0 (∀k ∈

J, k 6= s), µ(s)i > 0 (∀i ∈ I(x)) and γj(s)∈ R (∀j ∈ L) such that

0 ∈ co∂∗fs(x) + X

k∈J,k6=s

λ(s)k co∂∗fk(x)

i∈I(x)

µ(s)i co∂∗gi(x) +X

j∈L

γ(s)j ∇Ghj(x) + T0, (18)

Taking s = 1, , m in (18) and adding up both sides of the obtained inclusions, we arrive at

k∈J

λkco∂∗fk(x) + X

i∈I(x)

µico∂∗gi(x) +X

j∈L

γj∇Ghj(x) + T0,

where λk = 1 +P

s∈J,s6=kλ(s)k > 0 (∀k ∈ J ), µi = P

s∈Jµ(s)i > 0 (∀i ∈ I(x)), γj = P

s∈Jγj(s)∈ R (∀j ∈ L), as was to be shown

Remark 4.1 In Theorem 4.1, ∂∗gi(x)(i ∈ I(x)) may be unbounded The functions

gi(i ∈ I(x)), hj(j ∈ L) may be not continuous

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3, 208, 802 (1994)

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