DSpace at VNU: Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization

DSpace at VNU: Second-Order Optimality Conditions with the Envelope-Like Effect for Set-Valued Optimization tài liệu, gi...

DOI 10.1007/s10957-015-0728-6

Second-Order Optimality Conditions with the

Envelope-Like Effect for Set-Valued Optimization

P Q Khanh 1 · N M Tung 2

Received: 18 August 2014 / Accepted: 13 March 2015

© Springer Science+Business Media New York 2015

Abstract We consider Karush–Kuhn–Tucker second-order optimality conditions for

nonsmooth set-valued optimization with attention to the envelope-like effect Toanalyse the critical feasible directions, which produce this phenomenon, we usethe contingent derivatives, the adjacent derivatives and the corresponding asymp-totic derivatives, since directions are explicitly involved in these kinds of derivatives

To pursue strong multiplier rules, we impose cone-Aubin conditions to deal with theobjective and constraint maps separately In this way, we can invoke constraint qualifi-cations of the Kurcyusz–Robinson–Zowe type To our knowledge, some of the resultsare new; they will be indicated explicitly The paper also discusses improvements orextensions of known results

Keywords Optimality condition· Second-order contingent derivative ·

Kurcyusz–Robinson–Zowe constraint qualification· Weak minimizer ·

Firm minimizer

Mathematics Subject Classification 90C29· 49J52 · 90C46 · 90C48

Communicated by Jafar Zafarani.

2 Department of Mathematics and Computing, University of Science, Vietnam National University

Ho Chi Minh City, Ho Chi Minh City, Vietnam

1 Introduction

In recent decades, set-valued optimization has been received much attention fromresearchers Set-valued optimization is an expanding branch of applied mathematicsthat deals with optimization problems where the objective map and the constraint mapsare set-valued Until now, many derivative-like notions have been proposed and applied

to investigate optimality conditions in nonsmooth problems (see [1 5] for first-orderconditions, [6 10] for higher-order conditions and the references therein) In the path-breaking paper [2], Corley employed the contingent and circatangent derivatives toestablish a first-order Fritz John necessary optimality condition By using contingentepiderivatives, Götz and Jahn in [4] obtained a first-order Karush–Kuhn–Tucker (KKT)necessary optimality condition With the well-known Dubovitski–Milutin approach,Issac and Khan in [5] established a Lagrange multiplier rule for set-valued optimiza-tion with generalized inequality constraints Another fruitful approach in set-valuedoptimization is the dual space approach initiated by Mordukhovich (see [11,12]).Recently, second-order optimality conditions for scalar and vector optimizationproblems have been intensively developed because they refine the first-order bysecond-order information which is very helpful for recognizing optimal solutions

as well as for designing numerical algorithms for computing them We observe, inmost related contributions in the literature, that the core of second-order necessaryoptimality conditions is a direct extension of the classical result in calculus that thesecond derivative of the objective map (or the Lagrange map in constrained problems)

at minimizers is nonnegative Kawasaki in [13] discovered that the second derivative

of the Lagrangian at the minimal solution may be strictly negative in certain criticaldirections He called this phenomenon the envelope-like effect The Kawasaki resultwas developed in [14,15] for C2scalar programming, in [16–18] for nonsmooth multi-objective programming, and in [19] for infinite-dimensional nonsmooth optimization.However, for set-valued optimization, we observe only paper [20] dealing withthe envelope-like effect Let us mention first some papers on second-order optimal-ity conditions for set-valued optimization (not discussing the envelope-like effect)

In [3], Durea employed second-order contingent derivatives to establish such ditions In [6], Jahn et al proposed a second-order contingent epiderivative and ageneralized second-order contingent epiderivative and applied them to obtain opti-mality conditions in the primal form Following the Dubovitski–Milutin approach,

con-in [21], Khan and Tammer proved second-order optimality conditions in terms ofsecond-order asymptotic contingent derivatives Zhu et al [10] proposed and usedthe second-order composed contingent derivative to establish dual second-order con-ditions Higher-order necessary and sufficient optimality conditions for set-valuedoptimization can be seen in [7,9] In all the above results concerning second-ordernecessary optimality conditions, the envelope-like effect was not considered In [22],Studniarski introduced the concept of a higher-order local strict (known also as firm orisolated) minimizer for scalar programming and established necessary conditions andsufficient conditions for set-constrained minimization in infinite-dimensional spaces.This notion was extended to vector optimization in [23] and to set-valued vector opti-mization problems in [24], where conditions for local firm minimizers (of order 1)were obtained by using various generalized derivatives of set-valued maps Li et al

in [8] established a primal second-order sufficient optimality condition for local firmminimizers of order 2 for set-valued optimization with inclusion constraints Next,

it is worth noticing that [20] considered the envelope-like effect only for Fritz Johnsecond-order conditions, not KKT ones

Motivated by the above arguments, in this paper we consider second-order KKToptimality conditions for set-valued optimization under set constraints and general-ized inequality constraints with attention to the envelope-like effect Furthermore,

in [20], second-order approximations were used (as generalized derivatives), which

do not explicitly involve directions, while the envelope-like effect occurs only incertain critical directions Hence, we have chosen the second-order contingent andasymptotic contingent derivatives since they have close relations to useful kinds ofsecond-order tangent sets, which express approximating directions of the object underconsideration Moreover, in many papers, such as [3,7,9,10,16–19], the disjunctionmap, composed from the objective and the constraints, is often used in necessary con-ditions Under assumptions on cone-Aubin properties, we get some stronger resultsinvolving the objective map and the constraint map separately We obtain second-orderKKT multiplier rules under second-order qualification conditions of the Kurcyusz–Robinson–Zowe (KRZ) type We also compare these qualification conditions and someother existing ones In our second-order sufficient conditions for firm minimizers, noconvexity assumption is imposed

The organization of the paper is as follows In Sect.2, we collect definitions andpreliminary facts for our later use Section 3is devoted to second-order necessaryoptimality conditions in the primal and dual forms In Sect.4, we discuss second-order sufficient optimality conditions without any convexity assumptions

2 Preliminaries

Throughout the paper, if not otherwise stated, X, Y and Z will denote real Banachspaces ByN, BnandBn

+, we denote the set of the natural numbers, a n-dimensional

Banach space and its nonnegative orthant, respectively (resp) B X denotes the open

unit ball of X and B X (x, r) the open ball centred at x with radius r (similarly for other spaces) For A ⊆ X, intA, clA, bdA and convA stand for its interior, closure, boundary

and convex hull, resp The cone generated by A is cone A := {λx : λ ≥ 0, x ∈ A} X∗

stands for the dual space of X and .,  for the canonical pairing of any pair of dual

spaces For t ∈ R, t ↓ 0 means t > 0 and t → 0 Let C ⊆ Y be a closed and convex

cone, and let Y be partially ordered by C.

Given a set-valued map F : X ⇒ Y , the domain, graph and epigraph of F are

dom F

epiF := {(x, y) ∈ X × Y : y ∈ F(x) + C}, resp F(A) := x ∈A F (x) and the profile/epigraphic map F+ : X ⇒ Y is defined by F+(x) := F(x) + C for

x ∈ domF F is said to be upper semicontinuous (in short, u.s.c.) at x0 ∈ X iff,

for any neighbourhood V of F (x0), there exists a neighbourhood U of x0 such

that F (U) ⊆ V F is called a C-function iff, for all x1, x2 ∈ X and λ ∈ [0, 1],

λF(x1) + (1 − λ)F(x2) ⊆ F(λx1+ (1 − λ)x2 ) + C When C contains (or equals,

or is contained in) the nonnegative orthant, then “C-function” is specialized as

“C-convex” (or “convex” or “strictly C-convex”) The terms “C-concave”, cave” and “strictly C-concave” are similarly defined Clearly, F is a C-function if and only if gphF+ is convex F is said to be C-Aubin at (x0, y0) ∈ gphF (see

“con-[24,25]) iff there exist neighbourhoods U of x0, V of y0, and L > 0 such that

F (x) ∩ V ⊆ F(x ) + C + Lx − x clB Y , ∀x, x ∈ U If C = {0}, this is the

well-known Aubin property

We recall notions of tangency and corresponding generalized derivatives

Definition 2.1 Let M ⊆ X and x0 , u ∈ X.

(i) The contingent cone (interior tangent cone, resp) of M at x0is

T (M, x0) := {u ∈ X : ∃t n ↓ 0, ∃u n → u, ∀n, x0 + t n u n ∈ M}



I T (M, x0) := {u ∈ X : ∀t n ↓ 0, ∀u n → u, ∀n large, x0 + t n u n ∈ M} (ii) The second-order contingent set (resp, adjacent set and interior set) of M at x0in

Note that if x0 /∈ clM, then all above tangent sets are empty and if u /∈ T (M, x0),

then all above second-order tangent sets are empty Hence, we always assume

con-ditions such as x0 ∈ clM, u ∈ T (M, x0 ) Some known properties of second-order

tangent sets used later are collected in the following (see more in [14,16,17,19,26])

Proposition 2.1 Let M ⊆ X and x0 , u ∈ X.

(i) T2(M, x0, 0) = T (M, x0, 0) = T (M, x0).

(ii) If X is reflexible and u ∈ T (M, x0 ), then either T2(M, x0, u) or T (M, x0, u) is

Definition 2.2 Let F : X ⇒ Y , (x0 , y0) ∈ gphF and (u, v) ∈ X × Y

(i) The contingent derivative D F (x0, y0) of F at (x0, y0) is defined by gphDF(x0, y0)

:= T (gphF, (x0 , y0)).

(ii) The second-order contingent derivative D2F (x0, y0, u, v) of F at (x0, y0) in

direction(u, v) is defined by gphD2F (x0, y0, u, v):= T2(gphF, (x0, y0), (u, v)) The adjacent derivative D (2) F (x0, y0, u, v) is similar, with A2replacing T2

(iii) The second-order asymptotic contingent derivative D F (x0, y0, (u, v)) of

F at (x0, y0) in direction (u, v) is defined by gphD F (x0, y0, u, v) :=

T (gphF, (x0, y0), (u, v)) The adjacent derivative D ( ) F (x0, y0, u, v) is lar, with A replacing T

simi-(iv) The second-order composed contingent derivative D c (2) F (x0, y0, (u, v)) of

F at (x0, y0) in direction (u, v) is defined by gphD c (2) F (x0, y0, u, v) :=

T (T (gphF, (x0, y0)), (u, v)).

Remark 2.1 (i) If X × Y is reflexive and (u, v) ∈ T (gphF, (x0 , y0)), then, by

Propo-sition2.1(ii), T2(gphF, (x0, y0), (u, v))∪T (gphF, (x0, y0

(iii) Since T (gphF+, (x0, y0), (u, v)) is a cone, the second-order asymptotic gent derivative is strictly positive homogeneous, i.e D F+(x0, y0, u, v)(tx) =

contin-t D F+(x0, y0, u, v)(x) for all t > 0.

3 Second-Order Necessary Optimality Conditions

In this section, let C ⊆ Y be a closed and convex cone, which defines a partial order

on Y (C is not necessarily pointed) Let D be a convex cone in Z Our set-valued

vector optimization problem is

(P) MinC F

G : X ⇒ Z are nonempty-valued and S is a nonempty subset of X.

(F, G)(x) := F(x) × G(x).

The following minimizer notions of vector optimization are discussed in this paper

Definition 3.1 Let x0∈ , (x0 , y0) ∈ gphF, and m ∈ N.

(i) Supposing intC 0, y0) is said to be a local weak minimizer of (P),

denoted by(x0, y0) ∈ LWMin(P), iff there exists a neighbourhood U of x0suchthat

Proposition 3.1 Let (x0, y0) ∈ LWMin(P) and z0 ∈ G(x0 ) ∩ (−D) Then, for all

u ∈ X, v ∈ DF+(x0, y0) (u) ∩ (−bdC), and w ∈ DG+(x0, z0)(u) ∩ (−clD(z0)),

Proof We prove only part (i) because the proof of (ii) is similar Since (x0, y0) ∈

LWMin(P),∃U (a neighbourhood of x0),∀x ∈  ∩ U, (F(x) − y0 ) ∩ (−intC) = ∅ Suppose to the contrary the existence of x ∈ I T2(S, x0, u) and (y, z) ∈ Y × Z such that there are t n ↓ 0, x n → x, and (y n , z n ) → (y, z), with y ∈ I T (−C, v) and

n ≥ max{n1 , n2} Since I T (−C, v) = I T (−intC, v) and y ∈ I T (−C, v), for the

preceding sequences t n and y n , there exists n3∈ N such that t n v +1

Proposition 3.2 Let (x0, y0) ∈ LWMin(P) and z0 ∈ G(x0 ) ∩ (−D) Assume that

F+ is C-Aubin at (x0, y0) and G+ is D-Aubin at (x0, z0) Then, for all u ∈ X,

v ∈ DF+(x0, y0)(u) ∩(−bdC), and w ∈ DG+(x0, z0) (u) ∩ (−clD(z0)),

(i) for all x ∈ A2(S, x0, u),

Proof By reasons of similarity, we prove only part (i) There exists a neighbourhood

U of x0such that(F(x) − y0) ∩ (−intC) = ∅, ∀x ∈  ∩ U We define  := {x ∈

(I) if x ∈ I T2(, x0, u), then D2F+(x0, y0, u, v)(x) ∩ I T (−C, v) = ∅;

(II) if x /∈ I T2(, x0, u), then D (2) G+(x0, z0, u, w)(x) ∩ I T2(−D, z0, w) = ∅ First, we prove (I) Suppose there exists y ∈ D2F+(x0, y0, u, v)(x) ∩ I T (−C, v) Then, there are t n ↓ 0, x n → x, and y n → y such that, for all n ∈ N,

Since x ∈ A2(S, x0, u) ∩ I T2(, x0, u), for the preceding sequences t n, one has

x n → x and n1 ∈ N such that x0 + t n u+1

2t

2x n ∈ S ∩ , i.e x0 + t n u n+1

2t

2x n ∈ 

for all n ≥ n1 Since F+is C-Aubin at (x0, y0), there exist a neighbourhood V of y0

and L F > 0 such that, for large n,

As y n − L F x n − x n b n → y ∈ I T (−C, v) and I T (−C, v) = I T (−intC, v), there

exists n2 ∈ N such that t n v + 1

contradiction because(x0, y0) is a local weak minimizer of (P).

Next, we prove (II) Suppose there exists z ∈ D (2) G+(x0, z0, u, w)(x) ∩

I T2(−D, z0, w) For every t n ↓ 0, there exist sequences ¯x n → x and z n → z

such that, for all n∈ N,

Since G+is D-Aubin at (x0, z0), for every x n → x, there exist a neighbourhood V

of z0, L G > 0, and n3∈ N such that, for all n ≥ n3,

2x n ∈  and thus x ∈ I T2(, x0, u), a contradiction. 

Remark 3.1 (i) Propositions3.1and3.2can be modified for an arbitrary subset D of

Z (not a convex cone) Namely, we need only to replace G+ by G Indeed, the

inclusion (2) in the proof of Proposition3.1(resp, (4) in the proof of Proposition3.2) becomes z0+ t n w +1

rest of the proof is similar to the proof of the above propositions We note that

in [13,16–19], where single-valued maps( f, g) is in the place of (F, G), D is a

convex set and( f, g) is used instead of the profile map ( f+, g+).

(ii) By Remark 2.1(i), applying Propositions 3.1 and 3.2 with (u, (v, w)) = (0, (0, 0)), we immediately obtain the following first-order necessary optimal- ity condition in the primal form, for all x ∈ I T (S, x0 ),

D(F+, G+)(x0, (y0, z0))(x) ∩− int(C × D(z0 ))= ∅,

D F+(x0, y0)(x) × DG+(x0, z0)(x)∩− int(C × D(z0 ))= ∅.

Propositions3.1and3.2provide second-order information for local weak

min-imizers when direction u ∈ X satisfies the first-order condition critically in the

sense thatv ∈ DF+(x0, y0)(u)∩(−bdC) and w ∈ DG+(x0, z0)(u)∩−clD(z0).

We note that in many known second-order necessary conditions, such a criticaldirectionw is not considered For instance, w is only in −D in [7,9,10], in

−intD − R+z0in [3], and in−D(z0 ) in [27] Only in directionsw belonging

to the additional part−(clD(z0 )\D(z0)) can the so-called envelope-like effect

occur, as we will see in Theorems3.1and3.2below

(iii) Sincev ∈ −C, z0 ∈ −D, and w ∈ −clD(z0 ), one has −intC × (−intD −

R+z0) ⊆ I T (−C, v) × I T2(−D, z0, w) Hence, Proposition 3.1(i) improvesProposition 3.8 in [3] As−intC − v ⊆ I T (−C, v), Proposition3.1(i) sharpensTheorem3.1in [6] We note that in many higher-order necessary conditions forset-valued optimization, e.g [3,7,9,10], only−intD − R+z0or−D(z0 ) or −D are involved I T2(−D, z0, w) and I T (−D, z0, w) in Propositions3.1and3.2play an important role in establishing necessary conditions in the dual form, as

we will see in Theorems3.1-3.4below

(iv) With the use of asymptotic objects, Proposition3.1(ii) improves Theorem 6.2 in[21], and Proposition3.2(ii) improves Theorem 6.1 in [21], since the authors of[21] used(F, G), not the profile map (F+, G+) as in our results.

(v) If epiG is second-order derivable at (x0, y0, u, v), one has D (2) G+(x0, z0, u, w) (x) = D2G+(x0, z0, u, w)(x) Hence, because D2(F+, G+)(x0, (y0, z0), u, (v, w))(x) ⊆ D2F+(x0, y0, u, v)(x) × D (2) G+(x0, z0, u, w)(x), Proposition

3.2(i) is stronger than Proposition 3.1(i) Similarly, if epiG is second-order

asymptotic derivable at(x0, y0, u, v), then Proposition3.2(ii) sharpens sition3.1(ii) In [21], the Aubin property of F and G was imposed Here, we use the relaxed property that F+is C-Aubin at (x0, y0) and G+is D-Aubin at (x0, z0).

Propo-For the cone C ⊆ Y (resp, D ⊆ Z), C∗ := {c∗ ∈ Y∗ : c∗, c ≥ 0, ∀c ∈ C}

is the polar cone of C (resp, D∗) Then, it is not hard to check that, for z0 ∈ −D, [D(z0 )]∗ = N(−D, z0 ), the normal cone of −D at z0 Note that, if D is a convex cone, then N (−D, z0) = {d∗∈ D∗: d∗, z0 = 0}.

Now, we are able to establish a dual-form second-order necessary condition forlocal weak minimizers in terms of KKT multipliers.

Theorem 3.1 Let (x0, y0) ∈ LWMin(P) and z0 ∈ G(x0 ) ∩ (−D) Then, for all

u ∈ X, v ∈ DF+(x0, y0)(u) ∩ (−bdC), and w ∈ DG+(x0, z0)(u) ∩ (−clD(z0)),

the following statements hold.

(i) For all x ∈ I T2(S, x0, u) and (y, z) ∈ D2(F+, G+)(x0, (y0, z0), u, (v, w))(x),

there exists (c∗, d∗) ∈ C∗× N(−D, z0 )\{(0, 0)} such that c∗, v = d∗, w = 0

and

c∗, y + d∗, z ≥ sup d ∈A2(−D,z0,w) d∗, d.

(ii) In particular, for (u, v, w) such that D2(F+, G+)(x0, (y0, z0), u, (v, w))(I T2(S,

x0, u)) is a convex set, there exists a common (c∗, d∗) ∈ C∗× N(−D, z0 )\{(0, 0)}

such that c∗, v = d∗, w = 0 and

c∗, y + d∗, z ≥ sup d ∈A2(−D,z0,w) d∗, d

for all (x, y, z) mentioned in (i) Moreover, c∗

condition of the KRZ type is fulfilled:

{z ∈ Z : (y, z) ∈ cone(D2(F+, G+)(x0, (y0, z0), u, (v, w))(I T2(S, x0, u))

−{0} × A2(−D, z0, w))} + D(z0) = Z.

Proof (i) By Proposition3.1(i), for all x ∈ I T2(S, x0, u) and (y, z) ∈ D2(F+, G+) (x0, (y0, z0), u, (v, w))(x), one has (y, z) /∈ I T (−C, v) × I T2(−D, z0, w) By

the standard separation theorem, we obtain(c∗, d∗) ∈ Y∗× Z∗\{(0, 0)}, such that,

for all c ∈ I T (−C, v) and d ∈ I T2(−D, z0, w),

c∗, y + d∗, z ≥ c∗, c + d∗, d. (5)

Since C is a convex cone, one has I T (−C, v) = int(cone(−C − v))

(accord-ing to Proposition 2.4 in [14]) It follows from (5) that c∗, c ≤ 0 for all c ∈ cone(−C − v) This leads to c∗ ∈ [cone(C + v)]∗ Because

v ∈ −bdC, one has c∗ ∈ C∗ and c∗, v = 0 According to

Proposi-tion2.1, one has clI T2(−D, z0, w) = A2 (−D, z0, w) and A2(−D, z0, w) +

T (T (−D, z0), w) ⊆ A2(−D, z0, w) From (5), by taking c= 0, one has, for all

By the assumed convexity of D2(F+, G+)(x0, (y0, z0), u, (v, w))(I T2(S, x0, u)),

similarly as for part (i), we obtain(c∗, d∗) ∈ C∗× N(−D, z0 )\{(0, 0)} such that

c∗, v = d∗, w = 0 and

c∗, y + d∗, z ≥ sup d ∈A2(−D,z0,w) d∗, d

for every(y, z) ∈ D2(F+, G+)(x0, (y0, z0), u, (v, w))(I T2(S, x0, u)).

hasd∗, z ≥ sup d ∈A2(−D,z0,w) d∗, d for every (y, z) ∈ D2(F+, G+)(x0, (y0, z0), u, (v, w))(I T2(S, x0, u)) By the qualification condition, ∃¯z ∈ Z, ∃t1, t2 ≥ 0,

∃z ∈ {z ∈ Z : (y , z ) ∈ D2(F+, G+)(x0, (y0, z0), u, (v, w)) (I T2(S, x0, u))},

ˆd ∈ A2(−D, z0, w), ∃d ∈ D, ¯z = t1(z − ˆd) + t2(d + z0) Since d∗ ∈ D∗ and

d∗, z0 = 0,

d∗, ¯z=t1d ∗, z − ˆd + t2d ∗, d +z0≥t1(sup d ∈A2(−D,z0,w) d∗, d − d∗, ˆd) ≥ 0.

Since¯z ∈ Z is arbitrary, we have d∗= 0, a contradiction because (c∗, d∗

Under a cone-Aubin property, we derive a deeper dual-form second-order necessary

condition for local weak minimizers, where the assumptions on F and G are separate, and hence, we need only a constraint qualification involving G, not the objective F ,

as follows

Theorem 3.2 Let (x0, y0) ∈ gphF be a local weak minimizer of (P) and z0∈ G(x0 )∩ (−D) Assume that F+is C-Aubin at (x0, y0) and G+is D-Aubin at (x0, z0) Then, for

all u ∈ X, v ∈ DF+(x0, y0)(u) ∩ (−bdC), and w ∈ DG+(x0, z0)(u) ∩ (−clD(z0)),

the following statements hold.

(i) For all x ∈ A2(S, x0, u), y ∈ D2F+(x0, y0, u, v)(x), and z ∈ D (2) G+(x0, z0, u, w)(x), there exists (c∗, d∗) ∈ C∗× N(−D, z0 )\{(0, 0)} such that c∗, v =

d∗, w = 0 and

c∗, y + d∗, z ≥ sup d ∈A2(−D,z0,w) d∗, d.

(ii) In particular, for (u, v, w) such that (D2F+(x0, y0, u, v), D (2) G+(x0, z0, u, w)) (A2(S, x0, u)) is convex, there exists a common (c∗, d∗) ∈ C∗× N(−D, z0 )\{(0,

0)} satisfying c∗, v = d∗, w = 0 and c∗, y + d∗, z ≥ sup d ∈A2(−D,z0,w)

d∗, d for all (x, y, z) mentioned in (i) Moreover, c∗

straint qualification of the KRZ type is satisfied:

cone(D (2) G+(x0, z0, u, w)(A2(S, x0, u))− A2(−D, z0, w)) + D(z0)= Z (6)

Proof Argue similarly as for Theorem 3.1, applying Proposition 3.2(i) instead of

In the next example, Theorem3.1rejects a candidate and supd ∈A2(−D,z0,w) d∗, d

< 0.