In this work we present an extention of the well-known Approximated Gradient Projection (AGP) [8] property from the scalar problem with equality and inequality constraints to multiobjective problems. We prove that the condition called Multiobjective Approximate Gradient Projection (MAGP), is necessary for a point to be a local weak Pareto point and we study, under convex assumptions, sufficient conditions.Facultad de Ingenierí

Ramos, Alberto

Schuverdt, María Laura

Sánchez, María Daniela

SEDICI - Repositorio de la UNLP

APPROXIMATE GRADIENT PROJECTED CONDITION IN
MULTIOBJECTIVE OPTIMIZATION
Ramos, Alberto†, Sa´nchez, Marı´a Daniela‡ and Schuverdt, Marı´a Laura‡
†Department of Mathematics, Federal University of Parana, Curitiba-PR, Brazil,albertoramos@ufpr.br
‡Faculty of Engineering, University of La Plata, CP 172, 1900 La Plata Bs. As., Argentina,
dsanchez@mate.unlp.edu.ar
‡CONICET, University of La Plata, CP 172, 1900 La Plata Bs. As., Argentina, schuverd@mate.unlp.edu.ar
Abstract: In this work we present an extention of the well-known Approximated Gradient Projection (AGP) [8]
property from the scalar problem with equality and inequality constraints to multiobjective problems. We prove that
the condition called Multiobjective Approximate Gradient Projection (MAGP), is necessary for a point to be a local
weak Pareto point and we study, under convex assumptions, sufficient conditions.
Keywords: Sequential optimality conditions, multiobjective problems, Fritz-John conditions.
2000 AMS Subject Classification: 21A54 - 55P54
1 INTRODUCTION
We will consider the multiobjective optimization problem (MOP) of the form:
Minimize F (x) subject to x ∈ Ω (1)
where Ω = {x ∈ Rn : h(x) = 0, g(x) ≤ 0} and F : Rn → Rr, h : Rn → Rm, g : Rn → Rp are
continuously differentiable functions and F (x) = (f1(x), . . . , fr(x)).
In [8] the authors introduce a sequential optimality condition, for the scalar optimization problem (r =
1), called Approximate Gradient Projection (AGP). It is proved that the AGP property is satisfied by local
minimizers of constrained optimization problems independently of constraint qualification.
One of the attractiveness of AGP is that it does not involve Lagrange multipliers estimates. AGP is also
the natural optimality condition that fits a stopping criteria for algorithms based on inexact restoration, and
it is strictly stronger than the usual AKKT condition [7]. Consequently, the stopping criteria based on AGP
is more reliable that those based on AKKT.
In this paper, we will extend the AGP condition to multiobjective problems, in the same way as the
AKKT condition [7] has been extended recently in [5] for multiobjective problems.
Given the multiobjective optimization problem in the form (1), a point x∗ ∈ Ω is said to be a Pareto
optimal solution [9] or an efficient solution if there is no x ∈ Ω such that F (x) ≤ F (x∗) and F (x) 6=
F (x∗). A point x∗ ∈ Ω is said to be a weak Pareto optimal solution of (1) if there is no x ∈ Ω such that
F (x) < F (x∗). A point x∗ is a local weak Pareto optimal solution if there exists a neighbourhood N ∩ Ω
of x∗ such that there is no x ∈ N with F (x) < F (x∗). In this paper we are interested in the Fritz-John
necessary condition for weak Pareto optimality stated in the following theorem:
Theorem 1 [4] A necessary condition for x∗ ∈ Ω to be a local weak Pareto optimal solution is that there
exist vectors θ ∈ Rr, λ ∈ Rm, µ ∈ Rp such that θ > 0, µ > 0 and
r∑
l=1
θl∇fl(x∗) +
m∑
i=1
λi∇hi(x∗) +
p∑
j=1
µj∇gj(x∗) = 0, (2)
µjgj(x
∗) = 0, j = 1, . . . , p, (3)
(θ, λ, µ) 6= (0, 0, 0). (4)
The non-zero vector (θ, λ, µ) satisfying (2)-(4) is known as the Fritz-John multiplier vector.
Following the classification in [2], given x∗ ∈ Ω we say that x∗ is weak- regular if there exists (θ, λ, µ)
satisfying (2)-(4) with θ 6= 0. Observe that a weak-regular point is a Karush-Kuhn-Tucker point when r = 1.
2 APPROXIMATE GRADIENT PROPERTY FOR MULTIOBJECTIVE PROBLEMS
In this section we will extend rigorously the AGP condition for multiobjective problems and we will
analyze its properties.
Definition 1 Given γ ∈ [−∞, 0]. We say that x∗ ∈ Ω satisfies the Multiobjective Approximate Gradient
Projection (MAGP (γ)) condition, if and only if, there exist sequences {θk} ⊂ Rr, {xk} such that θk > 0,
r∑
l=1
θkl = 1, x
k → x∗ and
d(xk,γ) = PΩ(xk,γ)(x
k −
r∑
l=1
θkl ∇fl(xk))− xk → 0
where PΩ(xk,γ) is the orthogonal projection and Ω(x, γ) is the polyhedron defined by the points z ∈ Rn
such that:
1. gj(x) +∇gj(x)T (z − x) 6 0 if γ < gj(x) < 0,
2. ∇gj(x)T (z − x) 6 0 if gj(x) > 0,
3. ∇hi(x)T (z − x) = 0 for i = 1, . . . ,m.
Observe that Ω(x, 0) = {z ∈ Rn|∇h(x)T (z−x) = 0,∇gj(x)T (z−x) 6 0 if gj(x) > 0, j = 1, . . . , p}.
In order to establish necessary optimality conditions for problem (1) we are going to scalarize it. For this
aim, we consider the non-smooth function φ : Rr → R defined by
φ(y) = max
1≤l≤r
{yl}.
As φ is a non-smooth function we introduce the so-called Clarke subdifferential of ψ (see [3]): The upper
Clarke directional derivative of a locally Lipschitz function ψ : Rn → R at x ∈ Rn in the direction d ∈ Rn
is
ψo(x, d) = lim sup
y→x,t↓0
ψ(y + td)− ψ(y)
t
and the Clarke subdifferential of ψ at x is given by
∂Cψ(x) = {ξ ∈ Rn : 〈ξ, d〉 6 ψo(x, d)∀d ∈ Rn},
where 〈·, ·〉 stands for the Euclidean scalar product.
Lemma 1 If x∗ is a local weak Pareto optimal point of (1), then x∗ is a local minimizer of the problem
Minimize φ(F (x)− F (x∗)) subject to x ∈ Ω.
The following result says that, every local weak Pareto solution of (1) satisfies theMAGP (γ) condition.
Theorem 2 If x∗ ∈ Ω is a local weak Pareto solution of problem (1), then x∗ satisfies the MAGP (γ)
condition.
Proof. Let ε, δ > 0, and ρ ∈ (0, δ). By assumption and Lemma 1 we can conclude that x∗ is a global
minimizer of φ(F (x)−F (x∗)) on Ω∩B(x∗, ρ). In consequence, we may suppose that for δ small enough,
x∗ is the unique solution of problem
Minimize φ(F (x)− F (x∗)) + ε
2ρ
‖x− x∗‖2, subject to x ∈ Ω, ‖x− x∗‖ ≤ ρ.
We define, for each k ∈ N,
ψk(x) = φ(F (x)− F (x∗)) + ε
2ρ
‖x− x∗‖2 +
m∑
i=1
khi(x)
2 +
p∑
j=1
kgj(x)
2
+,
and let xk be the solution of the problem
Minimize ψk(x), subject to ‖x− x∗‖ ≤ ρ. (5)
By using the theory of external penalty, see for example [5, 6, 8], we have that ‖xk − x∗‖ < δ for all k
sufficiently large, xk → x∗ and xk is a solution of problem (5) and it is an interior point of the feasible set,
for k large enough. From Proposition 2.1 (ii) of [5] it follows that 0 ∈ ∂Cψk(xk). By applying Proposition
2.1 of [5], parts (i) and (iii), we have
0 ∈ co
( ⋃
1≤l≤r
{∇fl(xk)}
)
+
ε
ρ
(xk − x∗) +
m∑
i=1
khi(x
k)∇hi(xk) +
p∑
j=1
kgj(x
k)+∇gj(xk),
where co
( ⋃
1≤l≤r
{∇fl(xk)}
)
denote the convex hull. Hence, there exist θkl ≥ 0, l = 1, . . . , r such that
r∑
l=1
θkl = 1 and
r∑
l=1
θkl ∇fl(xk) + k
( m∑
i=1
hi(x
k)∇hi(xk) +
p∑
j=1
gj(x
k)+∇gj(xk)
)
=
ε
ρ
(x∗ − xk).
Then, since ‖x∗ − xk‖ < ρ < δ,∥∥∥∥ r∑
l=1
θkl ∇fl(xk) + k
( m∑
i=1
hi(x
k)∇hi(xk) +
p∑
j=1
gj(x
k)+∇gj(xk)
)∥∥∥∥ ≤ ε.
So, ∥∥∥∥[xk − r∑
l=1
θl∇fl(xk)
]
−
[
xk + k
( m∑
i=1
hi(x
k)∇hi(xk) +
p∑
j=1
gj(x
k)+∇gj(xk)
)∥∥∥∥ ≤ ε.
This implies, taking projections onto Ω(xk, γ), that
∥∥∥∥PΩ(xk,γ)(xk− r∑
l=1
θl∇fl(xk)
)
−PΩ(xk,γ)
(
xk+k
( m∑
i=1
hi(x
k)∇hi(xk)+
p∑
j=1
gj(x
k)+∇gj(xk)
))∥∥∥∥ ≤ ε.
It remains to prove that PΩ(xk,γ)
(
xk + k
( m∑
i=1
hi(x
k)∇hi(xk) +
p∑
j=1
gj(x
k)+∇gj(xk)
))
= xk.
To see that , we consider the Karush-Kuhn-Tucker conditions of the subproblem with linear constraints
Minimize y
∥∥∥∥y − [xk + k( m∑
i=1
hi(x
k)∇hi(xk) +
p∑
j=1
gj(x
k)+∇gj(xk)
)]∥∥∥∥2
subject to y ∈ Ω(xk, γ)
and observe that y = xk satisfies the KKT optimality conditions with multipliers λi = khi(xk) and µj =
kgj(x
k) with µj = 0 when gj(xk) ≥ 0. So,
∥∥∥∥PΩ(xk,γ)(xk − r∑
l=1
θkl ∇fl(xk)
)
− xk
∥∥∥∥ ≤ ε as we wanted to
prove. 
Note that, from Theorem 2, we can ensure thatMAGP (γ) is a necessary sequential optimality condition
for any local weak Pareto solution.
In the following theorem we establish that, under the Cone-Continuity Property (CCP), the MAGP con-
dition implies weak-regularity.
Given x∗ ∈ Rn such that h(x∗) = 0, g(x∗) ≤ 0, we consider the closed convex cone defined as
K(x) =
{ m∑
i=1
λi∇hi(x) +
∑
j∈A(x∗)
µj∇gj(x) : λi ∈ R, µj ∈ R, µj > 0
}
where A(x∗) = {j = 1, . . . , p : gj(x∗) = 0} is the set of active inequality constraints at x∗.
Given a set-valued mapping (multifunction) F : Rs ⇒ Rd, the outer limit of F (z) as z → z∗ is denoted
by
lim sup
z→z∗
F (z) = {w∗ ∈ Rd : ∃ (zk, wk)→ (z∗, w∗) with wk ∈ F (zk)}.
Then, the multifunction F is said to be outer semicontinuous at z∗ if lim sup
z→z∗
F (z) ⊂ F (z∗).
Definition 2 [1] We say that x∗ ∈ Rn such that h(x∗) = 0, g(x∗) ≤ 0 satisfies the Cone-Continuity
Property (CCP) if the multifunction x⇒ K(x) is outer semicontinuous at x∗, that is
lim sup
x→x∗
K(x) ⊂ K(x∗).
Theorem 3 Let x∗ be a feasible point and γ ∈ [−∞, 0]. Assuming that the MAGP (γ) condition holds at
x∗, and furthermore, suppose that x∗ satisfies CCP. Then, x∗ is a weak-regular point of the problem (1).
In [8] it is proved that AGP is a sufficient condition for the scalar problem. In the following theorem we
extend the same result in the multiobjective case.
Theorem 4 Suppose that, in the multiobjective optimization problem (1), fl, l = 1, . . . , r and gi i =
1, . . . , p are convex and h is an affine function. Let γ ∈ [−∞, 0]. Suppose that x∗ ∈ Ω, {xk} ⊂ Rn and
{θk} ⊂ Rr are such that
r∑
l=1
θkl = 1, θ
k
l > 0, xk → x∗, h(xk) = 0 for all k = 0, 1, 2, . . . and d(xk, γ) = 0.
Then, x∗ is a weak Pareto solution of (1).
REFERENCES
[1] R. ANDREANI, J. M. MARTI´NEZ, A. RAMOS, AND P. J. S. SILVA, A cone-continuity constraint qualification and algorithmic
consequences, SIAM Journal on Optimization, 26(1):96-110, 2016.
[2] G. BIGI, M. PAPPALARDO, Regularity conditions in vector optimization, Journal of Optimization Theory and Applications,
102:83-96, 1999.
[3] F.H.CLARKE, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.
[4] N.O. DA CUNHA, E. POLAK, Constraint minimization under vector-value criteria in finite dimensional spaces, Journal of
Mathematical Analysis and Applications, 19:103-124, 1967.
[5] G. GIORGI, B. JIME´NEZ, V. NOVO, Approximate Karush-Kuhn-Tucker Condition in Multiobjective Optimization,Journal of
Optimization Theory and Applications, 171 , no. 1, 70-89, 2016.
[6] G. HAESER, M. L. SCHUVERDT, On approximate KKT condition and its extension to continuous variational inequalities,
Journal of Optimization Theory and Applications, 23:707-716, 2011.
[7] J.M. MARTI´NEZ, E.G. BIRGIN, Practical Augmented Lagrangian Methods for Constrained Optimization, SIAM Publications,
2014.
[8] J.M. MARTI´NEZ, B.F. SVAITER , A practical optimality condition without constraint qualifications for nonlinear program-
ming. J. Optim. Theory Appl. 118, 117-133,2003.
[9] V. PARETO, Cours d’Economie Politique, Rouge, Lausanne, Switxerland, 1986.


English

Approximate gradient projected condition in multiobjective optimization

Servicio de Difusión de la Creación Intelectual

APPROXIMATE GRADIENT PROJECTED CONDITION INMULTIOBJECTIVE OPTIMIZATIONRamos, Alberto†, Sa´nchez, Marı´a Daniela‡ and Schuverdt, Marı´a Laura‡†Department of Mathematics, Federal University of Parana, Curitiba-PR, Brazil,albertoramos@ufpr.br‡Faculty of Engineering, University of La Plata, CP 172, 1900 La Plata Bs. As., Argentina,dsanchez@mate.unlp.edu.ar‡CONICET, University of La Plata, CP 172, 1900 La Plata Bs. As., Argentina, schuverd@mate.unlp.edu.arAbstract: In this work we present an extention of the well-known Approximated Gradient Projection (AGP) [8]property from the scalar problem with equality and inequality constraints to multiobjective problems. We prove thatthe condition called Multiobjective Approximate Gradient Projection (MAGP), is necessary for a point to be a localweak Pareto point and we study, under convex assumptions, sufficient conditions.Keywords: Sequential optimality conditions, multiobjective problems, Fritz-John conditions.2000 AMS Subject Classification: 21A54 - 55P541 INTRODUCTIONWe will consider the multiobjective optimization problem (MOP) of the form:Minimize F (x) subject to x ∈ Ω (1)where Ω = {x ∈ Rn : h(x) = 0, g(x) ≤ 0} and F : Rn → Rr, h : Rn → Rm, g : Rn → Rp arecontinuously differentiable functions and F (x) = (f1(x), . . . , fr(x)).In [8] the authors introduce a sequential optimality condition, for the scalar optimization problem (r =1), called Approximate Gradient Projection (AGP). It is proved that the AGP property is satisfied by localminimizers of constrained optimization problems independently of constraint qualification.One of the attractiveness of AGP is that it does not involve Lagrange multipliers estimates. AGP is alsothe natural optimality condition that fits a stopping criteria for algorithms based on inexact restoration, andit is strictly stronger than the usual AKKT condition [7]. Consequently, the stopping criteria based on AGPis more reliable that those based on AKKT.In this paper, we will extend the AGP condition to multiobjective problems, in the same way as theAKKT condition [7] has been extended recently in [5] for multiobjective problems.Given the multiobjective optimization problem in the form (1), a point x∗ ∈ Ω is said to be a Paretooptimal solution [9] or an efficient solution if there is no x ∈ Ω such that F (x) ≤ F (x∗) and F (x) 6=F (x∗). A point x∗ ∈ Ω is said to be a weak Pareto optimal solution of (1) if there is no x ∈ Ω such thatF (x) < F (x∗). A point x∗ is a local weak Pareto optimal solution if there exists a neighbourhood N ∩ Ωof x∗ such that there is no x ∈ N with F (x) < F (x∗). In this paper we are interested in the Fritz-Johnnecessary condition for weak Pareto optimality stated in the following theorem:Theorem 1 [4] A necessary condition for x∗ ∈ Ω to be a local weak Pareto optimal solution is that thereexist vectors θ ∈ Rr, λ ∈ Rm, µ ∈ Rp such that θ > 0, µ > 0 andr∑l=1θl∇fl(x∗) +m∑i=1λi∇hi(x∗) +p∑j=1µj∇gj(x∗) = 0, (2)µjgj(x∗) = 0, j = 1, . . . , p, (3)(θ, λ, µ) 6= (0, 0, 0). (4)The non-zero vector (θ, λ, µ) satisfying (2)-(4) is known as the Fritz-John multiplier vector.Following the classification in [2], given x∗ ∈ Ω we say that x∗ is weak- regular if there exists (θ, λ, µ)satisfying (2)-(4) with θ 6= 0. Observe that a weak-regular point is a Karush-Kuhn-Tucker point when r = 1.2 APPROXIMATE GRADIENT PROPERTY FOR MULTIOBJECTIVE PROBLEMSIn this section we will extend rigorously the AGP condition for multiobjective problems and we willanalyze its properties.Definition 1 Given γ ∈ [−∞, 0]. We say that x∗ ∈ Ω satisfies the Multiobjective Approximate GradientProjection (MAGP (γ)) condition, if and only if, there exist sequences {θk} ⊂ Rr, {xk} such that θk > 0,r∑l=1θkl = 1, xk → x∗ andd(xk,γ) = PΩ(xk,γ)(xk −r∑l=1θkl ∇fl(xk))− xk → 0where PΩ(xk,γ) is the orthogonal projection and Ω(x, γ) is the polyhedron defined by the points z ∈ Rnsuch that:1. gj(x) +∇gj(x)T (z − x) 6 0 if γ < gj(x) < 0,2. ∇gj(x)T (z − x) 6 0 if gj(x) > 0,3. ∇hi(x)T (z − x) = 0 for i = 1, . . . ,m.Observe that Ω(x, 0) = {z ∈ Rn|∇h(x)T (z−x) = 0,∇gj(x)T (z−x) 6 0 if gj(x) > 0, j = 1, . . . , p}.In order to establish necessary optimality conditions for problem (1) we are going to scalarize it. For thisaim, we consider the non-smooth function φ : Rr → R defined byφ(y) = max1≤l≤r{yl}.As φ is a non-smooth function we introduce the so-called Clarke subdifferential of ψ (see [3]): The upperClarke directional derivative of a locally Lipschitz function ψ : Rn → R at x ∈ Rn in the direction d ∈ Rnisψo(x, d) = lim supy→x,t↓0ψ(y + td)− ψ(y)tand the Clarke subdifferential of ψ at x is given by∂Cψ(x) = {ξ ∈ Rn : 〈ξ, d〉 6 ψo(x, d)∀d ∈ Rn},where 〈·, ·〉 stands for the Euclidean scalar product.Lemma 1 If x∗ is a local weak Pareto optimal point of (1), then x∗ is a local minimizer of the problemMinimize φ(F (x)− F (x∗)) subject to x ∈ Ω.The following result says that, every local weak Pareto solution of (1) satisfies theMAGP (γ) condition.Theorem 2 If x∗ ∈ Ω is a local weak Pareto solution of problem (1), then x∗ satisfies the MAGP (γ)condition.Proof. Let ε, δ > 0, and ρ ∈ (0, δ). By assumption and Lemma 1 we can conclude that x∗ is a globalminimizer of φ(F (x)−F (x∗)) on Ω∩B(x∗, ρ). In consequence, we may suppose that for δ small enough,x∗ is the unique solution of problemMinimize φ(F (x)− F (x∗)) + ε2ρ‖x− x∗‖2, subject to x ∈ Ω, ‖x− x∗‖ ≤ ρ.We define, for each k ∈ N,ψk(x) = φ(F (x)− F (x∗)) + ε2ρ‖x− x∗‖2 +m∑i=1khi(x)2 +p∑j=1kgj(x)2+,and let xk be the solution of the problemMinimize ψk(x), subject to ‖x− x∗‖ ≤ ρ. (5)By using the theory of external penalty, see for example [5, 6, 8], we have that ‖xk − x∗‖ < δ for all ksufficiently large, xk → x∗ and xk is a solution of problem (5) and it is an interior point of the feasible set,for k large enough. From Proposition 2.1 (ii) of [5] it follows that 0 ∈ ∂Cψk(xk). By applying Proposition2.1 of [5], parts (i) and (iii), we have0 ∈ co( ⋃1≤l≤r{∇fl(xk)})+ερ(xk − x∗) +m∑i=1khi(xk)∇hi(xk) +p∑j=1kgj(xk)+∇gj(xk),where co( ⋃1≤l≤r{∇fl(xk)})denote the convex hull. Hence, there exist θkl ≥ 0, l = 1, . . . , r such thatr∑l=1θkl = 1 andr∑l=1θkl ∇fl(xk) + k( m∑i=1hi(xk)∇hi(xk) +p∑j=1gj(xk)+∇gj(xk))=ερ(x∗ − xk).Then, since ‖x∗ − xk‖ < ρ < δ,∥∥∥∥ r∑l=1θkl ∇fl(xk) + k( m∑i=1hi(xk)∇hi(xk) +p∑j=1gj(xk)+∇gj(xk))∥∥∥∥ ≤ ε.So, ∥∥∥∥[xk − r∑l=1θl∇fl(xk)]−[xk + k( m∑i=1hi(xk)∇hi(xk) +p∑j=1gj(xk)+∇gj(xk))∥∥∥∥ ≤ ε.This implies, taking projections onto Ω(xk, γ), that∥∥∥∥PΩ(xk,γ)(xk− r∑l=1θl∇fl(xk))−PΩ(xk,γ)(xk+k( m∑i=1hi(xk)∇hi(xk)+p∑j=1gj(xk)+∇gj(xk)))∥∥∥∥ ≤ ε.It remains to prove that PΩ(xk,γ)(xk + k( m∑i=1hi(xk)∇hi(xk) +p∑j=1gj(xk)+∇gj(xk)))= xk.To see that , we consider the Karush-Kuhn-Tucker conditions of the subproblem with linear constraintsMinimize y∥∥∥∥y − [xk + k( m∑i=1hi(xk)∇hi(xk) +p∑j=1gj(xk)+∇gj(xk))]∥∥∥∥2subject to y ∈ Ω(xk, γ)and observe that y = xk satisfies the KKT optimality conditions with multipliers λi = khi(xk) and µj =kgj(xk) with µj = 0 when gj(xk) ≥ 0. So,∥∥∥∥PΩ(xk,γ)(xk − r∑l=1θkl ∇fl(xk))− xk∥∥∥∥ ≤ ε as we wanted toprove. Note that, from Theorem 2, we can ensure thatMAGP (γ) is a necessary sequential optimality conditionfor any local weak Pareto solution.In the following theorem we establish that, under the Cone-Continuity Property (CCP), the MAGP con-dition implies weak-regularity.Given x∗ ∈ Rn such that h(x∗) = 0, g(x∗) ≤ 0, we consider the closed convex cone defined asK(x) ={ m∑i=1λi∇hi(x) +∑j∈A(x∗)µj∇gj(x) : λi ∈ R, µj ∈ R, µj > 0}where A(x∗) = {j = 1, . . . , p : gj(x∗) = 0} is the set of active inequality constraints at x∗.Given a set-valued mapping (multifunction) F : Rs ⇒ Rd, the outer limit of F (z) as z → z∗ is denotedbylim supz→z∗F (z) = {w∗ ∈ Rd : ∃ (zk, wk)→ (z∗, w∗) with wk ∈ F (zk)}.Then, the multifunction F is said to be outer semicontinuous at z∗ if lim supz→z∗F (z) ⊂ F (z∗).Definition 2 [1] We say that x∗ ∈ Rn such that h(x∗) = 0, g(x∗) ≤ 0 satisfies the Cone-ContinuityProperty (CCP) if the multifunction x⇒ K(x) is outer semicontinuous at x∗, that islim supx→x∗K(x) ⊂ K(x∗).Theorem 3 Let x∗ be a feasible point and γ ∈ [−∞, 0]. Assuming that the MAGP (γ) condition holds atx∗, and furthermore, suppose that x∗ satisfies CCP. Then, x∗ is a weak-regular point of the problem (1).In [8] it is proved that AGP is a sufficient condition for the scalar problem. In the following theorem weextend the same result in the multiobjective case.Theorem 4 Suppose that, in the multiobjective optimization problem (1), fl, l = 1, . . . , r and gi i =1, . . . , p are convex and h is an affine function. Let γ ∈ [−∞, 0]. Suppose that x∗ ∈ Ω, {xk} ⊂ Rn and{θk} ⊂ Rr are such thatr∑l=1θkl = 1, θkl > 0, xk → x∗, h(xk) = 0 for all k = 0, 1, 2, . . . and d(xk, γ) = 0.Then, x∗ is a weak Pareto solution of (1).REFERENCES[1] R. ANDREANI, J. M. MARTI´NEZ, A. RAMOS, AND P. J. S. SILVA, A cone-continuity constraint qualification and algorithmicconsequences, SIAM Journal on Optimization, 26(1):96-110, 2016.[2] G. BIGI, M. PAPPALARDO, Regularity conditions in vector optimization, Journal of Optimization Theory and Applications,102:83-96, 1999.[3] F.H.CLARKE, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.[4] N.O. DA CUNHA, E. POLAK, Constraint minimization under vector-value criteria in finite dimensional spaces, Journal ofMathematical Analysis and Applications, 19:103-124, 1967.[5] G. GIORGI, B. JIME´NEZ, V. NOVO, Approximate Karush-Kuhn-Tucker Condition in Multiobjective Optimization,Journal ofOptimization Theory and Applications, 171 , no. 1, 70-89, 2016.[6] G. HAESER, M. L. SCHUVERDT, On approximate KKT condition and its extension to continuous variational inequalities,Journal of Optimization Theory and Applications, 23:707-716, 2011.[7] J.M. MARTI´NEZ, E.G. BIRGIN, Practical Augmented Lagrangian Methods for Constrained Optimization, SIAM Publications,2014.[8] J.M. MARTI´NEZ, B.F. SVAITER , A practical optimality condition without constraint qualifications for nonlinear program-ming. J. Optim. Theory Appl. 118, 117-133,2003.[9] V. PARETO, Cours d’Economie Politique, Rouge, Lausanne, Switxerland, 1986.

http://sedici.unlp.edu.ar/bitstream/handle/10915/81899/Documento_completo.pdf?sequence=1

Approximate gradient projected condition in multiobjective optimization

Abstract

Similar works

Full text

Available Versions

SEDICI - Repositorio de la UNLP

Servicio de Difusión de la Creación Intelectual