Inexact projected gradient methods for vector optimization problems on variable ordered spaces

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by  Jose Yunier Bello Cruz

Noname manuscript No.
(will be inserted by the editor)
Inexact projected gradient methods for vector
optimization problems on variable ordered spaces
7
1 Jose Yunier Bello Cruz · Gemayqzel Bouza
0 Allende
2

n
a
J
6 Received: date / Accepted: date

]
C
Abstract Variable order setting models situations in which the comparison between two
O
points depends on a point-to-cone application. In this paper an inexact projected method
.
h for solving smooth constrained vector optimization problems on variable ordered spaces is
t
a presented. It is shown that every accumulation point of the generated sequence satisfies a
m
first order necessary optimality condition. The behavior of this approach is also studied
[ under K–convexity of the objective function where the convergence is established to a

1 weakly efficient point. Moreover, the convergence results is derived in the particular case in
v which the problem is unconstrained and if exact directions are taken as descend directions.
0
Furthermore, we adapt the proposed method to optimization models in which the domain
1
7 ofthevariableorderapplication andtheobjective functionisthesame.Inthiscase, similar
1
concepts and convergence results are presented. Finally, some computational experiments
0
. designed to investigate and illustrate the behavior of the proposed methods are presented.
1
0
7
Keywords Gradient method K–convexity Variable order Vector optimization
1
· · · ·
: Weakly efficient points
v
i
X
r Mathematics Subject Classification (2010) 90C29 90C52 65K05
a · ·
Jose YunierBello Cruz, Corresponding author
Department of Mathematical Sciences, Northern Illinois University. Watson Hall 366, DeKalb, IL, USA -
60115.
[email protected]
Gemayqzel Bouza Allende
Facultad deMatema´tica y Computacio´n, Universidad deLa Habana, La Habana, Cuba - 10400.
[email protected]
2 Jose Yunier Bello Cruz, Gemayqzel Bouza Allende
1 Introduction
Variable order is a natural extension of the well-known fixed (partial) order given by a
closed, pointed and convex cone. This kindof orderings modelsituations in which the com-
parisonbetween two pointsdependsonasetvalued application. Research focusedinvector
optimization problemson variable ordered spaces and its applications has recently received
much attention from the optimization community due to its broad applications to several
different areas. Variable order structures given by a point-to-cone valued application, were
well studiedin [1],motivated by importantapplications which appearin medical diagnosis,
portfolio optimization, capability theory of wellbeing, psychological modeling, andlocation
theory; see, for instance, [2–7]. The main goal of these models is to find an element of a
certain set such that the evaluation of the objective function cannot be improved by the
image of any other feasible point with respect to the variable order. So, their mathematical
description corresponds to the called Optimization Problem on Variable Ordered Spaces
(OPVOS). For above reasons, it is important obtain efficient and implementable solution
algorithms for solving this kind of models.
OPVOSs have been treated in [8], in the sense of finding a minimizer of the image of
a vector function, with respect to an ordered structure depending on points in the im-
age. It is a particular case of the problem described in [9], where the goal of the model
is to find a minimum of a set. Here we will consider a partial (variable) order defined
by the cone-valued application which is used to define our problem - OPVOS. We want
to point out that OPVOS generalizes the classical vector optimization model. Indeed, it
corresponds to the case in which the order is defined by a constant cone valued applica-
tion. Many approaches have been proposed to solve classical vector optimization, such as
gradient methods, proximal points iterations, weighting techniques schemes, Newton-like
and subgradientmethods;see, for instance, [10–21].Theuseof extensions of theseiterative
algorithms to the variable ordering setting is currently a promising idea. It is worth noting
that, as far as we know, only few of these schemes mentioned above have been proposed
and studied in the variable ordering setting; as e.g., the steepest descent algorithm and
sub-gradient-like algorithms for unconstrained problems; see, for instance, [22,23]. In this
work, due to its simplicity and the adaptability to the structure of the vectorial problem,
we present an inexact projected gradient method for solving constrained variable order
vector problems. The properties of the accumulation points of the generated sequence are
studiedanditsconvergence is alsoanalyzed underconvexity. Moreover, wederivetheexact
projected gradient method and the inexact unconstrained case. Finally, analogous results
are obtained if the variable order is given by a point-to-cone application whose domain
coincides with the image of the objective function.
This work is organized as follows. The next section provides some notations and prelim-
inary results that will be used in the remainder of this paper. We also recall the concept of
K–convexity of a function on a variable ordered space and present some properties of this
Title Suppressed Due to Excessive Length 3
class. Section 3 is devoted to the presentation of the algorithm. The convergence of the
sequence generated by the projected gradient method is shown in Section 4. Then, under
the K–convexity of the objective function and the convexity of the set of feasible solutions,
we guarantee that the generated sequence is bounded and all its accumulation points are
solutions of the variable order problem. Section 5 discusses about the properties of this
algorithm when the variable order is taken as a cone value set from the image of objective
function. Section 6 introduces some examples illustrating the behavior of both proposed
methods. Finally, some final remarks are given.
2 Preliminaries
In this section we present some previous results and definitions. First we will define the
constrained vector optimization problems on variable ordered spaces which finds a K–
minimizer of F in the set C:
K minF(x), x C. (1)
− ∈
Here C is a convex closed subset of Rn and K : Rn ⇒ Rm is a point-to-cone map, where
K(x) is a nonempty pointed, convex and closed cone, for all x Rn. We say that the point
∗ ∈
x is a minimizer of (1) if for all x C,
∈
∗ ∗
F(x) F(x ) / K(x ) 0 .
− ∈ \{ }
∗
The set of all minimizers of problem (1) is denoted by S .
Next we introduce some useful notations: Throughout this paper, we write p := q to
indicate that p is defined to be equal to q and we write N for the nonnegative integers
0,1,2,... . The inner product in Rn will be denoted by , and the induced norm by
{ } h· ·i
. The closed ball centered at x with radius r is represented by B(x,r) := y Rn :
k·k { ∈
y x r and also the sphere by S(x,r) := y B(x,r) : y x = r . Given two
k − k ≤ } { ∈ k − k }
bounded sets A and B, we will consider d (A,B) as the Hausdorff distance, i.e.
H
d (A,B) := max sup inf d(a,b), sup inf d(a,b) ,
H
(cid:26)a∈Ab∈B b∈Ba∈A (cid:27)
or equivalently dH(A,B) = inf ǫ 0 : A Bǫ and B Aǫ , where Dǫ := d∈D x
{ ≥ ⊆ ⊆ } ∪ { ∈
Rn :dist(d,x) ǫ is the ǫ–enlargement of any set D. The set Dc denotes the complement
≤ }
of D. Given the partial order structure induced by a cone , the concept of infimum of
a sequence can be defined. Indeed, for a sequence (xk)k∈NKand a cone , the point x∗
is infk xk iff (xk x∗)k∈N , and there is not x such that x∗ x K and (xk
{ } − ⊂ K − ∈ K −
x)k∈N . We said that has the Daniell property if for all sequence (xk)k∈N such that
⊂ K K
(xk xk+1)k∈N and for some xˆ, (xk xˆ)k∈N , then limk→∞xk = infk xk . Here we
− ⊂ K − ⊂ K { }
assume that K(x), x Rn, is a convex, pointed, and closed cone, which guarantees that
∈
4 Jose Yunier Bello Cruz, Gemayqzel Bouza Allende
K(x) has the Daniell property as was shown in [24]. For each x Rn, the dual cone of
K(x) is defined as K∗(x) := w Rm : w,y 0, for all y K(x∈) . As usual, the graph
{ ∈ h i ≥ ∈ }
of a set valued application K is the set Gr(K) := (x,y) Rn Rm : y K(x) . We
{ ∈ × ∈ }
recall that the mapping K is closed if Gr(K) is a closed subset of Rn Rm.
×
As in the case of classical vector optimization, related solution concepts such as weakly
∗
efficient and stationary points can beextended. Thepoint x is a weak solution of problem
(1) if for all x C, F(x) F(x∗) / int(K(x∗)), Sw is the set of all weak solution points.
∈ − ∈ −
We want to point out that this definition corresponds with the concept of weak minimizer
∗
given in [9]. On the other hand, if F is a continuously differentiable function, the point x
is stationary, iff for all d C x∗ := d Rn : d = c x∗, for some c C , we have
∈ − { ∈ − ∈ }
∗ ∗
F(x )d / int(K(x )).
∇ ∈ −
Ss denotes the set of all stationary points.
Next we present a constrained version of Proposition 2.1 of [22] as follows:
∗
Proposition 2.1 Let x be a weak solution of problem (1). If F is a continuously differ-
∗
entiable function, then x is a stationary point.
∗ ∗
Proof Supposethatx isaweaksolutionofproblem(1).Fixd C x .Bydefinitionthere
∗ ∈ − ∗
exists c C, such that d = c x . Since C is a convex set, for all α [0,1], x +αd C.
∗ ∈ − ∗ ∗ ∗ ∈ ∈
As x is a weak solution of P, F(x +αd) F(x ) / int(K(x )). Hence,
− ∈ −
F(x∗+αd) F(x∗) ( int(K(x∗)))c. (2)
− ∈ −
∗ ∗ ∗ ∗
The Taylor expansion of F at x , leads us to F(x +αd) = F(x )+α F(x )d+o(α). The
above equation together with (2) implies α F(x∗)d+o(α) ( int(∇K(x∗)))c. Using that
( int(K(x∗)))c is a closed cone, and since α∇> 0, it follows t∈hat−
−
o(α)
F(x∗)d+ ( int(K(x∗)))c.
∇ α ∈ −
Taking limit in the above inclusion, when α goes to 0, and using the closedness of
( int(K(x∗)))c, we obtain that F(x∗)d ( int(K(x∗)))c, establishing that x∗ Ss.
− ∇ ∈ − ∈
⊓⊔
In classical optimization, stationarity is also a sufficient for weak minimality under con-
vexity. For vector optimization problems on variable ordered spaces, the convexity concept
was introduced in Definition 3.1 of [22] as follows:
Definition 2.1 We say that F is a K–convex function in C if for all λ [0,1], x,x¯ C,
∈ ∈
F(λx+(1 λ)x¯) λF(x)+(1 λ)F(x¯) K(λx+(1 λ)x¯).
− ∈ − − −
Title Suppressed Due to Excessive Length 5
It is important to note that the convexity of epi(F) is equivalent to the K–convexity of
F iff K(x) K for all x Rn; see Proposition 3.1 of [22]. As already shown in [22], K–
≡ ∈
convex functions have directional derivatives under natural assumptions; see Proposition
3.5 of [22]. In particular, if Gr(K) is closed and F 1 is K–convex, then we have the
∈ C
gradient inclusion inequality as follows:
F(x) F(x¯) F(x¯)(x x¯)+K(x¯), x C, x¯ C. (3)
− ∈ ∇ − ∈ ∈
In the next proposition we study the relation between stationarity, descent directions and
weak solution concept in the constrained sense for problem (1).
Proposition 2.2 Let K be a point-to-cone and closed mapping, and F be a K–convex
function. Then,
∗
(i) The point x is a weak solution of problem (1) iff it is a stationary point.
∗ ∗ ∗ ∗
(ii) If for all d C x , F(x )d / K(x ) 0 , then x is a minimizer.
∈ − ∇ ∈ − \{ }
Proof (i): Let x∗ Ss, where Ss is the set of the stationary points. If x∗ is not a weak
∈ ∗ ∗
minimizer then there exists x C such that F(x) F(x ) = k int(K(x )). By the
1
∈ ∗ − − ∈ −
convexity of F, for some k K(x ), we have
2
∈
∗ ∗ ∗
k = F(x) F(x ) = F(x )(x x )+k . (4)
1 2
− − ∇ −
It follows from (4) that
∗ ∗
F(x )(x x ) = (k +k ). (5)
1 2
∇ − −
∗ ∗ ∗
Moreover, since K(x ) is a convex cone, k int(K(x )) and k K(x ), it holds that
1 2
∗ ∈ ∗ ∈ ∗ ∗
k + k int(K(x )). Thus, (4) and (5) imply that F(x )(x x ) int(K(x )),
1 2
∈ ∗ ∇ − ∈ −
which contradicts the fact that x is a stationary point because x belongs to C and hence
∗ ∗
x x C x . The conversely implication was already shown in Proposition 2.1.
− ∈ −
∗
(ii): By contradiction suppose that there exists x C such that F(x) F(x ) = k ,
1
∗ ∈ − −
where k K(x ) 0 . Combining the previous condition with (5), it follows that
1
∈ \{ }
∗ ∗ ∗
F(x )(x x )= (k +k ) K(x ).
1 2
∇ − − ∈ −
∗ ∗ ∗ ∗
Usingthat F(x )(x x ) / K(x ) 0 , wegetthat(k +k )= 0,andask ,k K(x ),
1 2 1 2
∇ − ∈ − \{ } ∗ ∈
k = k .ItfollowsfromthepointednessoftheconeK(x )thatk = k = 0,contradicting
1 2 1 2
−
the fact that k = 0.
1
6 ⊓⊔
It is worth mentioning that the concept of K–convexity for F depends of the point-to-
cone mapping K. Thus, this general approach covers several convexity concepts, from the
scalar setting to the vector one and it can beused to model a large numberof applications;
6 Jose Yunier Bello Cruz, Gemayqzel Bouza Allende
see, for instance, [5–7]. In Section 5 we discuss another variable order when the point-to-
cone application dependsof the image setof F,such kindof variable orderswas introduced
and studied in [22,23].
After wehadbeenpresentingthetheory ofconvexity forconstrained vector optimization
problems on variable ordered spaces, a solution approach to solve problem (1) will be
studied in the next section.
3 The Inexact Projected Gradient Method
This section is devoted to present an inexact projected gradient method for solving con-
strained smooth problems equipped with variable order. This method uses an Armijo-type
line-search, which is done on descent feasible directions. The proposed scheme here has
two main differences with respect to the approach introduced in [22]. First constrained
problems as (1) are considered. Second the introduced method, instead of computing the
direction through the exact solution of subproblem (P ) given below, it accepts approx-
x
imate solutions of this subproblem with some tolerance. Thus, in the following, several
constrained concepts and results will be presented and proved, which will be used in the
convergence analysis of the proposed method below.
We start the section by presenting some definitions and basic properties of some auxil-
iary functions and sets, which will be useful in the convergence analysis of the proposed
algorithms. Firstly we define the set valued mapping G : Rn ⇒ Rm, which for each x,
defines the set of the normalized generators of K∗(x), i.e. G(x) K∗(x) S(0,1) is a
⊆∗ ∩
compact set such that the cone generated by its convex hull is K (x). Although the set
of the dual cone K∗(x) S(0,1) fulfils those properties, in general it is possible to take
∩
smaller sets; see, for instance, [25–27].
On the other hand, we consider function φ: Rn Rn R,
× →
φ(x,v) := max yT F(x)v, (6)
y∈G(x) ∇
and for each x Rn and β > 0, the auxiliary subproblem
∈
v 2
min k k +βφ(x,v) . (P )
x
v∈C−x 2
(cid:26) (cid:27)
Nextpropositionprovidesacharacterization ofthestationarity usingtheauxiliaryfunction
φ, defined in (6). The unconstrained version of the following proposition can be found in
Proposition 4.1 of [22].
Proposition 3.1 The following statements hold:
Title Suppressed Due to Excessive Length 7
(i) For each x Rn, max yTwˆ < 0 if and only if wˆ int(K(x)).
y∈G(x)
∈ ∈ −
(ii) The point x is not stationary iff there exists v C x such that φ(x,v) < 0.
∈ −
λv 2
(iii) If φ(x,v) < 0, then there exists λ > 0 such that k k + βφ(x,λv) < 0 for all
2
λ (0,λ].
∈
(iv) For each x Rn, subproblem (P ) has a unique solution, denoted by v(x).
x
∈
Proof (i): Defining
ρ(x,w) := max yTw (7)
y∈G(x)
the result of this item follows as in Proposition 4.1(i) of [22].
(ii): Note that fixed x, it follows from (6) that φ(x,v) = ρ(x, F(x)v). Then, by the
∇
definition of stationarity and item (i), the statement holds true.
(iii):Itfollowsfromthedefinitionofφ(x,v)thatφ(x, )isapositivehomogeneousfunction.
·
Thus, for all λ > 0,
λv 2 v 2
k k +βφ(x,λv) = λ λk k +βφ(x,v) . (8)
2 2
(cid:18) (cid:19)
v 2
Since φ(x,v) < 0, there exists λ¯ > 0 small enough such that λ¯k k +βφ(x,v) < 0. Hence,
2
λv 2
(8)together withtheabove inequality impliesthat k k +βφ(x,λv) < 0,forallλ (0,λ¯],
2 ∈
as desired.
(iv): Using the definition of φ(x,v), given in (8), it is easy to prove that φ is a sublinear
v 2
function as well. So, φ(x, ) is a convex function, and then, k k +βφ(x,v) is a strongly
· 2
convex function. Since C is a convex set, C x is also convex and hence, subproblem (P )
x
−
has a unique minimizer.
⊓⊔
Remark 3.1 As a consequence of (6), if F is locally Lipschitz, then φ(x, ) is also locally
∇ ·
Lipschitz and hence continuous. Indeed, Proposition 4.1(iv) of [22] implies that ρ(x,w),
defined in (7), is a Lipschitz function for all (x,w) Rn W for any bounded subset
∈ ×
W Rn. That is,
⊂
ρ(x ,w ) ρ(x ,w ) L M x x + w w , (9)
1 1 2 2 G 1 2 1 2
| − | ≤ k − k k − k
where L is defined in Proposition 3.2(iv) below and w M for all w W. Thus, It
G
k k ≤ ∈
follows from the same analysis done below until (13) that φ(x ,z) φ(x ,z) (L M +
1 2 G
| − | ≤
L Mˆ) x x , where L is the Lipschitz constant of F and z Mˆ.
F 1 2 F
k − k ∇ k k ≤ ⊔⊓
8 Jose Yunier Bello Cruz, Gemayqzel Bouza Allende
Based on Proposition 3.1(iii), we define v(x) as the unique solution of problem (P ) and
x
y(x,v) is an element of G(x) such that y(x,v)T F(x)v = φ(x,v). We will discuss about
∇
the continuity of
v(x) 2
θ (x) := k k +βφ(x,v(x)). (10)
β
2
The following proposition is the constrained version of Proposition 4.2 in [22].
Proposition 3.2 Consider x,xˆ C and fix β > 0, then the following hold
∈
(i) θ (x) 0 and x is a stationary point iff θ (x) = 0.
β β
≤
(ii) v(x) 2β F(x) .
k k ≤ k∇ k
(iii) If G is a closed application, then θ is an upper semi-continuous function.
β
(iv) If d (G(x),G(xˆ)) L x xˆ for some L > 0 and F is locally Lipschitz, then
H G G
≤ k − k ∇
θ is a lower semicontinuos function.
β
0 2
Proof (i): For the first part, note that as 0 C x for all x C, θ (x) k k +
β
∈ − ∈ ≤ 2
βφ(x,0) = 0. As shown in Proposition 3.1(ii), x is a non stationary point iff for some
v C x, φ(x,v) < 0. Then, by Proposition 3.1(iii), there exists vˆ C x such that
∈ − ∈ −
λ2
v 2 +λβφ(x,v) < 0 and hence θ (x) < 0.
β
2 k k
v(x) 2
(ii): By (i), 0 θ (x) = k k +βy(x,v(x))T F(x)v(x). Then, after some algebra, we
β
≥ 2 ∇
get
v(x) 2
k k βy(x,v(x))T F(x)v(x) β y(x,v(x))T F(x)v(x) .
2 ≤ − ∇ ≤ k ∇ k
Using that y(x,v(x)) = 1, it follows from the above inequality that
k k
v(x) 2
k k β F(x) v(x) ,
2 ≤ k∇ kk k
and the result follows after dividing by the positive term v(x) = 0.
k k 6
(iii):Nowweprovetheuppersemi-continuity ofthefunctionθβ.Let(xk)k∈N beasequence
converging to x.Take xˆ such thatv(x) = xˆ x.Itis clear thatas, forall k, xˆ xk C xk,
− − ∈ −
xˆ xk 2
θ (xk) k − k +βφ(xk,xˆ xk)
β
≤ 2 −
xˆ xk 2
= k − k +βyT F(xk)(xˆ xk). (11)
2 k∇ −
Since each y belongs to the compact set G(xk) K∗(xk) S(0,1) B(0,1) for all k,
k
⊆ ∩ ⊆
then the sequence (yk)k∈N is bounded because is in k∈NG(xk) B(0,1). Therefore, there
∪ ⊆
Title Suppressed Due to Excessive Length 9
exists a convergent subsequence of (yk)k∈N. We can assume without lost of generality that
limk→∞yk = y, and also since G is closed, y G(x). Taking limit in (11), we get
∈
xˆ xk 2
limsupθ (xk) limsupk − k +βyT F(xk)(xˆ xk)
k→∞ β ≤ k→∞ 2 k∇ −
xˆ x 2
= k − k +βyT F(x)(xˆ x)
2 ∇ −
xˆ x 2
k − k +βφ(x,xˆ x) = θ (x).
β
≤ 2 −
Then, the function θ , defined in (10), is upper semi-continuous.
β
(iv): We proceed as in item (ii) above. First we prove that φ(x,xˆk x) φ(xk,xˆk xk)
− − −
converges to 0. Since (xk)k∈N is a convergent sequence and F is a smooth function, the
sequences (xk)k∈N and ( F(xk))k∈N are bounded. Moreover, by (ii),
∇
xˆk xk 2β F(xk) . (12)
k − k≤ k∇ k
Hence, the sequences ( F(x)(xˆk xk))k∈N and ( F(xk)(xˆk xk))k∈N are bounded. By
∇ − ∇ −
Remark3.1,(9)holdsforx = x,x = xk,w = F(x)(xˆk xk)andw = F(xk)(xˆk xk).
1 2 1 2
∇ − ∇ −
That is,
ρ x, F(x)(xˆk xk) ρ xk, F(xk)(xˆk xk) Lˆ x xk + ( F(x) F(xk))(xˆk xk)
∇ − − ∇ − ≤ k − k k ∇ −∇ − k
(cid:12)(cid:12) (cid:0) (cid:1)Lˆ x(cid:0) xk + F(x) (cid:1)F(cid:12)(cid:12)(xk) xˆk xk ,
(cid:12) ≤ k − k k∇ −∇ (cid:12) kk − k
where Lˆ := L M and F(x)(xˆk xk) M for all k. Noting that
G
k∇ − k ≤
φ(x,xˆk xk) φ(xk,xˆk xk)= ρ x, F(x)(xˆk x) ρ xk, F(xk)(xˆk xk) ,
− − − ∇ − − ∇ −
(cid:16) (cid:17) (cid:16) (cid:17)
and due to F is locally Lipschitz, from (12) and Remark 3.1, it follows that
∇
φ(x,xˆk xk) φ(xk,xˆk xk) (Lˆ +L Mˆ) x xk , (13)
F
− − − ≤ k − k
(cid:12) (cid:12)
where LF is the (cid:12)(cid:12)Lipschitz constant of F and(cid:12)(cid:12) xˆk xk Mˆ for all k. In particular,
∇ k − k ≤
limk→∞φ(x,xˆk x) φ(xk,xˆk xk) = 0. Next we consider the function θβ(x). Note
− − −
further that
xˆk x 2
θ (x) βφ(x,xˆk x)+ k − k
β
≤ − 2
xˆk x 2 xˆk xk 2
=θ (xk)+β φ(x,xˆk x) φ(xk,xˆk xk) + k − k −k − k
β
− − − 2
h i 1
=θ (xk)+β φ(x,xˆk x) φ(xk,xˆk xk) + 2 xˆk,xk x + x 2 xk 2 .
β
− − − 2 − h − i k k −k k
h i h i
10 Jose Yunier Bello Cruz, Gemayqzel Bouza Allende
Thus, taking limit in the previous inequality and using that
lim φ(x,xˆk x) φ(xk,xˆk xk) = 0,
k→∞ − − −
and limk→∞ 12 kxk2 −kxkk2 −hxˆk,xk −xi= 0, we obtain that
(cid:2) (cid:3) x 2 xk 2
θ (x) liminf θ (xk)+β φ(x,xˆk x) φ(xk,xˆk xk) xˆk,xk x + k k −k k
β β
≤ k→∞ − − − −h − i 2
(cid:26) (cid:27)
h i
= liminfθ (xk),
β
k→∞
establishing the desired result.
⊓⊔
Now we introduce the concept of δ-approximate descent direction.
Definition 3.1 Let β > 0. Given δ (0,1], we say that v is a δ-approximate solution
∈
v 2
of problem (P ) if v C x and βφ(x,v)+ k k δθ (x). If v = 0 we say that v is a
x β
∈ − 2 ≤ 6
δ-approximate descent direction.
Hence, from a numerical point of view, it would be interesting to consider algorithms in
which the line-search is given if δ-approximate descent direction instead exact solution of
subproblem (P ).
x
Remark 3.2 Note that if the solution of (P ) is 0, then the only possible δ approximate
x
solution is v = 0. In other case, due to θ (x) < 0, there exists feasible directions such that
β
v 2
βφ(x,v)+ k k [θ (x),δθ (x)].
β β
2 ∈
In particular v, the solution of (P ), is always a δ-approximate solution.
x
⊔⊓
Next we present the inexact algorithm for solving problem (1).
Inexact Projected Gradient Method (IPG Method). Given δ (0,1] and σ,θ (0,1).
∈ ∈
Initialization: Take x0 Rn and β > 0.
0
∈
Iterative step: Given xk and βk, compute vk, δ-approximate solution of (Pxk). If vk = 0,
then stop. Otherwise compute
j(k) := min j Z : F(xk) F(xk +θjvk)+σθj F(xk)vk K(xk) . (14)
+
∈ − ∇ ∈
n o
Set xk+1 = xk +γ vk C, with γ = θj(k).
k k
∈