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Journal of Applied Mathematics
Volume 2012, Article ID 692325, 9 pages
doi:10.1155/2012/692325
Research Article
Sufficient Optimality and Sensitivity Analysis of
a Parameterized Min-Max Programming
Huijuan Xiong,1 Yu Xiao,2 and Chaohong Song1
1
2
College of Science, Huazhong Agricultural University, Wuhan 430070, China
School of Basic Science, East China Jiaotong University, Nanchang 330000, China
Correspondence should be addressed to Huijuan Xiong, [email protected]
Received 4 June 2012; Accepted 17 July 2012
Academic Editor: Jian-Wen Peng
Copyright q 2012 Huijuan Xiong et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Sufficient optimality and sensitivity of a parameterized min-max programming with fixed
feasible set are analyzed. Based on Clarke’s subdifferential and Chaney’s second-order directional
derivative, sufficient optimality of the parameterized min-max programming is discussed first.
Moreover, under a convex assumption on the objective function, a subdifferential computation
formula of the marginal function is obtained. The assumptions are satisfied naturally for some
application problems. Moreover, the formulae based on these assumptions are concise and
convenient for algorithmic purpose to solve the applications.
1. Introduction
In this paper, sufficient optimality and sensitivity analysis of a parameterized min-max
programming are given. The paper is triggered by a local reduction algorithmic strategy for
solving following nonsmooth semi-infinite min-max-min programming SIM3 P, see 1, 2,
etc. for related applications reference:
min
fx
s.t.
gx max min gi x, y ≤ 0.
x
y∈Y 1≤i≤q
1.1
2
Journal of Applied Mathematics
With the local reduction technique, the SIM3 P can be rewritten as a bilevel programming first,
where the lower problem is the following parameterized min-max programming Px see 3–
5 for related reference of local reduction strategy:
min
y
s.t.
g x, y max −gi x, y
1≤i≤q
1.2
y ∈ Y.
To make the bilevel strategy applicable to SIM3 P, it is essential to discuss the second-order
sufficient optimality of Px and give sensitivity analysis of the parameterized minimum yx
and corresponding marginal function gx, yx.
Sensitivity analysis of optimization problems is an important aspect in the field
of operation and optimization research. Based on different assumptions, many results on
kinds of parametric programming have been obtained 6–9, etc.. Among these, some
conclusions on parameterized min-max programming like 1.2 have also been given.
For example, based on variation analysis, parameterized continuous programming with
fixed constraint was discussed in 7. Problem like 1.2 can be seen as a special case.
Under the inf-compactness condition and the condition objective function is concave with
respect to the parameter, directional derivative computational formula of marginal function
for 1.2 can be obtained directly. However, concave condition cannot be satisfied for
many problems. Recently, Fréchet subgradients computation formula of marginal functions
for nondifferentiable programming in Asplund spaces was given 9. By using Fréchet
subgradients computation formula in 9, subgradient formula of marginal function for
1.2 is direct. But the formula is tedious, if utilizing the formula to construct optimality
system of 1.1, the system is so complex that it is difficult to solve the obtained optimality
system.
For more convenient computational purpose, the focus of this paper is to establish
sufficient optimality and simple computation formula of marginal function for 1.2. Based
on Clarke’s subdifferential and Chaney’s second-order directional derivative, sufficient
optimality of the parameterized programming Px is given first. And then Lipschitzian
continuousness of the parameterized isolated minimizer yx and the marginal function
gyx, x is discussed; moreover, subdifferential computation formula of the marginal
function is obtained.
2. Main Results
Let Y in 1.2 be defined as Y {y ∈ Rm : hi y ≤ 0, i 1, . . . , l}, where hi · and
i 1, . . . , l, are twice continuously differentiable functions on Rm , and gi ·, · in 1.2 are twice
continuously differentiable functions on Rn×m . In the following, we first give the sufficient
optimality condition of 1.2 based on Clarke’s subdifferential and Chaney’s second-order
directional derivative, and then make sensitivity analysis of the parameterized problem
Px .
Journal of Applied Mathematics
3
2.1. Sufficient Optimality Conditions of Px
Definition 2.1 see 10. For a given parameter x, a point y∗ ∈ Y is said to be an local
minimum of problem Px if there exists a neighborhood U of y∗ such that
g x, y ≥ g x, y∗ ,
∗
∀y ∈ U ∩ Y, y /y .
2.1
Assumption 2.2. For a given parameter x, suppose that Px satisfying the following constraint
qualification:
T
d ∈ Rm : ∇hi y d < 0, ∀i ∈ Ih y , y ∈ Y / ∅,
2.2
where Ih y {i {1, . . . , l} : hi y 0}.
For a given parameter x, denote the Lagrange function of Px as Lx, y, λ gx, y l
i1 λi hi y, then the following holds.
Theorem 2.3. For a given parameter x, if y∗ is a minimum of Px , Assumption 2.2 holds, then there
exists a λ∗ ∈ Rl such that 0 ∈ ∂y Lx, y∗ , λ, where ∂y Lx, y∗ , λ∗ denotes the Clarke’s subdifferential
of Lx, y∗ , λ∗ . Specifically, the following system holds:
l
0 ∈ ∂y g x, y∗ λi ∇hi y∗ ,
2.3
i1
where ∂y gx, y denotes Clarke’s subdifferential of gx, y with respect to y, it can be computed
as co{∇y gi x, y∗ : i ∈ Ix, y∗ }, co{·} is an operation of making convex hull of the elements,
Ix, y∗ {i ∈ {1, . . . , q} : gx, y∗ gx, y∗ }.
Proof. The conclusion is direct from Theorem 3.2.6 and Corollary 5.1.8 in 11.
Since gx, y max1≤i≤p {gi x, y} is a directional differentiable function Theorem
3.2.13 in 11, the directional derivative of gx, y with respect to y in direction d can be
computed as follows:
gy
x, y; d max ξT d : ∀ξ ∈ ∂y g x, y , ∀d ∈ Rm .
2.4
Definition 2.4 see 10. Let fx is a locally Lipschitzian function on Rn , u be a nonzero
vector in Rn . Suppose that
x
d ∈ ∂u fx υ ∈ Rn : ∃{xk }, {υk }, s. t. xk −→, υk −→ υ, υk ∈ ∂fxk for each k
2.5
define Chaney’s lower second-order directional derivative as follows:
f−
x, υ, u lim inf
fxk − fx − υT xk − x
t2k
,
2.6
4
Journal of Applied Mathematics
taking over all triples of sequences {xk }, {υk }, and {tk } for which
a tk > 0 for each k and {xk } → x;
b tk → 0 and xk − x∗ /tk converges to u;
c {υk } → υ with υk ∈ ∂fxk for each k.
Similarly, Chaney’s upper second-order directional derivative can be defined as
f
lim sup
fxk − fx − υT xk − x
t2k
2.7
,
taking over all triples of sequences {xk }, {υk }, and {tk } for which a, b, and c above hold.
For parameterized max-type function gx, y max1≤i≤p {−gi x, y}, where x is a
given parameter, its Chaney’s lower and upper second-order directional derivatives can be
computed as follows.
Proposition 2.5 see 12. For any given parameter x, Chaney’s lower and upper second-order
directional derivatives of gx, y with respect to y exist; moreover, for any given 0 / u ∈ Rq , υ ∈
∂u gx, y, it has
g−
q
1
T 2
aj u ∇y gi x, y u : a ∈ Tu g, y, υ ,
x, y; d min
2 i1
g
x, y; d max
q
1
2
2.8
aj uT ∇2y gi x, y u : a ∈ Tu g, y, υ ,
i1
where
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨a ∈ Rq :
Tu g, y, υ ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∃ yk , ak , υk , such that
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
k
⎪
1 y −→ y in direction u,
⎪
⎪
⎪
⎪
⎪
⎬
k
k
k
, k 1, 2, . . . , ,
2 υ −→ υ, and υ ∈ ∂y g x, y
⎪
⎪
p
⎪
⎪
⎪
k
k
k
k
k
⎪
,⎪
3 a −→ a, a ∈ Eq , υ ai ∇y gi x, y
⎪
⎪
⎪
i1
⎪
⎪
⎪
k
⎭
k
/K y
4 a 0, for j ∈
j
2.9
g
q
where Kg yk {i ∈ Q : gi x, yn gx, yn , ∃yn ∈ By, 1/n, ∀n ∈ N}, Eq {a ∈ R :
p
i1 ai 1}, Q {1, . . . , q}, and By, 1/n denotes the ball centered in y with radius 1/n.
Theorem 2.6 sufficiency theorem. For a given parameter x ∈ Rn , Assumption 2.2 holds, then
there exists y∗ ∈ Rm such that 2.3 holds. Moreover, for any feasible direction d ∈ Rm of Y , that is,
max{∇hi y∗ T d : 1 ≤ i ≤ l} ≤ 0, if d satisfying one of the following conditions:
Journal of Applied Mathematics
5
0;
1 gy
x, y∗ ; d /
2 gy
x, y∗ ; d 0,
l
i1
λi ∇hi yT d 0, that is, L
y x, y; d 0, and
1
ai dT ∇2y gi x, y∗ d : a ∈ Eq
min
2 i1
q
l
λi dT ∇2 hi y∗ d > 0,
2.10
i1
then y∗ is a local minimum of Px .
Proof. 1 If not, then there exists sequences tk ↓ 0, dk → d, yk y∗ tk dk ∈ Y such that
g x, yk < g x, y∗ .
2.11
As a result, gy
x, y∗ ; d limt↓0 gx, y∗ td − gx, y∗ /t limk → ∞ gx, y∗ tk dk −
gx, y∗ /tk ≤ 0. If gy
x, y∗ ; d /
0, then gy
x, y∗ ; d < 0. From 2.4, we know that ξT d <
∗
0 for all ξ ∈ ∂y gx, y . Hence, for the direction d ∈ Rm , we have
ξT d l
T
∇hi y∗ d < 0,
ξ ∈ ∂y g x, y∗ .
2.12
i1
On the other hand, from y∗ satisfying 2.3, we know that there exists a ξ ∈ ∂y gx, y∗ such
that
ξT d l
T
∇hi y∗ d 0,
2.13
i1
which leads to a contradiction to 2.12.
2 From Theorem 4 in 10 and Proposition 2.5, the conclusion is direct.
2.2. Sensitivity Analysis of Parameterized Px
In the following, we make sensitivity analysis of parameterized min-max programming Px ,
that is, study variation of isolated local minimizers and corresponding marginal function
under small perturbation of x.
For convenience of discussion, for any given parameter x, denote y∗ x as a minimizer
of Px , υx min{gx, y : y ∈ Y } as the corresponding marginal function value and make
the following assumptions first.
Assumption 2.7. For given x ∈ Rn , the parametric problem Px is a convex problem, specifically,
gi x, y and i 1, . . . , q are concave functions with respect to that variables y and hj y, j 1, . . . , l are convex functions.
Assumption 2.8. Let Ih y {i ∈ L : hi y 0}, {∇hi y : i ∈ Ih y} are linearly independent.
6
Journal of Applied Mathematics
Definition 2.9 see Definition 2.1, 13. For a given x, y ∈ Y is said to be an isolated local
minimum with order i i 1 or 2 of Px if there exists a real m > 0 and a neighborhood V of y
such that
i
1 g x, y > g x, y my − y ,
2
∀y ∈ V ∩ Y, y /
y.
2.14
Theorem 2.10. For a given x ∈ Rn , Assumptions 2.2–2.8 hold, then the following conclusions hold:
1 if y∗ x with corresponding multiplier λ∗ is the solution of 2.3, then y∗ x is a unique
first-order isolated minimizer of Px ;
2 for any minimum y∗ x, it is a locally Lipschitzian function with respect to x, that is, there
exists a L1 > 0, δ > 0 such that
∗ k
y x − y∗ x ≤ L1 xk − x,
∀xk ∈ Ux, δ, y∗ xk ∈ Y xk ,
2.15
where Y xk denotes minima set of Pxk ;
3 for any minimum y∗ x, marginal function υx gx, y∗ x is also a locally Lipschitz
function with respect to x, and ∂υx ⊆ Sx, where
Sx co ∇x gi x, y∗ x , i ∈ I x, y∗ x ,
2.16
and Ix, y∗ x {i ∈ {1, . . . , q} : gi x, y∗ x gx, y∗ x}. As a result,
∂υx ⎧
⎨
⎩
i∈I x,y ∗ x
λi ∇x gi x, y∗ x : λi ≥ 0,
i∈I x,y ∗ x
⎫
⎬
λi 1 .
⎭
2.17
Proof. 1 From Assumption 2.7, it is direct that y∗ x is a global minimizer of Px . We only
prove y∗ x is a first-order isolated minimizer.
If the conclusion does not hold, then there exists a sequence {yk } ∈ Y x converging
∗
y∗ x, and a sequence mk , mk > 0, and mk converges to 0 such that
to y x, yk /
1 g x, yk ≤ g x, y∗ x mk yk − y∗ ,
2
yk ∈ Y.
2.18
Take dk yk − y∗ x/yk − y∗ x, for simplicity, we suppose dk → d, with d 1. Let
tk yk − y∗ x, then from yk ∈ Y , dk → d and Y is compact, we have
y∗ x tk d ∈ Y,
tk −→ 0,
2.19
that is,
T
∇hi y∗ x d ≤ 0,
∀i ∈ I x, y∗ x .
2.20
Journal of Applied Mathematics
7
0. As a result, we have
From Assumption 2.8, we know that i∈Ix,y∗ x ∇hi y∗ xT d /
T
∗
i∈Ix,y∗ x ∇hi y x d < 0.
From the first equation of 2.3, we know that there exists a z ∈ ∂y gx, y∗ x such that
for any feasible direction d, zT d − i∈Ix,y∗ x λi ∇hi y∗ xT d > 0. Hence,
gy
x, y∗ x; d max ξT d : ξ ∈ ∂y g x, y∗ x ≥ zT d > 0.
2.21
On the other hand, from y∗ x is a minimizer, we know that gy
x, y∗ x; d ≥ 0, this leads to
a contradiction;
2 from Assumption 2.8 and Theorem 3.1 in 13, the conclusion is direct;
3 since gx, y is a locally Lipschitzian function with respect to x and y, then there
exists δ > 0, δ
> 0, and L2 > 0 such that for any x1 ∈ Ux, δ, y ∈ Uy∗ x, δ
, it has
g x1 , y∗ x − g x, y∗ x ≤ L2 x1 − x,
g x, y − g x, y∗ x ≤ L2 y − y∗ x.
2.22
As to x1 ∈ Ux, δ, from the conclusion in 1.2, there exists a a L1 > 0 such that
y∗ x1 − y∗ x ≤ L1 x1 − x. As a result,
|υx1 − υx|
g x1 , y∗ x1 − g x, y∗ x g x1 , y∗ x1 − g x1 , y∗ x g x1 , y∗ x − g x, y∗ x ≤ g x1 , y∗ x1 − g x1 , y∗ x g x1 , y∗ x − g x, y∗ x ≤ L2 y∗ x1 − y∗ x L2 x1 − x ≤ L2 1 L1 x1 − x.
2.23
Hence, the marginal function υx is a local Lipschitzian function with respect to x.
We prove that
Let Sx
{−∇x gi x, yx, i ∈ Ix, yx}, then Sx co{ξ, ξ ∈ Sx}.
k
n
k
k
k , zk → z, it
Sx
is closed first, that is, prove for any sequence {x } ⊂ R , x → x, z ∈ Sx
has z ∈ Sx.
k , there exist yk ∈ Y xk ; ik ∈ Ixk , yk such that zk −∇x gik xk , yk .
From zk ∈ Sx
Without loss of generality, suppose that {yk } converges to y; {ik } converges to i. From
Proposition 3.3 in 14, it has y ∈ Y x, and i ∈ Ix, y and from ∇x gi x, y is a continuous
As a result, Sx
function, it has z limk → ∞ zk limk → ∞ ∇x gik xk , yk ∇x gi x, y ∈ Sx.
is a closed set.
8
Journal of Applied Mathematics
From Theorem 3.2.16 in 11, for any ξ ∈ ∂υx, there exists xk ∈ Rn , xk → x such that
∇υx exists and ξ limk → ∞ ∇υxk . In addition, for arbitrary d ∈ Rn , it has
k
∇υ x
k
T
d
≤
υ xk td − υ xk
υ x ; d lim
t↓0
t
k
g x td, y∗ xk td − g xk , y∗ xk
lim
t↓0
t
k
g x td, y∗ xk − g xk , y∗ xk
lim
t↓0
t
T k
max −∇x gi x , y d .
i∈I xk ,y k
T
2.24
T
From the definition of Sxk , ∃zk ∈ Sxk such that zk d maxi∈Ixk ,y {−∇x gi xk , y d}.
k T
T
Hence, it has ∇υx d ≤ zk d.
T
T
⊂ Sx, ∇υxk → ξ and ∇υxk d ≤ zk d, it has ξT d ≤ zT d, that
From zk → z ∈ Sx
is, for arbitrary d ∈ Rn and ξ ∈ ∂υx, there exists z ∈ Sx such that ξT d ≤ zT d.
If ∂υx ⊂ Sx does not hold, then there exists a ξ ∈ ∂υx and ξ ∈
/ Sx. From Sx is a
compact convex set and separation theorem 15, there exists a d ∈ Rn such that ξT d < 0 and
for arbitrary z ∈ Sx, zT d ≥ 0, which leads to a contradiction. As a result, ∂υx ⊂ Sx holds.
From ∂υx ⊂ Sx and Sx co{∇x gi x, y∗ x, i ∈ Ix, y∗ x}, computation formula
2.17 is direct.
3. Discussion
In this paper, sufficient optimality and sensitivity analysis of a parameterized min-max
programming are given. A rule for computation the subdifferential of υx is established.
Though the assumptions in this paper are some restrictive compared to some existing work,
the assumptions hold naturally for some applications. Moreover, the obtained computation
formula is simple, it is beneficial for establishing a concise first-order necessary optimality
system of 1.1, and then constructing effective algorithms to solve the applications.
Acknowledgments
This research was supported by the National Natural Science Foundation of China no.
11001092 and the Fundamental Research Funds for the Central Universities no. 2011QC064.
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