Journal of Inequalities and Applications Volume 2010, Article ID 931590,6pages doi:10.1155/2010/931590
Research Article
Further Study on Strong Lagrangian Duality
Property for Invex Programs via Penalty Functions
J. Zhang
1and X. X. Huang
21School of Mathematics and Physics, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
2School of Economics and Business Administration, Chongqing University, Chongqing 400030, China
Correspondence should be addressed to X. X. Huang,[email protected]
Received 5 February 2010; Revised 23 June 2010; Accepted 30 June 2010
Academic Editor: Kok Teo
Copyrightq2010 J. Zhang and X. X. Huang. 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.
We apply the quadratic penalization technique to derive strong Lagrangian duality property for an inequality constrained invex program. Our results extend and improve the corresponding results in the literature.
1. Introduction
It is known that Lagrangian duality theory is an important issue in optimization theory and methodology. What is of special interest in Lagrangian duality theory is the so-called strong duality property, that is, there exists no duality gap between the primal problem and its Lagrangian dual problem. More specifically, the optimal value of the primal problem is equal to that of its Lagrangian dual problem. For a constrained convex program, a number of conditions have been obtained for its strong duality property, see, for example,1–3and the references therein. It is also well known that penalty method is a very popular method in constrained nonlinear programming 4. In 5, a quadratic penalization technique was applied to establish strong Lagrangian duality property for an invex program under the assumption that the objective function is coercive. In this paper, we will derive the same results under weaker conditions. So our results improve those of5.
Consider the following inequality constrained optimization problem:
min fx
s.t. x∈Rn, gjx≤0, j1, . . . , m,
P
The Lagrangian function forPis
Lx, μfx m
j1
μjgjx, x∈Rn, μμ1, . . . , μm∈Rm. 1.1
The Lagrangian dual function forPis
hμ inf x∈RnL
x, μ, ∀μ∈Rm
. 1.2
The Lagrangian dual problem forPis
sup u∈Rm
hμ. D
Denote byMPandMDthe optimal values ofPandD, respectively. It is known that weak dualityMP ≥ MDholds. However, there is usually a duality gap, that is,MP > MD. If MP MD, we say that strong Lagrangian duality property holdsor zero duality gap property holds.
Recall that a differentiable functionu:Rn → R1is invex if there exists a vector-valued functionη:Rn×Rn → Rnsuch thatux−uy≥ηTx, y∇uy,for all x, y∈Rn.Clearly, a differentiable convex functionuis invex withηx, y x−y.It is known from6that a differentiable convex functionuis invex if and only if each stationary point ofuis a global optimal solution ofuonRn.
LetX ⊂Rnbe nonempty.u:Rn → R1is said to be level bounded onXif for any real
numbert, the set{x∈X:ux≤t}is bounded.
It is easily checked thatuis level bounded, onX if and only ifX is bounded oruis coercive onXifXis unboundedi.e., limx∈X,x →∞ux ∞.
2. Main Results
In this section, we present the main results of this paper.
Consider the following quadratic penalty function and the corresponding penalty problem forP:
Pkx fx k m
j1
g
j2x, x∈Rn, 2.1
min
x∈RnPkx, Pk
where the integerk >0 is the penalty parameter. For anyt∈R1, denote that
Xt x∈Rn:g
jx≤t, j 1, . . . , m. 2.2
We need the following lemma.
Lemma 2.1. Assume thatfis level bounded onX0, then the solution set of Pis nonempty and compact.
Proof. It is obvious that problemPand the following unconstrained optimization problem have the same optimal value and the same solution set,
minfx, P
where
fx
⎧ ⎪ ⎨ ⎪ ⎩
fx, x∈X0,
∞, otherwise.
2.3
It is obvious thatf:Rn → R1∪{∞}is proper, lower semicontinuous, and level bounded. By
7, Theorem 1.9, the solution set ofPis nonempty and compact. Consequently, the solution set ofPis nonempty and compact.
Now we establish the next lemma.
Lemma 2.2. Suppose that there existst0 >0such thatf is level bounded onXt0, and there exists
k∗>0andm
0∈R1such that
Pk∗x≥m0, ∀x∈Rn. 2.4
Then
ithe optimal set ofPis nonempty and compact;
iithere existsk∗ > 0such that for eachk ≥ k∗, the penalty problemPkhas an optimal solutionxk; the sequence{xk}is bounded and all of its limiting points are optimal solutions ofP.
Proof. iSinceX0 ⊂ Xt0is nonempty andf is level bounded onXt0, we see thatf is
level bounded onX0. ByLemma 2.1, we conclude that the solution set ofPis nonempty and compact.
iiLetx0∈X0andk∗≥k∗1 satisfy
fx0 1−m0
k∗−k∗ ≤t20. 2.5
Note that whenk≥k∗,
Pkx fx k∗ m
j1
g
j2x k−k∗ m
j1
g
j2x≥m0 k−k∗
m
j1
g
Consequently,Pkxis bounded below bym0 onRn. For any fixedk ≥k∗1,suppose that
{yl}satisfiesPkyl → infx∈RnPkx.Then, whenlis sufficiently large,
fx0 1Pkx0 1≥pkylfylk m
j1
g j2
yl≥m0 k−k∗
m
j1
g j2
yl. 2.7
Thus,
fx0 1−m0
k−k∗ ≥
m
j1
g j2
yl≥gj2
yl, j1, . . . , m. 2.8
It follows that
g j
yl≤
fx
0 1−m0
k−k∗
1/2
≤
fx
0 1−m0
k∗ −k∗
1/2
≤t0, j1, . . . , m. 2.9
That is,yl∈Xt0, whenlis sufficiently large. From2.7, we have
fyl≤fx0 1, 2.10
when l is sufficiently large. By the level boundedness of f on Xt0, we see that {yl} is
bounded. Thus, there exists a subsequence{ylp} of{yl} such thatylp → xk asp → ∞.
Then
Pk
ylp
−→Pkxk xinf
∈RnPkx. 2.11
Moreover,xk∈Xt0.Thus,{xk}is bounded. Let{xki}be a subsequence which converges to x∗. Then, for any feasible solutionxofP, we have
fxki ki
m
j1
g
j2xki≤fx. 2.12
That is,
m0 ki−k∗
m
j1
g
j2xki≤fxki k∗
m
j1
g
j2xki ki−k∗
m
j1
g
j2xki≤fx, 2.13
namely,
m
j1
g
j2xki≤
fx−m0
Passing to the limit asi → ∞and noting thatxki → x∗, we have
m
i1
g
j2x∗≤0. 2.15
Hence,
g
jx∗ 0, j1, . . . , m. 2.16
It follows that
gjx∗≤0, j 1, . . . , m. 2.17
Consequently,x∗ ∈X0.Moreover, from2.12, we havefxki≤ fx.Passing to the limit
asi → ∞, we obtainfx∗≤fx.By the arbitrariness ofx∈X0, we conclude thatx∗is an optimal solution ofP.
Remark 2.3. Iffxis bounded below onRn, then for anyk >0,Pkxis bounded below on
Rn.
The next proposition presents sufficient conditions that guarantee all the conditions of Lemma 2.2.
Proposition 2.4. Any one of the following conditions ensures the validity of the conditions of
Lemma 2.2
i fxis coercive onRn;
iithe functionmax{fx, gjx, j 1, . . . , m}is coercive onRnand there existk∗>0and m0∈R1such that
Pk∗x≥m0, ∀x∈Rn. 2.18
Proof. We need only to show that ifiiholds, then the conditions ofLemma 2.1hold, since conditioniis stronger than conditionii. Lett0>0. We need only to show thatfis coercive
onXt0. Otherwise, there existsσ >0 and{yk} ⊂Xt0with yk → ∞satisfying
fyk≤σ. 2.19
From{yk} ⊂Xt0, we deduce
gjyk≤t0, j1, . . . , m. 2.20
It follows from2.19and2.20that
maxfyk, gj
yk, j 1, . . . , m
≤max{σ, t0}, 2.21
contradicting the coercivity of max{fx, g
jx, j 1, . . . , m} since yk → ∞ as k →
The next proposition follows immediately fromLemma 2.2andProposition 2.4.
Proposition 2.5. If one of the two conditions (i) and (ii) ofProposition 2.4holds, then the conclusions ofLemma 2.2hold.
The following theorem can be established similarly to5, Theorem 4by usingLemma 2.2.
Theorem 2.6. Suppose thatf, gjj 1, . . . , mare all invex with the sameηand the conditions of
Lemma 2.2hold, then,MP MD.
Corollary 2.7. Suppose that f, gjj 1, . . . , m are all invex with the same η and one of the conditions (i) and (ii) ofProposition 2.4holds, then,MP MD.
Example 2.8. Consider the following optimization problem
min x
s.t. x∈R1, x2 ≤0. P
It is easy to see that both the objective function and the constraint function are convex and thus invex. Note that the objective functionfx x → −∞asx → −∞. It follows that limx →∞fx ∞does not hold. Consequently, all the results in5are not applicable.
However, it is easily checked that the conditions of ourCorollary 2.7hold and, hence,MP
MD.
Acknowledgment
This work is supported by the National Science Foundation of China.
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