Journal of Inequalities and Applications Volume 2009, Article ID 258756,10pages doi:10.1155/2009/258756
Research Article
Optimality Conditions and Duality for DC
Programming in Locally Convex Spaces
Xianyun Wang
College of Mathematics and Computer Science, Jishou University, Jishou 416000, China
Correspondence should be addressed to Xianyun Wang,[email protected]
Received 10 February 2009; Revised 25 June 2009; Accepted 25 September 2009
Recommended by K. L. Teo
Consider the DC programming problemPAinfx∈X{fx−gAx}, wheref andg are proper
convex functions defined on locally convex Hausdorff topological vector spaces X and Y, respectively, andAis a linear operator fromXtoY. By using the properties of the epigraph of the conjugate functions, the optimality conditions and strong duality ofPAare obtained.
Copyrightq2009 Xianyun Wang. 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.
1. Introduction
Let X and Y be real locally convex Hausdorfftopological vector spaces, whose respective dual spaces,X∗ andY∗,are endowed with the weak∗-topologiesw∗X∗, Xand w∗Y∗, Y. Letf :X → R:R∪ {∞},g :Y → Rbe proper convex functions, and letA:X → Y be a linear operator such thatAdomf∩domg /∅. We consider the primal DCdifference of convexprogramming problem
PA inf x∈X
fx−gAx, 1.1
and its associated dual problem
DA inf y∗∈Y∗
A
g∗y∗−f∗A∗y∗, 1.2
defined by A∗y∗,· y∗, A·is continuous onX. Note that, in general,YA∗is not the whole spaceY∗becauseAis not necessarily continuous.
Problems of DC programming are highly important from both viewpoints of optimization theory and applications. They have been extensively studied in the literature; see, for example, 1–6 and the references therein. On one hand, such problems being heavily nonconvex can be considered as a special class in nondifferentiable programmingin particular, quasidifferentiable programming7and thus are suitable for applying advanced techniques of variational analysis and generalized differentiation developed, for example, in 7–10. On the other hand, the special convex structure of both plus functionf and minus functiong◦Ain the objective of1.1offers the possibility to use powerful tools of convex analysis in the study of DC Programming.
DC programming of type1.1 whenAis an identity operatorhas been considered in theRnspace in paper5, where the authors obtained some necessary optimality conditions for local minimizers to1.1by using refined techniques and results of convex analysis. In this paper, we extend these results to DC programming in topological vector spaces and also derive some new necessary and/or sufficient conditions for local minimizers to1.1. Finally, we consider the strong duality of problem1.1; that is, there is no duality gap between the problemPAand the dual problemDAandDAhas at least an optimal solution.
In this paper we study the optimality conditions and the strong duality betweenPA andDAin the most general setting, namely, whenfandgare proper convex functionsnot necessarily lower semicontinuousand Ais a linear operatornot necessarily continuous. The rest of the paper is organized as follows. InSection 2we present some basic definitions and preliminary results. The optimality conditions are derived inSection 3, and the strong duality of DC programming is obtained inSection 4.
2. Notations and Preliminary Results
The notation used in the present paper is standard cf. 11. In particular, we assume throughout the paper that X and Y are real locally convex Hausdorff topological vector spaces, and letX∗ denote the dual space, endowed with the weak∗-topologyw∗X∗, X.By
x∗, xwe will denote the value of the functionalx∗ ∈X∗atx ∈X, that is, x∗, x x∗x.
The zero of each of the involved spaces will be indistinctly represented by 0.
Letf:X → Rbe a proper convex function. The effective domain and the epigraph of
fare the nonempty sets defined by
dom f:x∈X:fx<∞,
epif:x, r∈X×R: fx≤r. 2.1
The conjugate function offis the functionf∗:X∗ → Rdefined by
f∗x∗:sup x∗, x −fx:x∈X. 2.2
Iffis lower semicontinuous, then the following equality holds:
Letx∈domf. For each≥0, the-subdifferential offatxis the convex set defined by
∂fx:x∗∈X∗:x∗, y−x−≤fy−fxfor eachy∈X. 2.4
Whenx /∈domf, we put∂fx:∅. If0 in2.4, the set∂fx:∂0fxis the classical subdifferential of convex analysis, that is,
∂fx:x∗∈X∗:x∗, y−x≤fy−fxfor eachy∈X. 2.5
Let >0, the following inequality holdscf.11, Theorem 2.4.2ii :
fx f∗x∗≤ x∗, x⇐⇒x∗∈∂fx. 2.6
Following12,
epif∗ ≥0
x∗, x∗, x −fx :x∗∈∂
fx. 2.7
The Young equality holds
fx f∗x∗ x∗, x ⇐⇒x∗∈∂fx. 2.8
As a consequence of that,
x∗, x∗, x −fx∈epi f∗ ∀x∗∈∂fx. 2.9
The following notion of Cartesian product map is used in13.
Definition 2.1. LetM1, M2, N1, N2be nonempty sets and consider mapsF :M1 → M2and
G:N1 → N2. We denote byF×G:M1×N1 → M2×N2the map defined by
F×Gx, y:Fx, Gy. 2.10
3. Optimality Conditions
Let idRdenote the identity map onR. We consider the image setA∗×idRepig∗of epig∗ through the mapA∗×idR:YA∗ ×R → X∗×R, that is,
x∗, r∈A∗×id
Repig∗⇐⇒ ∃y∗∈YA∗ such that
y∗, r∈epig∗ andA∗y∗x∗. 3.1
Lemma 3.1. Letφ1, φ2 be proper convex fucntions onX, and letφ φ1−φ2. Thenx0 is a local minimizer ofφif and only if, for each≥0
∂φ2x0⊆∂φ1x0. 3.2
Especially, ifx0is a local minimizer ofφ, then
∂φ2x0⊆∂φ1x0. 3.3
Theorem 3.2. The following statements are equivalent: iepig◦A∗ A∗×idRepig∗,
iiFor eachx0∈A−1dom gand each≥0,
∂g◦Ax0 A∗∂gAx0. 3.4
Moreover,x0is a local optimal solution to problemPAif and only if for each≥0,
A∗∂gAx0⊆∂g◦Ax0⊆∂fx0. 3.5
Proof. i⇒ii. Suppose thatiholds. Letx0 ∈A−1domg, ≥ 0, andu ∈ YA∗ ∩∂gAx0, then for eachx∈X,
A∗u, x−x0 u, Ax−Ax0 ≤gAx−gAx0 . 3.6
Therefore,A∗u∈∂g◦Ax0. Hence,A∗∂gAx0⊆∂g◦Ax0.
Conversely, letv∈YA∗∩∂g◦Ax0. Thenv, v, x0 −g◦Ax0 ∈epig◦A∗. Byi,
v, v, x0 −g◦Ax0 ∈A∗×idRepig∗. 3.7
Therefore, there existsw ∈YA∗ such thatA∗w vandg∗w≤ v, x0 −gAx0 . Noting
that A∗w, x0 w, Ax0, then
0≤ v, x0 −gAx0−g∗w w, Ax0 −gAx0−g∗w . 3.8
This impliesw∈∂gAx0thanks to2.6. Thus,vA∗w∈A∗∂gAx0and∂g◦Ax0⊆
A∗∂gAx0. Hence,3.4is seen to hold.
ii⇒i. Suppose that ii holds. To showi, it suffices to show that epig◦A∗ ⊆ A∗×idRepig∗. To do this, letx∗, r∈epi g◦A∗andx
0 ∈A−1dom g. By2.7, there exists≥0 such thatx∗∈∂g◦Ax0andr x∗, x0 −gAx0 .From3.4, there exists
y∗∈∂gAx0such thatx∗A∗y∗. Sincey∗∈∂gAx0, it follows from2.6that
g∗y∗gAx0≤y∗, Ax
0
that isg∗y∗≤ x∗, x0−gAx0r. Hence,x∗, r∈A∗×idRepig∗and so epig◦A∗⊆ A∗×id
Repig∗.
By the well-known characterization of optimal solution to DC problem see
Lemma 3.1,x0is a local optimal solution to problemPAif and only if, for each≥0,
∂g◦Ax0⊆∂fx0. 3.10
Obviously,A∗∂gAx0⊆∂g◦Ax0holds automatically. The proof is complete.
Letp∈Y. Define
f∗◦A∗∗
A
p: sup
y∗∈Y∗ A
p, y∗−f∗A∗y∗. 3.11
Theorem 3.3. The following statements are equivalent: iepif∗◦A∗∗A A×idRepif,
iiFor each≥0and eachy∗∈YA∗ ∩A∗domf∗,
∂f∗◦A∗y∗A∂f∗A∗y∗. 3.12
Moreover,y∗is a local optimal solution to problemDAif and only if, for each≥0,
A∂f∗A∗y∗⊆∂f∗◦A∗y∗⊆∂g∗y∗. 3.13
Proof. i⇒ii. Suppose thatiholds. Let≥0, y∗∈YA∗∩A∗domf∗andy∈∂f∗◦A∗y∗. Then one has
f∗◦A∗∗
A
yf∗◦A∗y∗≤y, y∗. 3.14
Hence,y, y, y∗ −f∗A∗y∗ ∈epif∗◦A∗∗A.By the given assumption,
y,y, y∗−f∗A∗y∗∈A×idRepif. 3.15
Therefore, there exists x ∈ X such thatAx y and x, y, y∗ −f∗A∗y∗ ∈ epif. Hence,fx≤ y, y∗ −f∗A∗y∗ , this meansx∈∂f∗A∗y∗and soAx∈A∂f∗A∗y∗. Consequently,∂f∗◦A∗y∗⊆A∂f∗A∗y∗. This completes the proof because the converse inclusion holds automatically.
ii⇒i. Suppose thatiiholds. To showi, it suffice to show that epif∗◦A∗∗A ⊆ A×idRepif. To do this, lety, r ∈ epif∗◦A∗∗A andy∗ ∈ YA∗ ∩A∗dom f∗. By2.7, there exists ≥0 such thaty∈∂f∗◦A∗y∗andr y∗, y −f∗A∗y∗ . From3.12, there existsx∈∂f∗A∗y∗such thatyAx. Sincex∈∂f∗A∗y∗, it follows from2.6that
that isfx≤ y∗, y−f∗A∗y∗ r. Hence,y, r∈A×idRepi fand so epif∗◦A∗∗A ⊆ A×idRepif.
Similar to the proof of3.5, one has that3.13holds.
4. Duality in DC Programming
This section is devoted to study the strong duality between the primal problem and its Toland dual, namely, the property that both optimal values coincide and the dual problem has at least an optimal solution.
Givenp∈X∗, we consider the DC programming problem given in the form
PA,pinf x∈X
fx−gAx−p, x, 4.1
and the corresponding dual problem
DA,p inf y∗∈Y∗
A
g∗y∗−f∗pA∗y∗. 4.2
Let vPA,p, vDA,p denote the optimal values of problems PA,p and DA,p,
respectively, that is
v PA,pinf
x∈X
fx−gAx−p, x,
v DA,p inf
y∗∈Y∗ A
g∗y∗−f∗pA∗y∗.
4.3
In the special case whenp 0, problemsPA,pandDA,pare just the problemPAand DA.
Before establishing the relationship between problemsPA,pandDA,p, we give
useful formula for computing the values of conjugate functions. The formula is an extension of a well-known result, called Toland duality, for DC problems. In this section, we always assume thatgandf∗are everywhere subdifferentible.
Proposition 4.1. Lethf−g◦A. Then the conjugate functionh∗ofhis given by
h∗x∗ sup
y∗∈Y∗ A
Proof. By the definition of conjugate function, it follows that
h∗x∗ sup
x∈X
x∗, x −f−g◦Ax
sup x∈X
x∗, x −fx gAx
sup x∈X
x∗, xA∗y∗, x−fx−A∗y∗, xgAx ∀y∗∈Y∗
A
≥sup x∈X
x∗A∗y∗, x−fx−sup
x∈X
y∗, Ax−gAx ∀y∗∈Y∗
A
≥ sup y∗∈Y∗ A
f∗x∗A∗y∗−g∗y∗.
4.5
Next, we prove that
h∗x∗≤ sup
y∗∈Y∗ A
f∗x∗A∗y∗−g∗y∗.
4.6
Suppose on the contrary thath∗x∗> supy∗∈Y∗ A{f
∗x∗A∗y∗−g∗y∗},that is, there exists
x0∈Xsuch that
x∗, x0 −fx0 gAx0> sup
y∗∈Y∗ A
f∗x∗A∗y∗−g∗y∗. 4.7
Lety0Ax0andy0∗∈∂gy0, then
g∗y∗
0
y∗
0, y0
−gy0
. 4.8
From this, it follows that
x∗, x0 −fx0 gAx0 x∗A∗y∗, x
0
−fx0−A∗y∗, x
0
−g∗y∗
0
≤f∗x∗A∗y∗−g∗y0∗, 4.9
which is contradiction to4.7, and so4.4holds.
Following fromProposition 4.1, we obtain the following proposition.
Proposition 4.2. For eachp∈X∗,
Proof. Letp∈X∗. Since infx∈X{fx−gAx− p, x}−f−g◦A∗p, it follows from4.4 that
v PA,p inf
x∈X
fx−gAx−p, x
−sup y∗∈Y∗ A
f∗pA∗y∗−g∗y∗
inf y∗∈Y∗
A
g∗y∗−f∗pA∗y∗
v DA,p.
4.11
Remark 4.3. In the special case when p 0 and A 0, formula 4.10 was first given by Pshenichnyisee10and related results on duality can be found in15–17.
Proposition 4.4. For eachp∈X∗,
iif x0 is an optimal solution to problemPA,p, then y∗0 ∈ YA∗ ∩∂gAx0is an optimal
solution to problemDA,p;
iisuppose thatf and g are lower semicontinuous. Ify0∗ is an optimal solution to problem DA,p, thenx0∈∂f∗A∗y∗0is an optimal solution to problemPA,p.
Proof. iLetx0be an optimal solution to problemPA,pand lety∗0 ∈ YA∗ ∩∂gAx0. Then
A∗y∗
0∈A∗∂gAx0. It follows from3.5thatA∗y∗0∈∂fx0. By the Young equality, we have
A∗y∗
0, x0
A∗y∗
0p, x0
−p, x0
f∗pA∗y∗
0
fx0−p, x0
,
y∗
0, Ax0
g∗y∗
0
gAx0. 4.12
Therefore,
g∗y∗
0
−f∗pA∗y∗
0
fx0−gAx0−p, x0
. 4.13
By4.10,y∗0is an optimal solution to problemDA,p.
iiLety0∗be an optimal solution to problemDA,pandx0∈∂f∗A∗y∗0. ThenAx0∈
A∂f∗A∗y∗
0and henceAx0∈∂g∗y∗0thanks toTheorem 3.3. By the Young equality, we have
A∗y∗
0, x0
A∗y∗
0p, x0
−p, x0
f∗∗x0 f∗pA∗y∗
0
−p, x0
,
y∗
0, Ax0
g∗∗Ax0 g∗y∗
0
. 4.14
Since the functionsfandg are lower semicontinuous, it follows from2.3thatf∗∗ fand
g∗∗g. Hence, by the above two equalities, one has
g∗y∗
0
−f∗pA∗y∗
0
fx0−gAx0−p, x0
. 4.15
Obviously, ifAis continuous, thenYA∗ Y∗and soYA∗∩∂gAx/∅for eachx∈X. By Propositions4.2and4.4, we get the following strong duality theorem straightforwardly.
Theorem 4.5. For eachp∈X∗,
isuppose that A is continuous. If the problem PA,p has an optimal solution, then vPA,p vDA,pandDA,phas an optimal solution;
iisuppose that f and g are lower semicontinuous. If the problemDA,phas an optimal solution, thenvPA,p vDA,pandPA,phas an optimal solution.
Corollary 4.6. (i) If the problemPAhas an optimal solution, thenvPA vDAandDAhas
an optimal solution.
(ii)Suppose that f and g are lower semicontinuous. If the problem DA has an optimal
solution, thenvPA vDAandPAhas an optimal solution.
Remark 4.7. As in13, ifvPA vDAandPAhas an optimal solution, then we say the converse duality holds betweenPAandDA.
Example 4.8. LetXY Rand letAid.Definef, g:X → Rby
fx x4, gx 2x2. 4.16
Then the conjugate functionsf∗andg∗are
f∗pp p
4
1/3 − p
4
4/3
, g∗p p2
8 , p∈R. 4.17
Obviously,vPA:infx∈R{fx−gx} −1 andPAattained the infimun at±1,vDA infp∈R{g∗p−f∗p} −1 and DAattained the infimum at±4. Hence,vP vD. It is easy to see that∂g1 {4}, ∂g−1 {−4}and ∂f∗4 {1}, ∂f∗−4 {−1}. Therefore,
Proposition 4.4is seen to hold andTheorem 4.5is applicable.
Acknowledgment
The author wish to thank the referees for careful reading of this paper and many valuable comments, which helped to improve the quality of the paper.
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