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Lagrange-type duality in DC programming
J. Math. Anal. Appl. 418 (2014) 415–424
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Lagrange-type duality in DC programming Ryohei Harada a,∗ , Daishi Kuroiwa b a b
Graduate School of Science and Engineering, Shimane University, Japan Major in Interdisciplinary Science and Engineering, Shimane University, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 27 December 2013 Available online 13 April 2014 Submitted by H. Frankowska
We show a Lagrange-type duality theorem for a DC programming problem, which is a generalization of previous results by J.-E. Martínez-Legaz, M. Volle [5] and Y. Fujiwara, D. Kuroiwa [1] when all constraint functions are real-valued. To the purpose, we decompose the DC programming problem into certain infinite convex programming problems. © 2014 Elsevier Inc. All rights reserved.
Keywords: DC programming problem Lagrange duality theorem
1. Introduction In the present work, our main purpose is to show a generalized Lagrange-type duality theorem, which is an extension of J.-E. Martínez-Legaz, M. Volle [5] and Y. Fujiwara, D. Kuroiwa [1], for the following DC programming problem (P): (P)
minimize f0 (x) − g0 (x) subject to fi (x) − gi (x) ≤ 0, i = 1, · · · , m,
where fi , gi : Rn → R ∪ {+∞} are lsc proper convex functions for each i = 0, 1, · · · , m, and we adopt the conventions (+∞) − (+∞) = +∞, 0 · (+∞) = 0. To this aim, we decompose the above DC programming problem (P) into certain infinite convex programming problems, and we propose a constraint qualification for the Lagrange-type duality. The outline of the paper is as follows. In Section 2, we introduce definitions and preliminary results that will be used later in this paper. In Section 3, we introduce previous Lagrange-type duality results for DC programming and canonical DC programming. In Section 4, we show a Lagrange-type duality theorem for DC programming problem (P), which is a generalization of the previous ones when all constraint functions are real-valued, we apply the main result to a DC programming problem with reverse convex constraints, and we give an example of the main result. * Corresponding author. http://dx.doi.org/10.1016/j.jmaa.2014.04.017 0022-247X/© 2014 Elsevier Inc. All rights reserved.
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2. Notations and preliminaries In this section, we describe our notations and present the preliminary results. The inner product of two vectors x and y in the n-dimensional real Euclidean space Rn will be denoted by x, y. For a set A ⊆ Rn we shall denote the closure, convex hull, conical hull of A by cl A, co A, and cone A, respectively. For the set A, the indicator function δA is defined by 0 x ∈ A, δA (x) = +∞ otherwise. For an extended real-valued function f : Rn → R ∪ {+∞}, the domain, the epigraph, and the conjugate function of f are defined by dom f = x ∈ Rn f (x) < +∞ , epi f = (x, r) ∈ Rn × R x ∈ dom f, f (x) ≤ r , f ∗ (y) = sup x, y − f (x) .
and
x∈Rn
For each x ∈ dom f , the subdifferential of the function f at x is the set ∂f (x) = x∗ ∈ Rn x∗ , y − x ≤ f (y) − f (x), ∀y ∈ Rn . For all x, y ∈ Rn , we have f (x) + f ∗ (y) ≥ y, x (the Young–Fenchel inequality) and it can be shown that if x ∈ dom f , then f (x) + f ∗ (y) = y, x
⇔
y ∈ ∂f (x).
For an arbitrary function g : Rn → R ∪ {+∞, −∞}, the level sets are defined by {g • y} = x ∈ Rn g(x) • y
(y ∈ R)
where • is a binary relation on R ∪ {+∞, −∞}. The following results are used in our main results. Theorem 1. (See M.A. Goberna, V. Jeyakumar, M.A. López [2].) Let I be an arbitrary index set. Let gi : Rn → R ∪ {+∞} be a lsc convex function for each i ∈ I, let C be a closed convex set. Assume that each gi is continuous at least at one point of A = {x ∈ C | gi (x) ≤ 0, ∀i ∈ I}. Then the following statements are equivalent: ∗ is closed. (i) cone co i∈I epi gi∗ + epi δC ∗ is (ii) For every lsc proper convex function f : Rn → R ∪ {+∞} with A ∩ dom f = ∅ and epi f ∗ + epi δA closed,
inf f (x) = max inf f (x) + λi gi (x) . x∈A
(I) λ∈R+ x∈C
i∈I
(iii) For every v ∈ Rn , inf v, x = max inf
x∈A
(I)
(I) λ∈R+ x∈C
v, x +
λi gi (x) ,
i∈I
where λ ∈ R+ if and only if λi ≥ 0 for each i ∈ I and the set {i ∈ I | λi = 0} is finite.
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Lemma 1. (See V. Jeyakumar, N. Dinh and G.M. Lee [4, Proposition 2.1].) Let gi : Rn → R be a convex function for each i = 1, · · · , m, let C be a closed convex set. If {gi | i = 1, · · · , m} satisfies the Slater constraint qualification on C (i.e. there exists x0 ∈ C such that gi (x0 ) < 0 for each i = 1, · · · , m), then cone co
m
∗ epi gi∗ + epi δC is closed.
i=1
3. Previous results of Lagrange-type duality in DC and canonical DC programming In this paper, we study the following DC programming problem (P): minimize f0 (x) − g0 (x)
(P)
subject to fi (x) − gi (x) ≤ 0, ∀i = 1, 2, · · · , m,
where fi , gi : Rn → R∪{+∞} are lsc proper convex functions for each i = 0, 1, · · · , m, and assume that the set of all feasible solutions S = {x ∈ Rn | fi (x)−gi (x) ≤ 0, ∀i = 1, 2, · · · , m} is nonempty. In this section, we give previous Lagrange-type duality results for (P). Theorem 2 and Theorem 3 give duality for the DC programming problem (P). We adopt the conventions 0·(+∞) = +∞ and 0·(−∞) = 0 in Theorem 2 and Theorem 3. Theorem 2. (See J.-E. Martínez-Legaz, M. Volle [5].) Let fi : Rn → R ∪ {+∞} be a convex function for each i = 0, 1, · · · , m, let g0 : Rn → R ∪ {+∞, −∞} satisfy g0 = g0∗∗ , and let gi : Rn → R ∪ {+∞, −∞} m be subdifferentiable on S for each i = 1, · · · , m. If for each (x∗1 , · · · , x∗m ) ∈ i=1 {gi∗ − fi∗ ≤ 0}, there exists x) − ¯ x, x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m, then x ¯ ∈ dom f0 such that fi (¯
inf
fi (x)−gi (x)≤0
=
f0 (x) − g0 (x)
inf
inf
g0∗
max
m ∗ ∗ x∗ ∈dom g0∗ gi∗ (x∗ i )−fi (xi )≤0 λ∈R+
m ∗ λi gi∗ x∗i − x +
f0 +
i=1
m
∗ ∗
λi fi
x +
i=1
m
λi x∗i
.
i=1
Theorem 3. (See J.-E. Martínez-Legaz, M. Volle [5].) Let fi : Rn → R ∪ {+∞} be a convex function for each i = 0, 1, · · · , m, let g0 : Rn → R ∪ {+∞, −∞} satisfy g0 = g0∗∗ , and let gi : Rn → R ∪ {+∞, −∞} be subdifferentiable on S for each i = 1, · · · , m. If for each (x∗1 , · · · , x∗m ) ∈ Ω = {(x∗1 , · · · , x∗m ) ∈ Rnm | ∗ (x∗m ) = ∅}, there exists x0 ∈ dom f0 such that fi (x0 ) − x0 , x∗i + gi∗ (x∗i ) < 0 for each ∂g1∗ (x∗1 ) ∩ · · · ∩ ∂gm i = 1, · · · , m, then
inf
fi (x)−gi (x)≤0
=
f0 (x) − g0 (x)
inf
g0∗
max
m ∗ ∗ ∗ (x∗ 0 ,x1 ,···,xm )∈dom g0 ×Ω λ∈R+
m ∗ x + λi gi∗ x∗i −
f0 +
i=1
m
∗ ∗
λi fi
x +
i=1
m
λi x∗i
.
i=1
Remark 1. The right-hand side of Theorem 2 and Theorem 3 can transform to a formulation of Lagrange-type duality. Indeed, ∗ m m m ∗ ∗ ∗ ∗ ∗ ∗ g0 x + λi gi xi − f0 + λi fi x + λi x i i=1
=
g0∗
x
∗
+
i=1 m
λi gi∗
∗ xi − sup ∗
= infn f0 (x) − x, x x∈R
∗
x, x +
x∈Rn
i=1
i=1
+
g0∗
x
∗
m
λi x∗i
−
f0 +
i=1
+
m i=1
λi fi (x) −
m
λi fi (x)
i=1
x, x∗i
+
gi∗
x∗i
.
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It is well-known that the DC programming problem (P) can be transformed into the following canonical DC programming problem (Q) where fi and gi are real-valued functions: (Q)
minimize a, x subject to f (x) ≤ 0, g(x) ≥ 0,
where f, g : Rn → R are convex functions and a ∈ Rn . Theorem 4 and Theorem 5 give duality in the canonical DC programming problem (Q). Theorem 4. (See Y. Fujiwara, D. Kuroiwa [1].) Let f, g : Rn → R be convex functions, and let a ∈ Rn , let S = {x ∈ Rn | f (x) ≤ 0, g(x) ≥ 0} be nonempty and let x∈S ∂g(x) ⊆ A. If for each z ∈ A ∩ dom g ∗ , {f ≤ 0} ∩ {−z, · + g ∗ (z) ≤ 0} is nonempty and cone co(epi f ∗ ∪ {−z} × [−g ∗ (z), +∞) ∪ {0} × [0, +∞)) is closed, then inf
a, x = inf sup infn a, x + λf (x) + μ −y, x + g ∗ (y)
f (x)≤0, g(x)≥0
y∈A λ,μ≥0 x∈R
holds and the supremum on λ, μ ≥ 0 being attained for all y ∈ A. Theorem 5. (See Y. Fujiwara, D. Kuroiwa [1].) Let f, g : Rn → R be convex functions, and let a ∈ Rn , S = {x ∈ Rn | f (x) ≤ 0, g(x) ≥ 0} is nonempty, Y = {y ∈ Rn | {f ≤ 0} ∩ {y, · > g ∗ (y)} = ∅} and ∗ x∈S ∂g(x) ⊆ A ⊆ Y . If cone epi f + {0} × [0, +∞) is closed, then inf
a, x = inf sup infn a, x + λf (x) + μ −y, x + g ∗ (y)
f (x)≤0, g(x)≥0
y∈A λ,μ≥0 x∈R
holds and the supremum on λ, μ ≥ 0 being attained for all y ∈ A. It is well-known that canonical DC programming problems are special cases of DC programming problems, see [3], so Theorems 2 and 3 have broader application areas than Theorems 4 and 5. However, the assumptions of Theorems 2 and 3 are stronger than Theorems 4 and 5 whenever the DC programming problem is canonical and constraint function f is real-valued. Our purpose of this paper is to give a Lagrangetype duality result for a general DC programming problem, which is a generalization of Theorems 2, 3, 4, and 5. 4. Main results We consider the following subproblems (P(y0 , (yi ))) of (P): P y0 , (yi )
minimize f0 (x) − x, y0 + g0∗ (y0 ) subject to fi (x) − x, yi + gi∗ (yi ) ≤ 0, ∀i = 1, · · · , m,
where (y0 , (yi )) = (y0 , y1 , · · · , ym ) ∈ Rn(m+1) . It is clear that all subproblems (P(y0 , (yi ))) are convex programming. Let Val(P) and Val(P(y0 , (yi ))) be the infimum values of (P) and (P(y0 , (yi ))), respectively. Lemma 2. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, (yi ) ∈ Rnm and S(yi ) = {x ∈ Rn | fi (x) − x, yi + gi∗ (yi ) ≤ 0, ∀i = 1, · · · , m}. Then S(yi ) ⊆ S,
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
furthermore if gi is subdifferentiable on S for each i = 1, · · · , m and
m
x∈S (
i=1
419
∂gi (x)) ⊆ D ⊆ Rnm , then
S(yi ) = S.
(yi )∈D
Proof. For each i = 1, · · · , m and x ∈ S(yi ), 0 ≥ fi (x) − x, yi + gi∗ (yi ) ≥ fi (x) − gi (x) by using the Young–Fenchel duality, so S(yi ) ⊆ S. Next, assume that all gi are subdifferentiable on S. Let z ∈ S. For each i = 1, · · · , m, since ∂gi (z) is a nonempty set, there exists yi ∈ ∂gi (z) and 0 ≥ fi (z) − gi (z) = fi (z) − z, yi + gi∗ (yi ), m so z ∈ S(yi ). Also, since (yi ) ∈ x∈S ( i=1 ∂gi (x)), we have z ∈ (yi )∈D S(yi ). Therefore S ⊆ (yi )∈D S(yi ). It is clear that the opposite inclusion holds. 2 Lemma 3 needs to the proof of Theorem 6. Lemma 3. Let h : Rn → R ∪ {+∞} be a function, let A be a nonempty set, and let B(x) be nonempty subsets of Rn for all x ∈ A. Then inf
inf h(y) =
x∈A y∈B(x)
h(y). inf y∈ x∈A B(x)
Proof. For each x ∈ A, since B(x) ⊆ x∈A B(x), inf y∈B(x) h(y) ≥ inf y∈x∈A B(x) h(y). Therefore we have inf x∈A inf y∈B(x) h(y) ≥ inf y∈x∈A B(x) h(y). Assume that inf x∈A inf y∈B(x) h(y) > inf y∈x∈A B(x) h(y). Then there exists β such that inf x∈A inf y∈B(x) h(y) > β > inf y∈x∈A B(x) h(y). From β > inf y∈x∈A B(x) h(y), there exists y0 ∈ x∈A B(x) such that h(y0 ) < β, and there exist x0 ∈ A such that y0 ∈ B(x0 ). Therefore inf
inf h(y) ≤
x∈A y∈B(x)
inf
y∈B(x0 )
h(y) ≤ h(y0 ) < β.
This contradicts to inf x∈A inf y∈B(x) h(y) > β. So, inf x∈A inf y∈B(x) h(y) = inf y∈x∈A B(x) h(y).
2
Theorem 6. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 0, 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, let gi be subdifferentiable on S for each i = 1, · · · , m, m n nm . Then x∈S ∂g0 (x) ⊆ D0 ⊆ R , and x∈S ( i=1 ∂gi (x)) ⊆ D ⊆ R Val(P) =
inf
(y0 ,(yi ))∈D0 ×D
Val P y0 , (yi ) .
Proof. Let x ∈ Rn . For each (yi ) ∈ D, we have gi (x)+gi∗ (yi ) ≥ x, yi for each i = 1, · · · , m. So fi (x)−gi (x) ≤ fi (x) − x, yi + gi∗ (yi ) for each i = 1, · · · , m, and by using Lemma 2, inf f0 (x) − g0 (x) ≤ inf f0 (x) − x, yi + g0∗ (yi ) x∈S x∈S ≤ inf f0 (x) − x, yi + g0∗ (yi ) . x∈S(yi )
This shows Val(P) ≤ inf (y0 ,(yi ))∈D0 ×D Val(P(y0 , (yi ))). Conversely, for each x ∈ S, pick yi ∈ ∂gi (x) for each i = 1, · · · , m, then fi (x) − x, yi + gi∗ (yi ) = fi (x) − gi (x). Therefore
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
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inf
(y0 ,(yi ))∈D0 ×D
Val P y0 , (yi ) = inf
inf
inf
y0 ∈D0 (yi )∈D z∈S(yi ) inf y0 ∈D0 z∈ (y )∈D S(yi )
= inf
f0 (z) − z, y0 + g0∗ (y0 )
f0 (z) − z, y0 + g0∗ (y0 )
i
= inf inf f0 (z) − z, y0 + g0∗ (y0 ) y0 ∈D0 z∈S
= inf inf
z∈S y0 ∈D0
f0 (z) − z, y0 + g0∗ (y0 )
≤ inf
≤ inf
inf
= inf
inf
f0 (z) − z, y0 + g0∗ (y0 )
inf z∈S y0 ∈ x∈S ∂g0 (x)
z∈S y0 ∈∂g0 (z)
f0 (z) − z, y0 + g0∗ (y0 )
z∈S y0 ∈∂g0 (z)
f0 (z) − g0 (z)
= inf f0 (z) − g0 (z) z∈S
= Val(P). The second and third equalities hold from Lemma 2 and Lemma 3, respectively. When S = ∅, this reverse inequality is clear since Val(P) = +∞. This completes the proof. 2 The following is the main result of this paper: Theorem 7. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 0, 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, let gi be subdifferentiable on S for each i = 1, · · · , m, m n nm . If each fi is continuous at least at one point x∈S ∂g0 (x) ⊆ D0 ⊆ R and x∈S ( i=1 ∂gi (x)) ⊆ D ⊆ R n ∗ of S(yi ) = {x ∈ R | fi (x) − x, yi + gi (yi ) ≤ 0, ∀i = 1, · · · , m} and cone co
m
epi fi∗ − yi , gi∗ (yi ) + {0} × [0, +∞) is closed,
i=1
for each (yi ) ∈ D ∩
m i=1
dom gi∗ , then
Val(P) =
inf
max inf
(y0 ,(yi ))∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
g0∗ (y0 )
+
m
∗ λi fi (x) − x, yi + gi (yi ) .
i=1
Proof. For each (yi ) ∈ D, ∗ epi fi − ·, yi + gi∗ (yi ) = epi fi∗ (· + yi ) − gi∗ (yi ) = epi fi∗ − yi , gi∗ (yi ) for each i = 1, · · · , m. Also it is easy to check that epi δR∗ n = {0} × [0, +∞). From the assumption of this theorem and Theorem 1, we have inf
x∈S(yi )
f0 (x) − x, y0 +
g0∗ (y0 )
= max infn f0 (x) − x, y0 + λi ≥0 x∈R
Therefore we conclude the final equality by using Theorem 6.
g0∗ (y0 )
+
m
λi fi (x) − x, yi + gi (yi ) .
i=1
2
Remark 2. We will show that Theorem 7 is a generalization of Theorem 2 and Theorem 3 when all fi (i = 1, · · · , m) are real-valued functions.
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421
[Theorem 7 ⇒ Theorem 2] Assume the assumption of Theorem 2, that is, for each (x∗1 , · · · , x∗m ) ∈ m ∗ ∗ ¯ ∈ Rn such that fi (¯ x) − ¯ x, x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m. i=1 {gi − fi ≤ 0}, there exists x ∗ ∗ ∗ Therefore {fi − ·, xi + gi (xi ) | i = 1, · · · , m} holds the Slater constraint qualification. From Lemma 1, m m cone co i=1 epi(fi − ·, x∗i + gi∗ (x∗i ))∗ + {0} × [0, +∞) = cone co i=1 (epi fi∗ − (x∗i , gi∗ (x∗i ))) + {0} × [0, +∞) m is closed. Let D0 = dom g0∗ and D = i=1 {gi∗ − fi∗ ≤ 0}. It is clear that x∈S ∂g0 (x) ⊆ D0 . Now we m m show x∈S ( i=1 ∂gi (x)) ⊆ D. For each (y1 , · · · , ym ) ∈ x∈S ( i=1 ∂gi (x)), there exists x0 ∈ S such that m (y1 , · · · , ym ) ∈ i=1 ∂gi (x0 ). Then, for each i = 1, · · · , m, gi (x0 ) + gi∗ (yi ) = x0 , yi , that is, gi (x0 ) + gi∗ (yi ) = fi (x0 ) + x0 , yi − fi (x0 ) ≤ fi (x0 ) + fi∗ (yi ), so we have gi∗ (yi ) − fi∗ (yi ) ≤ fi (x0 ) − gi (x0 ) ≤ 0. This shows m (y1 , · · · , ym ) ∈ D, that is, x∈S ( i=1 ∂gi (x)) ⊆ D. The assumption of Theorem 7 is satisfied. [Theorem 7 ⇒ Theorem 3] Assume the assumption of Theorem 3, that is, for each (x∗1 , · · · , x∗m ) ∈ Ω = m {(x∗1 , · · · , x∗m ) ∈ Rnm | i=1 ∂gi∗ (x∗i ) = ∅}, there exists x0 ∈ Rn such that fi (x0 ) − x0 , x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m. Therefore {fi − ·, x∗i + gi∗ (x∗i ) | i = 1, · · · , m} holds the Slater constraint qualification. m m From Lemma 1, cone co i=1 epi(fi −·, x∗i +gi∗ (x∗i ))∗ +{0}×[0, +∞) = cone co i=1 (epi fi∗ −(x∗i , gi∗ (x∗i )))+ {0} × [0, +∞) is closed. Let D0 = dom g0∗ and D = Ω. It is clear that x∈S ∂g0 (x) ⊆ D0 . Now we m m show x∈S ( i=1 ∂gi (x)) ⊂ D. For each (y1 , · · · , ym ) ∈ x∈S ( i=1 ∂gi (x)), there exists x0 ∈ S such that m m (y1 , · · · , ym ) ∈ i=1 ∂gi (x0 ). Since gi = gi∗∗ , x0 ∈ ∂gi (yi ) for each i = 1, · · · , m. Therefore x0 ∈ i=1 ∂gi (yi ), m and we have i=1 ∂gi (yi ) = ∅, that is, (y1 , · · · , ym ) ∈ Ω = D. Consequently the assumption of Theorem 7 is satisfied. Also, Theorem 7 is a strongly generalization of Theorem 2 and Theorem 3 when all fi (i = 1, · · · , m) are real-valued functions. For example, let n = 1, m = 1, ⎧ ⎨ −x − 1 x < −1, x ∈ [−1, 1], and g1 (x) = 0. f1 (x) = 0 ⎩ x−1 x > 1, Then f1∗ (x∗ ) = |x∗ | + δ[−1,1] (x∗ ) and g1∗ (x∗ ) = δ{0} (x∗ ). cone epi f1∗ + {0} × [0, +∞) is closed. So, the assumption of Theorem 7 holds. However, both the assumptions of Theorems 2 and 3 do not hold because f1 (x) − x, x∗ + g1∗ (x∗ ) ≥ 0 for each x, x∗ ∈ Rn . But, Theorem 7 is not a generalization of Theorem 2 and Theorem 3 when some fi (i = 1, · · · , m) is not a real-valued function. For example, let n = 1, m = 1, f0 (x) = x,
g0 (x) = 0,
f1 (x) =
−1 x ∈ [−1, 1], +∞ x ∈ / [−1, 1],
and g1 (x) = 0.
Then, f1∗ (x∗ ) = |x∗ | + 1 and g1∗ (x∗ ) = δ{0} (x∗ ). So, f0 , g0 , f1 and g1 satisfy assumptions of Theorem 2 and Theorem 3. However, cone epi f1∗ + {0} × [0, +∞) is not closed. So, f1 and g1 do not hold the assumption of Theorem 7. Remark 3. We will show that Theorem 7 is a generalization of Theorem 4 and Theorem 5. [Theorem 7 ⇒ Theorem 4] Assume the assumption to Theorem 4. Let f0 = a, ·, g0 = 0, f1 = f , g1 = 0, f2 = 0, and g2 = g. Then ∂g0 (x) = {0} for each x ∈ S and g0∗ = δ{0} . Let D0 = {0}, D = {0} × A, where A is the set satisfying x∈S ∂g(x) ⊆ A. Then
∂g0 (x) ⊆ D0
and
x∈S
∂g1 (x) × ∂g2 (x) ⊆ D. x∈S
Also S(y1 , y2 ) is nonempty from Lemma 2 and ∗ ∗ cone co epi f1 − ·, y1 + g1∗ (y1 ) ∪ epi f2 − ·, y2 + g2∗ (y2 ) + {0} × [0, +∞) = cone co epi f ∗ ∪ {−y2 } × −g ∗ (y2 ), +∞ ∪ {0} × [0, +∞)
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422
is closed for each (y1 , y2 ) ∈ D. By using Theorem 7, we have inf
a, x =
f (x)≤0, g(x)≥0
=
inf
fi (x)−gi (x)≤0
inf
f0 (x) − g0 (x)
max inf
(y0 ,y1 ,y2 )∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
g0∗ (y0 )
+
2
∗
λi fi (x) − x, yi + g (yi )
i=1
= inf max infn a, x + λ1 f (x) + λ2 −x, y + g ∗ (y) . y∈A λi ≥0 x∈R
This completes the proof of Theorem 4. [Theorem 7 ⇒ Theorem 5] Assume the assumption to Theorem 5. For any z ∈ A, where A satisfies ∗ ∗ x∈S ∂g(x) ⊆ A ⊆ Y , there exists x0 ∈ {f ≤ 0} such that −z, x0 + g (z) < 0, that is, {−z, · + g (z)} holds the Slater constraint qualification on {f ≤ 0}. From Lemma 1, ∗ ∗ cone epi −·, z + g ∗ (z) + epi δ{f ≤0} is closed. So, for every v ∈ Rn ,
f (x)≤0,
inf
−x,z+g ∗ (z)≤0
v, x = max inf
λ≥0 f (x)≤0
v, x + λ −x, z + g ∗ (z)
by using Theorem 1. Since cone epi f ∗ + {0} × [0, +∞) is closed, inf
f (x)≤0
v, x + λ −x, z + g ∗ (z) = max infn v, x + λ −x, z + g ∗ (z) + μf (x) , μ≥0 x∈R
by using Theorem 1 again. Therefore for every v ∈ Rn ,
f (x)≤0,
inf
−x,z+g ∗ (z)≤0
v, x = max max infn v, x + λ −x, z + g ∗ (z) + μf (x) , λ≥0 μ≥0 x∈R
and this shows ∗ cone co epi f ∗ ∪ epi −·, z + g ∗ (z) + {0} × [0, +∞) is closed from Theorem 1 once again. This is the assumption of Theorem 4, and the proof is completed from the previous discussion. So, Theorem 7 is a strongly generalization of Theorem 4 and Theorem 5 because DC programming problems are not canonical DC programing problems in general. Remark 4. When fi = 0 for each i = 1, · · · , m, the minimization problem (P) of Theorem 7 becomes a DC programming problem with a reverse convex inequality system. In this case, epi fi∗ = {0} × [0, +∞) for each i = 1, · · · , m, and then the characteristic cone is always closed. Therefore,
Val(P) =
inf
max inf
(y0 ,(yi ))∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
=
inf
max
(y0 ,(yi ))∈D0 ×D λi ≥0
g0∗ (y0 )
+
m i=1
g0∗ (y0 )
−
f0∗
λi −x, yi +
i=1
λi gi∗ (yi )
+
m
y0 +
m i=1
λi yi
.
gi∗ (yi )
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
423
Example 1. In this example, we calculate the optimal value of the following DC programming problem by using Theorem 7: minimize x − |y|
(P)
subject to x2 + y 2 − 1 − |x| ≤ 0.
Let f0 (x, y) = x, g0 (x, y) = |y|, f1 (x, y) = x2 +y 2 −1, g1 (x, y) = |x|, D0 = {0}×[−1, 1], and D = [−1, 1]×{0}. Then we have g0∗ = δ{0}×[−1,1] and g1∗ = δ[−1,1]×{0} . For each y1 ∈ D, we can check that f1 is continuous at least at one point of S(y1 ) because S(y1 ) = {x ∈ Rn | f1 (x) − x, y1 + g1∗ (y1 ) ≤ 0} is nonempty, and cone co epi f1∗ − y1 , g1∗ (y1 ) + {0} × [0, +∞) is closed. By using Theorem 7, Val(P) =
inf
max
inf
(t1 ,t2 )∈D0 , (t3 ,t4 )∈D λ≥0 (x,y)∈R2
f0 (x, y) − (x, y), (t1 , t2 ) + g0∗ (t1 , t2 )
+ λ f1 (x, y) − (x, y), (t3 , t4 ) + g1∗ (t3 , t4 ) max inf 2 x − t2 y + λ x2 + y 2 − 1 − t3 x = inf t2 ,t3 ∈[−1,1] λ≥0 (x,y)∈R
=
inf
max
t2 ,t3 ∈[−1,1] λ>0
inf
(x,y)∈R2
x − t 2 y + λ x2 + y 2 − 1 − t3 x
2 2 2 2
λt3 − 1 t2 λt3 − 1 t2 λ x − + y − − − − 1 2λ 2λ 2λ 2λ t2 ,t3 ∈[−1,1] λ>0 (x,y)∈R2 2 2
λt3 − 1 t2 max λ − − −1 = inf 2λ 2λ t2 ,t3 ∈[−1,1] λ>0 2
t +4 t 2 + 1 t3 λ+ 2 − − min 3 = inf λ>0 4 4λ 2 t2 ,t3 ∈[−1,1] 2
t + 4 t22 + 1 t3 −2 3 · + = inf 4 4 2 t2 ,t3 ∈[−1,1] 2
t + 4 12 + 1 t3 −2 3 · + = inf 4 4 2 t3 ∈[−1,1] √ 1 10 =− − . 2 2 2 In the seventh equality, min is attained when λ = tt22 +1 . +4 =
inf
max
inf
3
Acknowledgments The authors are grateful to the anonymous referee for many comments and suggestions which improved the quality of the paper. This work has been partially supported by JSPS KAKENHI Grant Number 25400205. References [1] Y. Fujiwara, D. Kuroiwa, Lagrange duality in canonical DC programming, J. Math. Anal. Appl. 408 (2013) 476–483. [2] M.A. Goberna, V. Jeyakumar, M.A. López, Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities, Nonlinear Anal. 68 (2008) 1184–1194.
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[3] R. Horst, N.V. Thoai, DC programming: overview, J. Optim. Theory Appl. 103 (1999) 1–43. [4] V. Jeyakumar, N. Dinh, G.M. Lee, A new closed cone constraint qualification for convex optimization, Research Report AMR 04/8, Department of Applied Mathematics, University of New South Wales, 2004. [5] J.-E. Martínez-Legaz, M. Volle, Duality in DC programming: the case of several DC constraints, J. Math. Anal. Appl. 237 (1999) 657–671.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Lagrange-type duality in DC programming Ryohei Harada a,∗ , Daishi Kuroiwa b a b
Graduate School of Science and Engineering, Shimane University, Japan Major in Interdisciplinary Science and Engineering, Shimane University, Japan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 27 December 2013 Available online 13 April 2014 Submitted by H. Frankowska
We show a Lagrange-type duality theorem for a DC programming problem, which is a generalization of previous results by J.-E. Martínez-Legaz, M. Volle [5] and Y. Fujiwara, D. Kuroiwa [1] when all constraint functions are real-valued. To the purpose, we decompose the DC programming problem into certain infinite convex programming problems. © 2014 Elsevier Inc. All rights reserved.
Keywords: DC programming problem Lagrange duality theorem
1. Introduction In the present work, our main purpose is to show a generalized Lagrange-type duality theorem, which is an extension of J.-E. Martínez-Legaz, M. Volle [5] and Y. Fujiwara, D. Kuroiwa [1], for the following DC programming problem (P): (P)
minimize f0 (x) − g0 (x) subject to fi (x) − gi (x) ≤ 0, i = 1, · · · , m,
where fi , gi : Rn → R ∪ {+∞} are lsc proper convex functions for each i = 0, 1, · · · , m, and we adopt the conventions (+∞) − (+∞) = +∞, 0 · (+∞) = 0. To this aim, we decompose the above DC programming problem (P) into certain infinite convex programming problems, and we propose a constraint qualification for the Lagrange-type duality. The outline of the paper is as follows. In Section 2, we introduce definitions and preliminary results that will be used later in this paper. In Section 3, we introduce previous Lagrange-type duality results for DC programming and canonical DC programming. In Section 4, we show a Lagrange-type duality theorem for DC programming problem (P), which is a generalization of the previous ones when all constraint functions are real-valued, we apply the main result to a DC programming problem with reverse convex constraints, and we give an example of the main result. * Corresponding author. http://dx.doi.org/10.1016/j.jmaa.2014.04.017 0022-247X/© 2014 Elsevier Inc. All rights reserved.
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
416
2. Notations and preliminaries In this section, we describe our notations and present the preliminary results. The inner product of two vectors x and y in the n-dimensional real Euclidean space Rn will be denoted by x, y. For a set A ⊆ Rn we shall denote the closure, convex hull, conical hull of A by cl A, co A, and cone A, respectively. For the set A, the indicator function δA is defined by 0 x ∈ A, δA (x) = +∞ otherwise. For an extended real-valued function f : Rn → R ∪ {+∞}, the domain, the epigraph, and the conjugate function of f are defined by dom f = x ∈ Rn f (x) < +∞ , epi f = (x, r) ∈ Rn × R x ∈ dom f, f (x) ≤ r , f ∗ (y) = sup x, y − f (x) .
and
x∈Rn
For each x ∈ dom f , the subdifferential of the function f at x is the set ∂f (x) = x∗ ∈ Rn x∗ , y − x ≤ f (y) − f (x), ∀y ∈ Rn . For all x, y ∈ Rn , we have f (x) + f ∗ (y) ≥ y, x (the Young–Fenchel inequality) and it can be shown that if x ∈ dom f , then f (x) + f ∗ (y) = y, x
⇔
y ∈ ∂f (x).
For an arbitrary function g : Rn → R ∪ {+∞, −∞}, the level sets are defined by {g • y} = x ∈ Rn g(x) • y
(y ∈ R)
where • is a binary relation on R ∪ {+∞, −∞}. The following results are used in our main results. Theorem 1. (See M.A. Goberna, V. Jeyakumar, M.A. López [2].) Let I be an arbitrary index set. Let gi : Rn → R ∪ {+∞} be a lsc convex function for each i ∈ I, let C be a closed convex set. Assume that each gi is continuous at least at one point of A = {x ∈ C | gi (x) ≤ 0, ∀i ∈ I}. Then the following statements are equivalent: ∗ is closed. (i) cone co i∈I epi gi∗ + epi δC ∗ is (ii) For every lsc proper convex function f : Rn → R ∪ {+∞} with A ∩ dom f = ∅ and epi f ∗ + epi δA closed,
inf f (x) = max inf f (x) + λi gi (x) . x∈A
(I) λ∈R+ x∈C
i∈I
(iii) For every v ∈ Rn , inf v, x = max inf
x∈A
(I)
(I) λ∈R+ x∈C
v, x +
λi gi (x) ,
i∈I
where λ ∈ R+ if and only if λi ≥ 0 for each i ∈ I and the set {i ∈ I | λi = 0} is finite.
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417
Lemma 1. (See V. Jeyakumar, N. Dinh and G.M. Lee [4, Proposition 2.1].) Let gi : Rn → R be a convex function for each i = 1, · · · , m, let C be a closed convex set. If {gi | i = 1, · · · , m} satisfies the Slater constraint qualification on C (i.e. there exists x0 ∈ C such that gi (x0 ) < 0 for each i = 1, · · · , m), then cone co
m
∗ epi gi∗ + epi δC is closed.
i=1
3. Previous results of Lagrange-type duality in DC and canonical DC programming In this paper, we study the following DC programming problem (P): minimize f0 (x) − g0 (x)
(P)
subject to fi (x) − gi (x) ≤ 0, ∀i = 1, 2, · · · , m,
where fi , gi : Rn → R∪{+∞} are lsc proper convex functions for each i = 0, 1, · · · , m, and assume that the set of all feasible solutions S = {x ∈ Rn | fi (x)−gi (x) ≤ 0, ∀i = 1, 2, · · · , m} is nonempty. In this section, we give previous Lagrange-type duality results for (P). Theorem 2 and Theorem 3 give duality for the DC programming problem (P). We adopt the conventions 0·(+∞) = +∞ and 0·(−∞) = 0 in Theorem 2 and Theorem 3. Theorem 2. (See J.-E. Martínez-Legaz, M. Volle [5].) Let fi : Rn → R ∪ {+∞} be a convex function for each i = 0, 1, · · · , m, let g0 : Rn → R ∪ {+∞, −∞} satisfy g0 = g0∗∗ , and let gi : Rn → R ∪ {+∞, −∞} m be subdifferentiable on S for each i = 1, · · · , m. If for each (x∗1 , · · · , x∗m ) ∈ i=1 {gi∗ − fi∗ ≤ 0}, there exists x) − ¯ x, x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m, then x ¯ ∈ dom f0 such that fi (¯
inf
fi (x)−gi (x)≤0
=
f0 (x) − g0 (x)
inf
inf
g0∗
max
m ∗ ∗ x∗ ∈dom g0∗ gi∗ (x∗ i )−fi (xi )≤0 λ∈R+
m ∗ λi gi∗ x∗i − x +
f0 +
i=1
m
∗ ∗
λi fi
x +
i=1
m
λi x∗i
.
i=1
Theorem 3. (See J.-E. Martínez-Legaz, M. Volle [5].) Let fi : Rn → R ∪ {+∞} be a convex function for each i = 0, 1, · · · , m, let g0 : Rn → R ∪ {+∞, −∞} satisfy g0 = g0∗∗ , and let gi : Rn → R ∪ {+∞, −∞} be subdifferentiable on S for each i = 1, · · · , m. If for each (x∗1 , · · · , x∗m ) ∈ Ω = {(x∗1 , · · · , x∗m ) ∈ Rnm | ∗ (x∗m ) = ∅}, there exists x0 ∈ dom f0 such that fi (x0 ) − x0 , x∗i + gi∗ (x∗i ) < 0 for each ∂g1∗ (x∗1 ) ∩ · · · ∩ ∂gm i = 1, · · · , m, then
inf
fi (x)−gi (x)≤0
=
f0 (x) − g0 (x)
inf
g0∗
max
m ∗ ∗ ∗ (x∗ 0 ,x1 ,···,xm )∈dom g0 ×Ω λ∈R+
m ∗ x + λi gi∗ x∗i −
f0 +
i=1
m
∗ ∗
λi fi
x +
i=1
m
λi x∗i
.
i=1
Remark 1. The right-hand side of Theorem 2 and Theorem 3 can transform to a formulation of Lagrange-type duality. Indeed, ∗ m m m ∗ ∗ ∗ ∗ ∗ ∗ g0 x + λi gi xi − f0 + λi fi x + λi x i i=1
=
g0∗
x
∗
+
i=1 m
λi gi∗
∗ xi − sup ∗
= infn f0 (x) − x, x x∈R
∗
x, x +
x∈Rn
i=1
i=1
+
g0∗
x
∗
m
λi x∗i
−
f0 +
i=1
+
m i=1
λi fi (x) −
m
λi fi (x)
i=1
x, x∗i
+
gi∗
x∗i
.
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
418
It is well-known that the DC programming problem (P) can be transformed into the following canonical DC programming problem (Q) where fi and gi are real-valued functions: (Q)
minimize a, x subject to f (x) ≤ 0, g(x) ≥ 0,
where f, g : Rn → R are convex functions and a ∈ Rn . Theorem 4 and Theorem 5 give duality in the canonical DC programming problem (Q). Theorem 4. (See Y. Fujiwara, D. Kuroiwa [1].) Let f, g : Rn → R be convex functions, and let a ∈ Rn , let S = {x ∈ Rn | f (x) ≤ 0, g(x) ≥ 0} be nonempty and let x∈S ∂g(x) ⊆ A. If for each z ∈ A ∩ dom g ∗ , {f ≤ 0} ∩ {−z, · + g ∗ (z) ≤ 0} is nonempty and cone co(epi f ∗ ∪ {−z} × [−g ∗ (z), +∞) ∪ {0} × [0, +∞)) is closed, then inf
a, x = inf sup infn a, x + λf (x) + μ −y, x + g ∗ (y)
f (x)≤0, g(x)≥0
y∈A λ,μ≥0 x∈R
holds and the supremum on λ, μ ≥ 0 being attained for all y ∈ A. Theorem 5. (See Y. Fujiwara, D. Kuroiwa [1].) Let f, g : Rn → R be convex functions, and let a ∈ Rn , S = {x ∈ Rn | f (x) ≤ 0, g(x) ≥ 0} is nonempty, Y = {y ∈ Rn | {f ≤ 0} ∩ {y, · > g ∗ (y)} = ∅} and ∗ x∈S ∂g(x) ⊆ A ⊆ Y . If cone epi f + {0} × [0, +∞) is closed, then inf
a, x = inf sup infn a, x + λf (x) + μ −y, x + g ∗ (y)
f (x)≤0, g(x)≥0
y∈A λ,μ≥0 x∈R
holds and the supremum on λ, μ ≥ 0 being attained for all y ∈ A. It is well-known that canonical DC programming problems are special cases of DC programming problems, see [3], so Theorems 2 and 3 have broader application areas than Theorems 4 and 5. However, the assumptions of Theorems 2 and 3 are stronger than Theorems 4 and 5 whenever the DC programming problem is canonical and constraint function f is real-valued. Our purpose of this paper is to give a Lagrangetype duality result for a general DC programming problem, which is a generalization of Theorems 2, 3, 4, and 5. 4. Main results We consider the following subproblems (P(y0 , (yi ))) of (P): P y0 , (yi )
minimize f0 (x) − x, y0 + g0∗ (y0 ) subject to fi (x) − x, yi + gi∗ (yi ) ≤ 0, ∀i = 1, · · · , m,
where (y0 , (yi )) = (y0 , y1 , · · · , ym ) ∈ Rn(m+1) . It is clear that all subproblems (P(y0 , (yi ))) are convex programming. Let Val(P) and Val(P(y0 , (yi ))) be the infimum values of (P) and (P(y0 , (yi ))), respectively. Lemma 2. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, (yi ) ∈ Rnm and S(yi ) = {x ∈ Rn | fi (x) − x, yi + gi∗ (yi ) ≤ 0, ∀i = 1, · · · , m}. Then S(yi ) ⊆ S,
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
furthermore if gi is subdifferentiable on S for each i = 1, · · · , m and
m
x∈S (
i=1
419
∂gi (x)) ⊆ D ⊆ Rnm , then
S(yi ) = S.
(yi )∈D
Proof. For each i = 1, · · · , m and x ∈ S(yi ), 0 ≥ fi (x) − x, yi + gi∗ (yi ) ≥ fi (x) − gi (x) by using the Young–Fenchel duality, so S(yi ) ⊆ S. Next, assume that all gi are subdifferentiable on S. Let z ∈ S. For each i = 1, · · · , m, since ∂gi (z) is a nonempty set, there exists yi ∈ ∂gi (z) and 0 ≥ fi (z) − gi (z) = fi (z) − z, yi + gi∗ (yi ), m so z ∈ S(yi ). Also, since (yi ) ∈ x∈S ( i=1 ∂gi (x)), we have z ∈ (yi )∈D S(yi ). Therefore S ⊆ (yi )∈D S(yi ). It is clear that the opposite inclusion holds. 2 Lemma 3 needs to the proof of Theorem 6. Lemma 3. Let h : Rn → R ∪ {+∞} be a function, let A be a nonempty set, and let B(x) be nonempty subsets of Rn for all x ∈ A. Then inf
inf h(y) =
x∈A y∈B(x)
h(y). inf y∈ x∈A B(x)
Proof. For each x ∈ A, since B(x) ⊆ x∈A B(x), inf y∈B(x) h(y) ≥ inf y∈x∈A B(x) h(y). Therefore we have inf x∈A inf y∈B(x) h(y) ≥ inf y∈x∈A B(x) h(y). Assume that inf x∈A inf y∈B(x) h(y) > inf y∈x∈A B(x) h(y). Then there exists β such that inf x∈A inf y∈B(x) h(y) > β > inf y∈x∈A B(x) h(y). From β > inf y∈x∈A B(x) h(y), there exists y0 ∈ x∈A B(x) such that h(y0 ) < β, and there exist x0 ∈ A such that y0 ∈ B(x0 ). Therefore inf
inf h(y) ≤
x∈A y∈B(x)
inf
y∈B(x0 )
h(y) ≤ h(y0 ) < β.
This contradicts to inf x∈A inf y∈B(x) h(y) > β. So, inf x∈A inf y∈B(x) h(y) = inf y∈x∈A B(x) h(y).
2
Theorem 6. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 0, 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, let gi be subdifferentiable on S for each i = 1, · · · , m, m n nm . Then x∈S ∂g0 (x) ⊆ D0 ⊆ R , and x∈S ( i=1 ∂gi (x)) ⊆ D ⊆ R Val(P) =
inf
(y0 ,(yi ))∈D0 ×D
Val P y0 , (yi ) .
Proof. Let x ∈ Rn . For each (yi ) ∈ D, we have gi (x)+gi∗ (yi ) ≥ x, yi for each i = 1, · · · , m. So fi (x)−gi (x) ≤ fi (x) − x, yi + gi∗ (yi ) for each i = 1, · · · , m, and by using Lemma 2, inf f0 (x) − g0 (x) ≤ inf f0 (x) − x, yi + g0∗ (yi ) x∈S x∈S ≤ inf f0 (x) − x, yi + g0∗ (yi ) . x∈S(yi )
This shows Val(P) ≤ inf (y0 ,(yi ))∈D0 ×D Val(P(y0 , (yi ))). Conversely, for each x ∈ S, pick yi ∈ ∂gi (x) for each i = 1, · · · , m, then fi (x) − x, yi + gi∗ (yi ) = fi (x) − gi (x). Therefore
R. Harada, D. Kuroiwa / J. Math. Anal. Appl. 418 (2014) 415–424
420
inf
(y0 ,(yi ))∈D0 ×D
Val P y0 , (yi ) = inf
inf
inf
y0 ∈D0 (yi )∈D z∈S(yi ) inf y0 ∈D0 z∈ (y )∈D S(yi )
= inf
f0 (z) − z, y0 + g0∗ (y0 )
f0 (z) − z, y0 + g0∗ (y0 )
i
= inf inf f0 (z) − z, y0 + g0∗ (y0 ) y0 ∈D0 z∈S
= inf inf
z∈S y0 ∈D0
f0 (z) − z, y0 + g0∗ (y0 )
≤ inf
≤ inf
inf
= inf
inf
f0 (z) − z, y0 + g0∗ (y0 )
inf z∈S y0 ∈ x∈S ∂g0 (x)
z∈S y0 ∈∂g0 (z)
f0 (z) − z, y0 + g0∗ (y0 )
z∈S y0 ∈∂g0 (z)
f0 (z) − g0 (z)
= inf f0 (z) − g0 (z) z∈S
= Val(P). The second and third equalities hold from Lemma 2 and Lemma 3, respectively. When S = ∅, this reverse inequality is clear since Val(P) = +∞. This completes the proof. 2 The following is the main result of this paper: Theorem 7. Let fi , gi : Rn → R ∪ {+∞} be lsc proper convex functions for each i = 0, 1, · · · , m, S = {x ∈ Rn | fi (x) − gi (x) ≤ 0, ∀i = 1, · · · , m}, let gi be subdifferentiable on S for each i = 1, · · · , m, m n nm . If each fi is continuous at least at one point x∈S ∂g0 (x) ⊆ D0 ⊆ R and x∈S ( i=1 ∂gi (x)) ⊆ D ⊆ R n ∗ of S(yi ) = {x ∈ R | fi (x) − x, yi + gi (yi ) ≤ 0, ∀i = 1, · · · , m} and cone co
m
epi fi∗ − yi , gi∗ (yi ) + {0} × [0, +∞) is closed,
i=1
for each (yi ) ∈ D ∩
m i=1
dom gi∗ , then
Val(P) =
inf
max inf
(y0 ,(yi ))∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
g0∗ (y0 )
+
m
∗ λi fi (x) − x, yi + gi (yi ) .
i=1
Proof. For each (yi ) ∈ D, ∗ epi fi − ·, yi + gi∗ (yi ) = epi fi∗ (· + yi ) − gi∗ (yi ) = epi fi∗ − yi , gi∗ (yi ) for each i = 1, · · · , m. Also it is easy to check that epi δR∗ n = {0} × [0, +∞). From the assumption of this theorem and Theorem 1, we have inf
x∈S(yi )
f0 (x) − x, y0 +
g0∗ (y0 )
= max infn f0 (x) − x, y0 + λi ≥0 x∈R
Therefore we conclude the final equality by using Theorem 6.
g0∗ (y0 )
+
m
λi fi (x) − x, yi + gi (yi ) .
i=1
2
Remark 2. We will show that Theorem 7 is a generalization of Theorem 2 and Theorem 3 when all fi (i = 1, · · · , m) are real-valued functions.
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[Theorem 7 ⇒ Theorem 2] Assume the assumption of Theorem 2, that is, for each (x∗1 , · · · , x∗m ) ∈ m ∗ ∗ ¯ ∈ Rn such that fi (¯ x) − ¯ x, x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m. i=1 {gi − fi ≤ 0}, there exists x ∗ ∗ ∗ Therefore {fi − ·, xi + gi (xi ) | i = 1, · · · , m} holds the Slater constraint qualification. From Lemma 1, m m cone co i=1 epi(fi − ·, x∗i + gi∗ (x∗i ))∗ + {0} × [0, +∞) = cone co i=1 (epi fi∗ − (x∗i , gi∗ (x∗i ))) + {0} × [0, +∞) m is closed. Let D0 = dom g0∗ and D = i=1 {gi∗ − fi∗ ≤ 0}. It is clear that x∈S ∂g0 (x) ⊆ D0 . Now we m m show x∈S ( i=1 ∂gi (x)) ⊆ D. For each (y1 , · · · , ym ) ∈ x∈S ( i=1 ∂gi (x)), there exists x0 ∈ S such that m (y1 , · · · , ym ) ∈ i=1 ∂gi (x0 ). Then, for each i = 1, · · · , m, gi (x0 ) + gi∗ (yi ) = x0 , yi , that is, gi (x0 ) + gi∗ (yi ) = fi (x0 ) + x0 , yi − fi (x0 ) ≤ fi (x0 ) + fi∗ (yi ), so we have gi∗ (yi ) − fi∗ (yi ) ≤ fi (x0 ) − gi (x0 ) ≤ 0. This shows m (y1 , · · · , ym ) ∈ D, that is, x∈S ( i=1 ∂gi (x)) ⊆ D. The assumption of Theorem 7 is satisfied. [Theorem 7 ⇒ Theorem 3] Assume the assumption of Theorem 3, that is, for each (x∗1 , · · · , x∗m ) ∈ Ω = m {(x∗1 , · · · , x∗m ) ∈ Rnm | i=1 ∂gi∗ (x∗i ) = ∅}, there exists x0 ∈ Rn such that fi (x0 ) − x0 , x∗i + gi∗ (x∗i ) < 0 for each i = 1, · · · , m. Therefore {fi − ·, x∗i + gi∗ (x∗i ) | i = 1, · · · , m} holds the Slater constraint qualification. m m From Lemma 1, cone co i=1 epi(fi −·, x∗i +gi∗ (x∗i ))∗ +{0}×[0, +∞) = cone co i=1 (epi fi∗ −(x∗i , gi∗ (x∗i )))+ {0} × [0, +∞) is closed. Let D0 = dom g0∗ and D = Ω. It is clear that x∈S ∂g0 (x) ⊆ D0 . Now we m m show x∈S ( i=1 ∂gi (x)) ⊂ D. For each (y1 , · · · , ym ) ∈ x∈S ( i=1 ∂gi (x)), there exists x0 ∈ S such that m m (y1 , · · · , ym ) ∈ i=1 ∂gi (x0 ). Since gi = gi∗∗ , x0 ∈ ∂gi (yi ) for each i = 1, · · · , m. Therefore x0 ∈ i=1 ∂gi (yi ), m and we have i=1 ∂gi (yi ) = ∅, that is, (y1 , · · · , ym ) ∈ Ω = D. Consequently the assumption of Theorem 7 is satisfied. Also, Theorem 7 is a strongly generalization of Theorem 2 and Theorem 3 when all fi (i = 1, · · · , m) are real-valued functions. For example, let n = 1, m = 1, ⎧ ⎨ −x − 1 x < −1, x ∈ [−1, 1], and g1 (x) = 0. f1 (x) = 0 ⎩ x−1 x > 1, Then f1∗ (x∗ ) = |x∗ | + δ[−1,1] (x∗ ) and g1∗ (x∗ ) = δ{0} (x∗ ). cone epi f1∗ + {0} × [0, +∞) is closed. So, the assumption of Theorem 7 holds. However, both the assumptions of Theorems 2 and 3 do not hold because f1 (x) − x, x∗ + g1∗ (x∗ ) ≥ 0 for each x, x∗ ∈ Rn . But, Theorem 7 is not a generalization of Theorem 2 and Theorem 3 when some fi (i = 1, · · · , m) is not a real-valued function. For example, let n = 1, m = 1, f0 (x) = x,
g0 (x) = 0,
f1 (x) =
−1 x ∈ [−1, 1], +∞ x ∈ / [−1, 1],
and g1 (x) = 0.
Then, f1∗ (x∗ ) = |x∗ | + 1 and g1∗ (x∗ ) = δ{0} (x∗ ). So, f0 , g0 , f1 and g1 satisfy assumptions of Theorem 2 and Theorem 3. However, cone epi f1∗ + {0} × [0, +∞) is not closed. So, f1 and g1 do not hold the assumption of Theorem 7. Remark 3. We will show that Theorem 7 is a generalization of Theorem 4 and Theorem 5. [Theorem 7 ⇒ Theorem 4] Assume the assumption to Theorem 4. Let f0 = a, ·, g0 = 0, f1 = f , g1 = 0, f2 = 0, and g2 = g. Then ∂g0 (x) = {0} for each x ∈ S and g0∗ = δ{0} . Let D0 = {0}, D = {0} × A, where A is the set satisfying x∈S ∂g(x) ⊆ A. Then
∂g0 (x) ⊆ D0
and
x∈S
∂g1 (x) × ∂g2 (x) ⊆ D. x∈S
Also S(y1 , y2 ) is nonempty from Lemma 2 and ∗ ∗ cone co epi f1 − ·, y1 + g1∗ (y1 ) ∪ epi f2 − ·, y2 + g2∗ (y2 ) + {0} × [0, +∞) = cone co epi f ∗ ∪ {−y2 } × −g ∗ (y2 ), +∞ ∪ {0} × [0, +∞)
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is closed for each (y1 , y2 ) ∈ D. By using Theorem 7, we have inf
a, x =
f (x)≤0, g(x)≥0
=
inf
fi (x)−gi (x)≤0
inf
f0 (x) − g0 (x)
max inf
(y0 ,y1 ,y2 )∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
g0∗ (y0 )
+
2
∗
λi fi (x) − x, yi + g (yi )
i=1
= inf max infn a, x + λ1 f (x) + λ2 −x, y + g ∗ (y) . y∈A λi ≥0 x∈R
This completes the proof of Theorem 4. [Theorem 7 ⇒ Theorem 5] Assume the assumption to Theorem 5. For any z ∈ A, where A satisfies ∗ ∗ x∈S ∂g(x) ⊆ A ⊆ Y , there exists x0 ∈ {f ≤ 0} such that −z, x0 + g (z) < 0, that is, {−z, · + g (z)} holds the Slater constraint qualification on {f ≤ 0}. From Lemma 1, ∗ ∗ cone epi −·, z + g ∗ (z) + epi δ{f ≤0} is closed. So, for every v ∈ Rn ,
f (x)≤0,
inf
−x,z+g ∗ (z)≤0
v, x = max inf
λ≥0 f (x)≤0
v, x + λ −x, z + g ∗ (z)
by using Theorem 1. Since cone epi f ∗ + {0} × [0, +∞) is closed, inf
f (x)≤0
v, x + λ −x, z + g ∗ (z) = max infn v, x + λ −x, z + g ∗ (z) + μf (x) , μ≥0 x∈R
by using Theorem 1 again. Therefore for every v ∈ Rn ,
f (x)≤0,
inf
−x,z+g ∗ (z)≤0
v, x = max max infn v, x + λ −x, z + g ∗ (z) + μf (x) , λ≥0 μ≥0 x∈R
and this shows ∗ cone co epi f ∗ ∪ epi −·, z + g ∗ (z) + {0} × [0, +∞) is closed from Theorem 1 once again. This is the assumption of Theorem 4, and the proof is completed from the previous discussion. So, Theorem 7 is a strongly generalization of Theorem 4 and Theorem 5 because DC programming problems are not canonical DC programing problems in general. Remark 4. When fi = 0 for each i = 1, · · · , m, the minimization problem (P) of Theorem 7 becomes a DC programming problem with a reverse convex inequality system. In this case, epi fi∗ = {0} × [0, +∞) for each i = 1, · · · , m, and then the characteristic cone is always closed. Therefore,
Val(P) =
inf
max inf
(y0 ,(yi ))∈D0 ×D λi ≥0 x∈Rn
f0 (x) − x, y0 +
=
inf
max
(y0 ,(yi ))∈D0 ×D λi ≥0
g0∗ (y0 )
+
m i=1
g0∗ (y0 )
−
f0∗
λi −x, yi +
i=1
λi gi∗ (yi )
+
m
y0 +
m i=1
λi yi
.
gi∗ (yi )
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Example 1. In this example, we calculate the optimal value of the following DC programming problem by using Theorem 7: minimize x − |y|
(P)
subject to x2 + y 2 − 1 − |x| ≤ 0.
Let f0 (x, y) = x, g0 (x, y) = |y|, f1 (x, y) = x2 +y 2 −1, g1 (x, y) = |x|, D0 = {0}×[−1, 1], and D = [−1, 1]×{0}. Then we have g0∗ = δ{0}×[−1,1] and g1∗ = δ[−1,1]×{0} . For each y1 ∈ D, we can check that f1 is continuous at least at one point of S(y1 ) because S(y1 ) = {x ∈ Rn | f1 (x) − x, y1 + g1∗ (y1 ) ≤ 0} is nonempty, and cone co epi f1∗ − y1 , g1∗ (y1 ) + {0} × [0, +∞) is closed. By using Theorem 7, Val(P) =
inf
max
inf
(t1 ,t2 )∈D0 , (t3 ,t4 )∈D λ≥0 (x,y)∈R2
f0 (x, y) − (x, y), (t1 , t2 ) + g0∗ (t1 , t2 )
+ λ f1 (x, y) − (x, y), (t3 , t4 ) + g1∗ (t3 , t4 ) max inf 2 x − t2 y + λ x2 + y 2 − 1 − t3 x = inf t2 ,t3 ∈[−1,1] λ≥0 (x,y)∈R
=
inf
max
t2 ,t3 ∈[−1,1] λ>0
inf
(x,y)∈R2
x − t 2 y + λ x2 + y 2 − 1 − t3 x
2 2 2 2
λt3 − 1 t2 λt3 − 1 t2 λ x − + y − − − − 1 2λ 2λ 2λ 2λ t2 ,t3 ∈[−1,1] λ>0 (x,y)∈R2 2 2
λt3 − 1 t2 max λ − − −1 = inf 2λ 2λ t2 ,t3 ∈[−1,1] λ>0 2
t +4 t 2 + 1 t3 λ+ 2 − − min 3 = inf λ>0 4 4λ 2 t2 ,t3 ∈[−1,1] 2
t + 4 t22 + 1 t3 −2 3 · + = inf 4 4 2 t2 ,t3 ∈[−1,1] 2
t + 4 12 + 1 t3 −2 3 · + = inf 4 4 2 t3 ∈[−1,1] √ 1 10 =− − . 2 2 2 In the seventh equality, min is attained when λ = tt22 +1 . +4 =
inf
max
inf
3
Acknowledgments The authors are grateful to the anonymous referee for many comments and suggestions which improved the quality of the paper. This work has been partially supported by JSPS KAKENHI Grant Number 25400205. References [1] Y. Fujiwara, D. Kuroiwa, Lagrange duality in canonical DC programming, J. Math. Anal. Appl. 408 (2013) 476–483. [2] M.A. Goberna, V. Jeyakumar, M.A. López, Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities, Nonlinear Anal. 68 (2008) 1184–1194.
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