DOI 10.1007/s10898-009-9459-2
A recurrence method for a special class of continuous
time linear programming problems
Ching-Feng Wen · Yung-Yih Lur · Yan-Kuen Wu
Received: 15 May 2008 / Accepted: 15 July 2009 / Published online: 29 July 2009 © Springer Science+Business Media, LLC. 2009
Abstract This article studies a numerical solution method for a special class of continuous time linear programming problems denoted by (S P). We will present an efficient method for finding numerical solutions of(SP). The presented method is a discrete approximation algo-rithm, however, the main work of computing a numerical solution in our method is only to solve finite linear programming problems by using recurrence relations. By our constructive manner, we provide a computational procedure which would yield an error bound intro-duced by the numerical approximation. We also demonstrate that the searched approximate solutions weakly converge to an optimal solution. Some numerical examples are given to illustrate the provided procedure.
Keywords Continuous time linear programming problems· Infinite-dimensional linear programming problems
1 Introduction
Let T> 0 and q ∈ N, and let L∞+[0, T ] be the set of nonnegative real-valued, Lebesgue measurable, essentially bounded functions on the closed interval[0, T ]. We consider an
Research of Ching-Feng Wen is partially supported by a grant from the Kaohsiung Medical University Research Foundation (Q097016) and NSC 97-2115-M-037-001.
Research of Yung-Yih Lur is partially supported by NSC 97-2115-M-238-001. Research of Yan-Kuen Wu is partially supported by NSC 97-2410-H-238-004. C.-F. Wen
General Education Center, Kaohsiung Medical University, Kaohsiung 807, Taiwan, ROC e-mail: [email protected]
Y.-Y. Lur· Y.-K. Wu (
B
)Department of Industrial Management, Vanung University, Taoyuan 320, Taiwan, ROC e-mail: [email protected]
Y.-Y. Lur
infinite-dimensional linear programming problem denoted by(SP) and defined as follows: (S P): maximize q j=1 T 0 fj(t)xj(t)dt subject to q j=1 ⎡ ⎣βjxj(t) − t 0 γjxj(s)ds ⎤ ⎦ ≤ g(t), ∀ t ∈ [0, T ] (1) xj(t) ∈ L∞+[0, T ], for 1 ≤ j ≤ q,
whereβj andγj are given constants, fj(t) : [0, T ] → R and g(t) : [0, T ] → R are given
functions. xj(·) : [0, T ] → R (1 ≤ j ≤ q) is a decision variable. It is well known (refer to [3]) that the dual problem (D S P) of (S P) is defined as follows:
(D S P): minimize T 0 g(t)w(t)dt subject toβjw(t) − T t γjw(s)ds ≥ fj(t), (2) ∀ 1 ≤ j ≤ q, t ∈ [0, T ], w(t) ∈ L∞+[0, T ],
wherew(·) : [0, T ] → R is the decision variable.
(SP) is a special case of the so called continuous time linear programming problems (C L P), first introduced by Bellman [6] to model some production planning problems. The model of(C L P) has a wide range of applications (e.g., [6,8,21]), but is notoriously difficult to solve in general. In the literature, many researches have been proposed to consider(C L P). Studying the duality of(C L P), Grinold [12,13], Levinson [17] and Tyndall [22,23] have established strong duality theorems. Investigating a solution algorithm for(C L P), Anstrei-cher [5], Drews [9], Hartberger [14], Lehman [16], Perold [18] and Segers [20] have attempted to extend the simplex method to(C L P), however, the emerged theory is highly complex, and there remain substantial difficulties that make an implementation of the method unlikely to be successful. Studying a special case of(C L P), Anderson [1] introduced the separated continuous linear programs(SC L P) to model job-shop scheduling problems. Since then many researches concerned with(C L P) have focused on (SC L P) [2,4,10,19,24]. On the other hand, Buie and Abrham [7] proposed a discrete approximation method for finding numerical solutions of(C L P). As a solution technique, however, the provided method in [7] has some drawbacks. For instance, the searched numerical solutions may not be feasible; one cannot know how accurate the searched solution is; the termination criterion is not provided. Therefore, it would be useful to have a computational procedure which would yield bounds on the error introduced by the numerical approximation.
In this paper, we propose an efficient approximation method to approach the optimal value of(SP) by using recurrence relations. This method can be employed not only to easily solve
(SP), but also to provide an error bound of the optimal value as well. Moreover, we also
prove that our searched approximate solutions can converge weakly to an optimal solution of(SP).
For improving the readability, we define the notations F(P) and V (P) to be the feasible set and the optimal value of a linear programming problem(P), respectively. If S1, S2⊆ R,
then we denote by C(S1, S2) the space of all continuous functions from S1 to S2. And the superscript “ ” denotes the transpose operation.
This paper is organized as follows. In Sect.2, we develop a recurrence method for solv-ing discretization problems(Pn) and (Dn) of (SP) and (DSP), respectively. In Sect.3, we provide methods to construct approximate solutions for(SP) and (DSP). Moreover, we also establish an estimation for the error bounds of approximate values evaluated by the proposed method. In Sect.4, we demonstrate that the searched approximate solutions weakly converge to an optimal solution. Finally, in Sect.5, we provide some numerical examples to show the quality of the proposed error bound.
2 A recurrence method for solving discretization problems(Pn) and (Dn) In the sequel of this paper we will make the following assumptions:
Assumption
(A1) fj(t) ∈ C([0, T ], R) for all 1 ≤ j ≤ q and g ∈ C([0, T ], R+), where R+is the set
of all nonnegative real numbers. (A2) βj > 0 and γj≥ 0 for all 1 ≤ j ≤ q.
To solve(SP) and (DSP), for each n ∈ N, we let P2n = {0, 1
2nT,22nT, . . . ,2 n−1
2n T, T } be a partition on[0, T ] into 2n subintervals with equal length2Tn. For 1≤ l ≤ 2n, let
bl(n)= min g(t) : t ∈ l− 1 2n T, l 2nT (3) and c(n)jl = min fj(t) : t ∈ l− 1 2n T, l 2nT , (4)
for 1≤ j ≤ q. We define step functions g(n)(t) and f(n)j (t) as follows:
g(n)(t) = b(n)l , if t ∈ l−1 2n T,2lnT , b(n)2n, if t = T (5) and fj(n)(t) = c(n)jl, if t ∈ l−1 2n T,2lnT , c(n)j 2n, if t = T , (6)
where 1≤ l ≤ 2n. Consider the following programming problem:
(S Pn) : maximize q j=1 T 0 f(n)j (t)xj(t)dt subject to q j=1 ⎡ ⎣βjxj(t) − t 0 γjxj(s)ds ⎤ ⎦ ≤ g(n)(t), ∀ t ∈ [0, T ], xj(t) ∈ L∞+[0, T ], for 1 ≤ j ≤ q.
And its dual problem is defined as follows: (DS Pn) : minimize T 0 g(n)(t)w(t)dt subject toβjw(t) − T t γjw(s)ds ≥ fj(n)(t), ∀ 1 ≤ j ≤ q, t ∈ [0, T ], and w(t) ∈ L∞+[0, T ]. Remark 1
(1a) Under assumption (A1),(SP) and (S Pn) are feasible for all n ∈ N. Indeed, the zero vector functions is a common feasible solution of(SP) and (S Pn).
(1b) Under assumptions (A1) and (A2),(DSP) and (DS Pn) are feasible for all n ∈ N. To see this, we chooseα > 0 such that αβj ≥ γj andαβj ≥ maxt∈[0,T ]{ fj(t)} for all
1 ≤ j ≤ q. Define w(t) = αeα(T −t), thenw(t) ≥ 0. Besides, for 1 ≤ j ≤ q and t ∈ [0, T ], we have βjw(t) − T t γjw(s)ds = βjαeα(T −t)− γj T t αeα(T −s)ds = βjαeα(T −t)+ γj− γjeα(T −t) = αβj− γj eα(T −t)+ γj ≥ αβj− γj+ γj = αβj ≥ fj(t) ≥ fj(n)(t),
for all n∈ N. Hence (SP) and (DS Pn) are feasible for all n ∈ N.
(1c) It is well known (refer to [3]) that(SP) and (S Pn) have the weak duality property, that is, V(S P) ≤ V (DS P) and V (S Pn) ≤ V (DS Pn). Moreover, the strong duality for
(SP) will be demonstrated by our constructive method, although the strong duality for
general problem(C L P) has been established by Tyndall [22]. It is remarkable that the strong duality theorem has been extended to different versions by Grinold [12,13], Levinson [17] and Tyndall [23].
(1d) Because
g(1)(t) ≤ g(2)(t) ≤ · · · ≤ g(n)(t) ≤ · · · ≤ g(t), and
for all 1≤ j ≤ q and t ∈ [0, T ], we have F(S P1) ⊆ F(S P2) ⊆ · · · ⊆ F(S P), and F(DS P1) ⊇ F(DS P2) ⊇ · · · ⊇ F(DS P), which implies − ∞ < V (S P1) ≤ V (S P2) ≤ · · · ≤ V (S P) < ∞ (7) and −∞ < V (DS P1) ≤ V (DS P2) ≤ · · · ≤ V (DS P) < ∞. Hence lim n→∞V(S Pn) ≤ V (S P) and lim n→∞(DS Pn) ≤ V (DS P). (8)
Now we consider the finite dimensional linear programming problem which is due to
(S Pn). For each n ∈ N, let b(n)l and c(n)jl be defined as in (3) and (4). We define the following linear programming problem and use the convention that “empty sum,”01 has the value zero. (Pn) : maximize q j=1 2n l=1 T 2nc (n) jl xjl subject to q j=1 βjxjl− T 2nγj l−1 α=1 xjα ≤ bl(n), l = 1, 2, . . . , 2n, xjl≥ 0, j = 1, 2, . . . , q, l = 1, 2, . . . , 2n.
The dual problem(Dn) of (Pn) is defined as follows:
(Dn) : minimize 2n l=1 T 2nb (n) l wl subject toβjwl− T 2nγj 2n α=l+1 wα≥ c(n)jl , j = 1, 2, . . . , q, l = 1, 2, . . . , 2n, wl≥ 0, l = 1, 2, . . . , 2n,
where “empty sum,”22nn+1has the value zero. Remark 2
(2a) Under assumptions (A1) and (A2), the feasible set F(Pn) is nonempty for all n ∈ N, since the zero vector is a feasible solution of(Pn).
(2b) Under assumptions (A1) and (A2), for all n∈ N, the feasible set F(Dn) is nonempty by the following Theorem1.
(2c) By (2a),(2b) and the strong duality theorem of finite linear programming, under assumptions (A1) and (A2), both(Pn) and (Dn) have optimal solutions and −∞ < V(Pn) = V (Dn) < ∞.
The following results provide a recurrence method for solving(Pn) and (Dn). Let L := max 1≤ j≤q0max≤t≤T{ fj(t), 0}, σ := min 1≤ j≤q{βj} and ¯γ := max 1≤ j≤q{γj}.
Theorem 1 Suppose that assumptions (A1) and (A2) hold. Let the vector ¯w(n) =
( ¯w(n)1 , . . . , ¯w(n)2n ) be defined by ¯w(n)2n := max 1≤ j≤q max c(n)j 2n βj , 0 , and ¯w(n)l := max 1≤ j≤q max c(n)jl +2Tnγj 2n α=l+1 ¯wα(n) βj , 0 , for l= 2n− 1, 2n− 2, . . . , 2, 1. Then
(i) ¯w(n)is an optimal solution of(Dn). (ii) For 1 ≤ l ≤ 2n 0≤ ¯wl(n)≤ L σ 1+ T¯γ 2nσ 2n−l ≤ L σ 1+ T¯γ 2nσ 2n ≤ L σe T¯γ σ . (9)
Proof It is easy to check that ¯w(n) is feasible for (Dn). Now we claim that if w(n) =
(w(n)1 , . . . , w(n)2n ) ∈ F(Dn) then w(n)l ≥ ¯wl(n)for all 1 ≤ l ≤ 2n. We prove it by induc-tion. Obviously,w(n)2n ≥ ¯w2(n)n for allw(n) ∈ F(Dn). Suppose that wl(n)≥ ¯wl(n)for all l = k+ 1, k + 2, . . . , 2n. We will show thatw(n)
k ≥ ¯w(n)k . Since w(n) ∈ F(Dn), βjw(n)k − T 2nγj 2n α=k+1w(n)α ≥ c(n)j k. This implies wk(n)≥ c(n)j k +2Tnγj2 n α=k+1w(n)α βj ,
for all j . Hence
wk(n)≥ max 1≤ j≤q max c(n)j k +2Tnγj 2n α=k+1 ¯wα(n) βj , 0 = ¯wk(n).
By induction on k, we show that our claim is valid. In view of (A1) and the fact that b(n)l ≥ 0 for all l, we obtain that2l=1n 2Tnb(n)l wl(n)≥
2n
l=12Tnb(n)l ¯wl(n). Sincew(n)∈ F(Dn) is arbitrary, we see that ¯w(n)is an optimal solution of(Dn).
On the other hand, we assert that ¯wl(n)≤ L σ 1+ T¯γ 2nσ 2n−l ,
for all 1≤ l ≤ 2n. It is obvious that ¯w(n)2n = max 1≤ j≤q max c(n)j 2n βj , 0 ≤ L σ and ¯w(n)2n−1≤ L+T2n¯γ Lσ σ = L σ 1+ T¯γ 2nσ . Suppose that ¯w(n)k ≤σL 1+2Tn¯γσ 2n−k
for all k= l + 1, . . . , 2n. Then
¯w(n)l = max 1≤ j≤q max c(n)jl +2Tnγj 2n α=l+1 ¯wα(n) βj , 0 ≤ max 1≤ j≤q c(n)jl βj , 0 + max 1≤ j≤q T 2nγj2 n α=l+1 ¯w(n)α βj ≤ L σ + T¯γ 2nσ 2n α=l+1 ¯w(n) α ≤ L σ + L σ 1+ T¯γ 2nσ 2n−l − 1 = L σ 1+ T¯γ 2nσ 2n−l .
By induction the assertion is valid, and this implies that for all 1≤ l ≤ 2n ¯wl(n)≤ L σ 1+ T¯γ 2nσ 2n−l ≤ L σ 1+ T¯γ 2nσ 2n ≤ L σe T¯γ σ , since 1+2Tnσ¯γ 2n
↑ eTσ¯γ as n→ ∞. We complete this proof. By the complementary slackness theorem, it is well known that x(n)∈ F(Pn) and w(n)∈ F(Dn) become an optimal solution pair if and only if x(n) andw(n)satisfy the following equations: ⎛ ⎝b(n) l − q j=1 βjx(n)jl −Tγj 2n l−1 α=1 x(n)jα ⎞ ⎠ w(n) l = 0, ∀ 1 ≤ l ≤ 2n, (10) and ⎛ ⎝βjwl(n)− Tγj 2n 2n α=l+1 wα(n)− c(n)jl ⎞ ⎠ x(n) jl = 0, ∀ 1 ≤ j ≤ q, 1 ≤ l ≤ 2 n. (11)
Let us recall that an optimal solution ¯w(n)of(Dn) can be found easily by Theorem 1. Now we want to construct a feasible solution ¯x(n)of(Pn) which corresponds to the dual optimal solution ¯w(n)by the complementary slackness theorem.
Let := { l : 1 ≤ l ≤ 2n and ¯w(n) l > 0}, (12) and for 1≤ l ≤ 2n, 1≤ j ≤ q djl := c(n)jl +2Tnγj 2n α=l+1 ¯wα(n) βj . For l∈ , we let (l):= {α : 1 ≤ α ≤ l − 1 and α ∈ }, (13) where(1)= ∅, and j(l):= argmaxj{djl : djl > 0}. (14)
(If more than one index j maximize djl, we let j(l)be the smallest such index.) Construct a q× 2n matrix ¯x(n)= ¯x(n) jl q×2n, where ¯x(n)jl := ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 if l /∈ , 0 if l∈ and j = j(l), bl(n)+2nT α∈(l)γj(α)¯x(n) j(α)α βj(l) if l∈ and j = j(l), (15)
for 1≤ j ≤ q and 1 ≤ l ≤ 2n. We first show that ¯x(n)∈ F(Pn). If l /∈ then
b(n)l − q j=1 βj¯x(n)jl − T 2nγj l−1 α=1 ¯x(n)jα = bl(n)+ l−1 α=1 q j=1 T 2nγj¯x (n) jα ≥ 0. If l∈ then b(n)l − q j=1 βj¯x(n)jl − T 2nγj l−1 α=1 ¯x(n)jα = b(n)l − βj(l)¯x(n)j(l)l+ l−1 α=1 q j=1 T 2nγj¯x (n) jα = b(n)l − βj(l)¯x(n)j(l)l+ α∈(l) T 2nγj(α)¯x (n) j(α)α = 0. (16)
Hence ¯x(n)∈ F(Pn). Now we assert that ¯x(n)and ¯w(n)satisfy equations (10) and (11). We verify this assertion by the following two cases.
Case 1. ¯w(n)l = 0. Then ¯x(n)jl = 0 for all j = 1, 2, . . . , q. Hence ⎛ ⎝b(n) l − q j=1 βj¯x(n)jl − T 2nγj l−1 α=1 ¯x(n)jα ⎞ ⎠ ¯w(n) l = 0, and ⎛ ⎝βj ¯wl(n)− T 2nγj 2n α=l+1 ¯w(n) α − c(n)jl ⎞ ⎠ ¯x(n) jl = 0, for all j = 1, 2, . . . , q. Case 2. ¯w(n)l > 0. Then ¯w(n)l = c(n)j(l)l+2Tnγj(l) 2n α=l+1 ¯w(n)α βj(l) , and hence βj(l)¯wl(n)− T 2nγj(l) 2n α=l+1 ¯wα(n)− c(n)j(l)l= 0.
According to¯x(n)jl = 0 for all j = j(l), we have
⎛ ⎝βj¯w(n)l − T 2nγj 2n α=l+1 ¯w(n)α − c(n)jl ⎞ ⎠ ¯x(n) jl = 0, ∀ 1 ≤ j ≤ q.
Note that l∈ , by (16), we have b(n)l − q j=1 βj¯x(n)jl − T 2nγj l−1 α=1 ¯x(n)jα = 0.
Hence equations (10) and (11) hold. By Case 1 and 2, our assertion is valid, and hence
¯x(n)is an optimal solution of(Pn).
Based on the above discussion, we have an algorithm for solving(Pn) and (Dn). Algorithm 1 Given n, q ∈ N. Set l = 2n and = ∅.
Step 1: Set j = 1. Step 2: Compute djl:= c(n)jl +2nTγj2nα=l+1¯wα(n) βj 2n α=2n+1 ¯w(n)α := 0 . Step 3: If djl≤ 0, set ¯x(n)jl = 0.
Step 4: If j= q, update j ← j + 1 and go to Step 2. Step 5: If djl≤ 0 for all j = 1, 2, . . . , q, set ¯w(n)l = 0.
Otherwise, update ← ∪ {l} and set ¯wl(n)= max{djl : djl > 0} and ¯x(n)jl = 0
for all j= j(l), where j(l)is defined in (14). Step 6: If l= 1, update l ← l − 1 and go to Step 1.
Step 7: If l∈ , then find ¯x(n)j(l)lby solving the following recurrence relation:
¯x(n)j(l)l= b (n) l +2Tn α∈(l)γj(α)¯x(n)j(α)α βj(l) ,
where(l)= {α : 1 ≤ α ≤ l − 1 and α ∈ } and (1)= ∅. Step 8: If l= 2n, then STOP; otherwise update l← l + 1 and go to Step 7.
Moreover, the feasible set F(Pn) is uniformly bounded. To see this, let M:= max 0≤t≤Tg(t). Lemma 1 Let x(n)= x1(n), x2(n), . . . , x2(n)n ∈ F(Pn), where xl(n)= x1l(n), x2l(n), . . . , xql(n) and 1≤ l ≤ 2n, then 0≤ x(n)jl ≤ M σ e q¯γ T σ , (17)
for all 1≤ j ≤ q and 1 ≤ l ≤ 2nand n∈ N.
Proof We claim that x(n)jl ≤ Mσ
1+q2n¯γ Tσ
l−1
for all 1≤ j ≤ q and 1 ≤ l ≤ 2n. Since
q j=1βjx
(n) j 1 ≤ b
(n)
1 andβj> 0, it follows that
x(n)j 1 ≤b (n) 1 βj ≤ M σ
for all j . Suppose that x(n)jl ≤ Mσ
1+q2n¯γ Tσ l−1 for l= 1, 2, . . . , k − 1. Then k−1 l=1 x(n)jl ≤ k−1 l=1 M σ 1+q¯γ T 2nσ l−1 = 2nM1+q2n¯γ Tσ k−1 − 1 q¯γ T (18)
for all j . Sinceqj=1
βjx(n)j k − Tγj 2n k−1 α=1x(n)jα
≤ bk(n)and by (18), we have that βjx(n)j k ≤ b(n)k + q j=1 Tγj 2n k−1 α=1 x(n)jα ≤ M + M 1+q¯γ T 2nσ k−1 − 1 .
This implies that x(n)j k ≤ Mσ
1+q2n¯γ Tσ
k−1
. Hence by induction we show that our claim is valid. Hence x(n)jl ≤ Mσ
1+q2n¯γ Tσ
2n
for all 1 ≤ j ≤ q, 1 ≤ l ≤ 2n and n ∈ N. Since
M σ 1+q2n¯γ Tσ 2n ↑ M σe q¯γ T σ as n→ ∞, we get 0≤ x(n)jl ≤ M σe q¯γ T σ .
3 Approximate solutions of(SP) and (DSP)
In one of early papers, Tyndall [22] conjectured that the solutions to continuous time program-ming problems would be piecewise smooth functions under proper conditions. J´ohannesson and Hanson [15] confirmed this conjecture. However, it seems that no practical methods have been provided to find approximate solutions and values. In this section, we provide methods to construct approximate solutions for(SP) and (DSP) by virtue of the optimal solutions of
(Pn) and (Dn).
Let¯x(n)be defined in (15). Recall ¯x(n)is an optimal solution of(Pn). Define a step func-tion ˆx(n)(·) : [0, T ] → Rqas follows: ˆx(n)(t) = ( ˆx1(n)(t), ˆx2(n)(t), . . . , ˆxq(n)(t)) , where for
1≤ j ≤ q ˆx(n)j (t) = ¯x(n)jl , if t ∈ [l−12n T,2lnT) for some 1 ≤ l ≤ 2n ¯x(n)j 2n, if t= T . (19)
Then we have the following result.
Lemma 2 Letˆx(n)(t) be defined as in (19). Thenˆx(n)(t) ∈ F(S Pn) ⊆ F(S P) for all n ∈ N.
Proof Since¯x(n)is an optimal solution of(Pn),
q j=1 βj¯x(n)jl − Tγj 2n l−1 α=1 ¯x(n)jα ≤ b(n)l , (20)
for 1≤ l ≤ 2n. Consider the following two cases, we have thatˆx(n)(t) ∈ F(S Pn). Case 1. t ∈ l−12n T,2lnT
, for some 1≤ l ≤ 2n. Then we have
q j=1 ⎡ ⎣βjˆx(n)j (t) − t 0 γjˆx(n)j (s)ds ⎤ ⎦ = q j=1 ⎡ ⎢ ⎢ ⎣βjˆx(n)j (t) − l−1 α=1 α 2nT α−1 2n T γjˆx(n)j (s)ds − t l−1 2nT γjˆx(n)j (s)ds ⎤ ⎥ ⎥ ⎦ ≤ q j=1 βj¯x(n)jl − Tγj 2n l−1 α=1 ¯x(n)jα ⎛ ⎜ ⎜ ⎝since t l−1 2nT γjˆx(n)j (s)ds ≥ 0 ⎞ ⎟ ⎟ ⎠ ≤ b(n)l (by(20)) = g(n)(t).
Case 2. t = T . Then we have q j=1 ⎡ ⎣βjˆx(n)j (T ) − T 0 γjˆx(n)j (s)ds ⎤ ⎦ = q j=1 ⎡ ⎣βj¯x(n)j 2n − 2n α=1 Tγj 2n ¯x (n) jα ⎤ ⎦ ≤ q j=1 ⎡ ⎣βj¯x(n)j 2n− Tγj 2n 2n−1 α=1 ¯x(n)jα ⎤ ⎦ (since γj≥ 0) ≤ b(n)2n (by(20)) = g(n)(T ).
We complete this proof.
Moreover, it is obvious that
q j=1 T 0 f(n)j (t) ˆx(n)j (t)dt = q j=1 2n l=1 T 2nc (n) jl ¯x (n) jl = V (Pn). (21)
Therefore, V(S Pn) ≥ V (Pn) and hence V (DS Pn) ≥ V (S Pn) ≥ V (Pn) = V (Dn). Hence, by inequality (7), one can easily see that
V(DS P) ≥ V (S P) ≥ V (S Pn) ≥ V (Pn) = V (Dn), (22) for all n∈ N.
Furthermore, we assert that limn→∞V(Dn) = V (DS P). To see this, we first need the
following notations and lemma. Let
n := max 1≤ j≤qt∈[0,T ]sup { fj(t) − f (n) j (t)}, (23) ¯n := sup t∈[0,T ]{g(t) − g (n)(t)} (24) and ρ := max 1≤ j≤q γj βj, 1 βj . (25)
Let ¯w(n)= ( ¯w1(n), ¯w2(n), . . . , ¯w(n)2n ) be the optimal solution of(Dn) defined as in Theorem 1. Define a function ˆw(n)(t) : [0, T ] → R as follows:
ˆw(n)(t) = ¯w(n)l + δ2nρeρ(T −t), if l−1 2n T ≤ t <2lnT for some 1≤ l ≤ 2n, ¯w(n)2n + δ2nρ, if t = T, (26) where δ2n := max 1≤l≤2n T 2n ¯w (n) l . (27) Moreover, define w(n)(t) = ˆw(n)(t) + nρeρ(T −t) (28)
for all t∈ [0, T ], where nis defined as in (23). It can be shown by the following lemma that
ˆw(n)(t) and w(n)(t) are feasible solutions of (DS Pn) and (DSP), respectively.
Lemma 3 Suppose that assumptions (A1) and (A2) hold. Let ˆw(n)(t) and w(n)(t) be defined as above. Then (i) ˆw(n)(t) ∈ F(DS Pn) and 0≤ T 0 g(n)(t) ˆw(n)(t)dt − V (Dn) ≤ δ2n T 0 ρeρ(T −t)g(t)dt. (29) (ii) w(n)(t) ∈ F(DS P) and 0≤ T 0 g(t)w(n)(t)dt − T 0 g(n)(t) ˆw(n)(t)dt ≤ ¯n T 0 ˆw(n)(t)dt + n T 0 ρeρ(T −t)g(t)dt, (30)
wherenand ¯n are defined as in (23) and (24), respectively. Proof
(i). We verify that ˆw(n)(t) ∈ F(DS Pn) by the following two cases. Case 1. t ∈ l−12n T,2lnT
for some 1≤ l ≤ 2n. Then, for 1≤ j ≤ q,
βjˆw(n)(t) − T t γjˆw(n)(s)ds = βjˆw(n)(t) − ⎡ ⎢ ⎢ ⎣ l 2nT t γj ˆw(n)(s)ds + 2n α=l+1 α 2nT α−1 2n T γj ˆw(n)(s)ds ⎤ ⎥ ⎥ ⎦ = βj¯wl(n)+δ2nρeρ(T −t) − ⎡ ⎢ ⎢ ⎣ l 2nT−t ¯w(n)l γj+γjδ2n l 2nT t ρeρ(T −s)ds + T 2nγj 2n α=l+1 ¯w(n) α +γjδ2n 2n α=l+1 α 2nT α−1 2n T ρeρ(T −s)ds ⎤ ⎥ ⎥ ⎦ = βj¯wl(n)+ βjδ2nρeρ(T −t)− l 2nT− t γj ¯wl(n)− T 2nγj 2n α=l+1 ¯w(n) α − γjδ2n T t ρeρ(T −s)ds
= βj¯wl(n)− T 2nγj 2n α=l+1 ¯wα(n)+ βjδ2nρeρ(T −t)− l 2nT− t γj ¯wl(n) − γjδ2n(eρ(T −t)− 1) ≥ c(n)jl +&ρβj− γj δ2neρ(T −t)− l 2nT− t γj ¯wl(n)+ γjδ2n ≥ c(n)jl +&ρβj− γj δ2n− T 2nγj ¯w (n) l + γjδ2n ≥ c(n)jl + & ρβj− γj δ2n sinceδ2n ≥ T 2n ¯w (n) l ≥ c(n)jl = f(n)j (t).
Case 2. t = T . Then for 1 ≤ j ≤ q,
βjˆw(n)(T ) − T T γjˆw(n)(s)ds = βjˆw(n)(T ) = β j( ¯w(n)2n + δ2nρ) ≥ βj¯w2(n)n ≥ c(n)j 2n = fj(n)(T ). Hence ˆw(n)(t) ∈ F(DS Pn).
Moreover, we observe that
T 0 g(n)(t) ˆw(n)(t)dt = 2n l=1 b(n)l l 2nT l−1 2nT ¯wl(n)dt+ δ2n T 0 ρeρ(T −t)g(n)(t)dt = 2n l=1 b(n)l T 2n ¯w (n) l + δ2n T 0 ρeρ(T −t)g(n)(t)dt = V (Dn) + δ2n T 0 ρeρ(T −t)g(n)(t)dt, (31) which implies 0 ≤ T ' 0 g(n)(t) ˆw(n)(t)dt − V (Dn) ≤ δ2n T ' 0 ρeρ(T −t)g(t)dt, since g(n)(t) ≤ g(t).
(ii). Observe thatw(n)(t) ≥ ˆw(n)(t) ≥ 0 for all t ∈ [0, T ]. For 1 ≤ j ≤ q we have
βjw(n)(t) − T
t
= βjˆw(n)(t) − T t γjˆw(n)(s)ds + n ⎡ ⎣βjρeρ(T −t)− T t γjρeρ(T −s)ds ⎤ ⎦ ≥ fj(n)(t) + n βjρeρ(T −t)+ γj− γjeρ(T −t) = fj(n)(t) + n (ρβj− γj)eρ(T −t)+ γj ≥ fj(n)(t) + n ρβj− γj+ γj (since ρβj ≥ γj) = fj(n)(t) + nρβj ≥ fj(n)(t) + n (since ρβj ≥ 1) ≥ fj(t) for all t ∈ [0, T ]. Hencew(n)(t) ∈ F(DS P).
Sincew(n)(t) ≥ ˆw(n)(t) ≥ 0 and g(t) ≥ g(n)(t) ≥ 0, we have
0≤ T 0 g(t)w(n)(t)dt − T 0 g(n)(t) ˆw(n)(t)dt = T 0 [g(t) − g(n)(t)] ˆw(n)(t)dt + n T 0 g(t)ρeρ(T −t)dt ≤ ¯n T 0 ˆw(n)(t)dt + n T 0 ρeρ(T −t)g(t)dt.
We complete this proof.
Note that, by (30), we have
V(DS P) − T 0 g(n)(t) ˆw(n)(t)dt ≤ T 0 g(t)w(n)(t)dt − T 0 g(n)(t) ˆw(n)(t)dt ≤ ¯n T 0 ˆw(n)(t)dt + n T 0 g(t)ρeρ(T −t)dt. (32)
Hence, by (22), (29) and (32), we have 0≤ V (DS P) − V (Dn) = V (DS P) − T 0 g(n)(t) ˆw(n)(t)dt + T 0 g(n)(t) ˆw(n)(t)dt − V (Dn) ≤ ¯n T 0 ˆw(n)(t)dt + n T 0 ρeρ(T −t)g(t)dt + δ 2n T 0 ρeρ(T −t)g(t)dt
= ¯n T 0 ˆw(n)(t)dt + (n+ δ2n) T 0 ρeρ(T −t)g(t)dt = ¯n ⎡ ⎣2 n l=1 T 2n ¯w (n) l + δ2n(eρT− 1) ⎤ ⎦ + (n+ δ2n) T 0 ρeρ(T −t)g(t)dt ≤ ¯n 2nδ2n+ δ2n(eρT − 1) + (n+ δ2n) T 0 ρeρ(T −t)g(t)dt (by(27)) = ¯nδ2n(2n+ eρT− 1) + (n+ δ2n) T 0 ρeρ(T −t)g(t)dt. (33)
Note thatn → 0 and ¯n → 0 as n → ∞, since fj(t) and g(t) are uniformly continuous
on[0, T ]. Accordingly, by (27) and Theorem1-(ii),δ2n → 0 and
¯nδ2n2n ≤ ¯nTL
σe
T¯γ
σ → 0 as n → ∞. Thus we have limn→∞V(Dn) = V (DS P).
Based on the above discussion, we have the following result which provides the estimation for the error between V(DS P) and V (Dn) and the error between V (S P) and V (Pn). Theorem 2 Suppose that assumptions (A1) and (A2) hold. Then the sequence{V (Dn)} is convergent to V(DS P). Moreover, we have
0≤ V (DS P) − V (Dn) ≤ εn, where εn := ¯nδ2n(2n+ eρT − 1) + (n+ δ2n) T 0 ρeρ(T −t)g(t)dt, (34)
n,¯nandδ2nare defined as in (23), (24) and (27), respectively. Note that, by inequality (22) and Theorem2, we have
V(DS P) ≥ V (S P) ≥ lim
n→∞V(Dn) = V (DS P).
Therefore, V(DS P) = V (S P) = limn→∞V(Dn) = limn→∞V(Pn), and 0≤ V (S P) − V (Pn) ≤ εn,
whereεnis defined by (34). Moreover, we can establish the estimation for the error bound of objective values of approximate solutionsˆx(n)(t) and w(n)(t) to (SP) and (DSP), respectively. Theorem 3 Suppose that assumptions (A1) and (A2) hold. Letˆx(n)(t) and w(n)(t) be defined as in (19) and (28), respectively. Then the error between the optimal value of(SP) and the objective value ofˆx(n)(t) and the error between the optimal value of (DSP) and the objective value ofw(n)(t) are both less than or equal to εn.
Proof By Lemma2,ˆx(n)(t) ∈ F(S P). Since fj(n)(t) ≤ fj(t) for every j and q j=1 T 0 f(n)j (t) ˆx(n)j (t)dt = q j=1 2n l=1 T 2nc (n) jl ¯x (n) jl = V (Pn) = V (Dn), we have 0≤ V (S P) − q j=1 T 0 fj(t) ˆx(n)j (t)dt ≤ V (S P) − q j=1 T 0 f(n)j (t) ˆx(n)j (t)dt = V (DS P) − V (Dn) ≤ εn, by Theorem2.
On the other hand, since V(Dn) ≤ V (DS P), we have
0≤ T 0 g(t)w(n)(t)dt − V (DS P) ≤ T 0 g(t)w(n)(t)dt − V (Dn) = T 0 g(t)w(n)(t)dt − T 0 g(n)(t) ˆw(n)(t)dt + T 0 g(n)(t) ˆw(n)(t)dt − V (Dn) ≤ ¯n T 0 ˆw(n)(t)dt + (n+ δ2n) T 0
ρeρ(T −t)g(t)dt (by (29) and (30))
≤ εn (by (33)).
We complete this proof.
4 Algorithm and convergence of approximate solutions
We summarize the preceding discussions to form the following solution procedure for finding the approximate solutions of(SP) and (DSP).
Algorithm 2 Letδ be the accuracy of tolerance and an initial number n0∈ N be given. Step 1: Set n← n0.
Step 2: Calculate¯x(n)and ¯w(n)by Algorithm 1. Compute the error boundεndefined as in (34).
Step 3: Ifεn ≤ δ, then STOP. By (19) and (28) , construct ˆx(n)(t) and w(n)(t) as the approximate solutions of(SP) and (DSP), respectively.
In what follows, we will demonstrate the convergent properties of the sequences{ ˆx(n)(t)} and{w(n)(t)} derived by Algorithm 2.
Let L1[0, T ] be the family of equivalence classes of real-valued Lebesgue measurable functions on[0, T ] with finite L1 norm. The dual space of the separable Banach space L1[0, T ] can be identified with L∞[0, T ]. An important property enjoyed by the dual of a separable Banach space is weak-star sequential compactness for sets bounded in the strong topology.
By [11, Theorem 4.12.3] and [17, Lemma 2.1], we have the following useful lemma. Lemma 4 Letλn ∈ L∞[0, T ]. If there exists a constant κ > 0 such that λn∞ ≤ κ for n= 1, 2, . . . . Then
(i) there existλ ∈ L∞[0, T ] and a subsequence {λnk} such that λnk → λ (weak∗), that is,
T 0 λnk(t)h(t)dt → T 0 λ(t)h(t)dt for all h(t) ∈ L1[0, T ]; (ii) we have λ(t) ≤ lim sup nk→∞
λnk(t) for almost all t ∈ [0, T ] and
λ(t) ≥ lim inf
nk→∞λnk(t) for almost all t ∈ [0, T ].
Remark 3 Note that ifλn(t) ≥ 0 for all t ∈ [0, T ] and λn → λ (weak∗), then λ(t) ≥ 0 for almost all t∈ [0, T ] by Lemma4(ii).
We also note that both the feasible domains of(SP) and (DSP) are in L∞[0, T ], which is normally regarded as a family of equivalence classes, however, the original formulations require that the feasible solutions must satisfy the constraints for all t not only for almost everywhere. In this section, based on our constructed method for approximate solutions of
(SP) and (DSP), we can show that there exist two functions ˆx (t) and w (t), as shown in
Theorem 4, such that they have the same objective value and satisfy the constraints of(SP) and(DSP), respectively, for all t not only for almost everywhere. Hence, by the weak duality property shown in Remark (1c), they are optimal solutions of(SP) and (DSP), respectively. To see this, the following Lemma5developed by Tyndall [22, Lemma5] is needed. Let
FL(S P) := {x(t) = (x1(t), . . . , xq(t)) | xj(t) ∈ L∞+[0, T ] (1 ≤ j ≤ q) and q j=1 ⎛ ⎝βjxj(t) − t 0 γjxj(s)ds ⎞
⎠ ≤ g(t) for almost all t ∈ [0, T ]}
and FL(DS P) := {w(t) | w(t) ∈ L∞+[0, T ] and βjw(t) − T t γjw(s)ds ≥ fj(t) (1 ≤ j ≤ q)
for almost all t∈ [0, T ]}.
Lemma 5 Given x(t) = (x1(t), . . . , xq(t)) ∈ FL(S P) and w(t) ∈ FL(DS P). Then there exist ˆx(t) = ( ˆx1(t), . . . , ˆxq(t)) ∈ F(S P) and ˜w(t) ∈ F(DS P) such that xj(t) =
Let ˆx(n)(t) and w(n)(t) be defined as in (19) and (28), respectively. Then we have the following result.
Theorem 4 Suppose that assumptions (A1) and (A2) hold. Then there exist subsequences
{ ˆx(nk)(t)} and {w(nk)(t)} such that ˆx(nk)→ ˆx (weak∗) and w(nk) → w (weak∗). More-over,ˆx (t) and w (t) are the optimal solutions of (SP) and (DSP), respectively.
Proof Note that, by Theorem1 and Lemma 1, there exists a constantκ > 0 such that
ˆx(n)j ∞ < κ and w(n)∞ < κ for all n and 1 ≤ j ≤ q. By Lemma 4, there exist ˆxj(t) ∈ L∞[0, T ], w(t) ∈ L∞[0, T ] and subsequences { ˆx(nk)
j (t)} and {w(nk)(t)} such that ˆx(nk)
j → ˆxj (weak∗) for all 1 ≤ j ≤ q (35)
and w(nk)→ w (weak∗). (36) Besides, we have ˆxj(t) ≤ lim sup nk→∞ ˆx(nk)
j (t) for almost all t ∈ [0, T ], (37)
and
w(t) ≥ lim inf nk→∞w
(nk)(t) for almost all t ∈ [0, T ]. (38)
Sinceˆx(nk)(t) ∈ F(S P) and w(nk)(t) ∈ F(DS P), we have for all t ∈ [0, T ]
q j=1 ⎡ ⎣βjˆx(nj k)(t) − t 0 γjˆx(nj k)(s)ds ⎤ ⎦ ≤ g(t), (39) ˆx(nk) j (t) ≥ 0
and for all t∈ [0, T ]
βjw(nk)(t) − T t γjw(nk)(s)ds ≥ fj(t) (1 ≤ j ≤ q), (40) w(nk)(t) ≥ 0. Sinceˆx(nk)
j (t) ≥ 0 and w(nk)(t) ≥ 0, it follows, by (35), (36) and Remark 3, that ˆxj(t) ≥ 0
andw(t) ≥ 0 for almost all t ∈ [0, T ]. From ( 39) and (40), by taking the limit superior and inferior, we obtain, for almost all t∈ [0, T ],
q j=1 βjˆxj(t) ≤ lim sup nk→∞ q j=1 βjˆx(nj k)(t) (by (37)) ≤ lim sup nk→∞ q j=1 t 0 γjˆx(nj k)(s)ds + g(t) (by (39)) = q j=1 t 0 γjˆxj(s)ds + g(t) (by (35)) and βjw(t) ≥ lim inf nk→∞βjw (nk)(t) (by (38)) ≥ lim inf nk→∞ T t γjw(nk)(s)ds + fj(t) (by (40)) = T t
γjw(s)ds + f j(t) for all j (by (36)).
Hence ˆx(t) = ( ˆx1(t), . . . , ˆxq(t)) ∈ FL(S P) and w(t) ∈ FL(DS P). By Lemma5, there existˆx (t) = ( ˆx1 (t), . . . , ˆxq (t)) ∈ F(S P) and w (t) ∈ F(DS P) such that ˆx j(t) = ˆxj(t)
(1≤ j ≤ q) and w (t) = w(t) for almost all t ∈ [0, T ]. Thus, by (35) and (36), we have
ˆx(nk)→ ˆx (weak∗) and w(nk)→ w (weak∗).
Next, we will prove that ˆx (t) and w (t) are the optimal solutions of (SP) and (DSP), respectively. We observe that
T 0 q j=1 fj(t) ˆx(nj k)(t)dt = q j=1 T 0 fj(t) − f(nk) j (t) ˆx(nk) j (t)dt + q j=1 T 0 fj(nk)(t) ˆx(nj k)(t)dt = q j=1 T 0 fj(t) − f(nk) j (t) ˆx(nk) j (t)dt + V (Pnk). (by (21)) This implies T 0 q j=1 fj(t) ˆx(nj k)(t)dt − q j=1 T 0 fj(t) − f(nj k)(t) ˆx(nk) j (t)dt = V (Pnk). (41) Besides, since T 0 g(t)w(nk)(t)dt
= T 0 g(t) ˆw(nk)(t)dt + T 0 g(t)nkρeρ(T −t)dt = T 0 [g(t) − g(nk)(t)] ˆw(nk)(t)dt + T 0 g(nk)(t) ˆw(nk)(t)dt + n k T 0 g(t)ρeρ(T −t)dt = T 0 [g(t) − g(nk)(t)] ˆw(nk)(t)dt + V (Dn k) + δ2nk T 0 ρeρ(T −t)g(n)(t)dt + nk T 0 g(t)ρeρ(T −t)dt, we have T 0 g(t)w(nk)(t)dt − T 0 [g(t) − g(nk)(t)] ˆw(nk)(t)dt − δ2nk T 0 ρeρ(T −t)g(nk)(t)dt − n k T 0 g(t)ρeρ(T −t)dt= V (Dnk). (42)
Hence, by (41) and (42), we obtain
T 0 q j=1 fj(t) ˆx(nj k)(t)dt − q j=1 T 0 fj(t) − fj(nk)(t) ˆx(nk) j (t)dt = T 0 g(t)w(nk)(t)dt − T 0 [g(t) − g(nk)(t)] ˆw(nk)(t)dt − δ2nk T 0 ρeρ(T −t)g(nk)(t)dt − n k T 0 g(t)ρeρ(T −t)dt. (43)
Taking the limit nk → ∞ on both sides of (43), by (35), (36) and Lebesgue’s bounded
convergence theorem, we obtain
q j=1 T 0 fj(t) ˆxj(t)dt = T 0 g(t)w(t)dt,
and this implies
q j=1 T 0 fj(t) ˆx j(t)dt = T 0 g(t)w (t)dt,
that is, the objective values of ˆx (t) and w (t) are equal. Hence ˆx (t) and w (t) are the optimal solutions of(SP) and (DSP), respectively. We complete this proof.
Let{ ˆx(n)(t)} and {w(n)(t)} be any subsequences of { ˆx(n)(t)} and {w(n)(t)}, respectively. Then, by the proof of Theorem4, there exist weak∗ convergent subsequences{ ˆx(nk)(t)} and{w(nk)(t)}. Hence if (SP) and (DSP) have the unique optimal solutions ˆx (t) and w (t), respectively. Then{ ˆx(nk)(t)} and {w(nk)(t)} are weak∗convergent toˆx (t) and w (t), respec-tively. In other words, all subsequences of{ ˆx(n)(t)} and {w(n)(t)} have further subsequences that are weak∗convergent toˆx (t) and w (t), respectively. Therefore, we have the following result.
Theorem 5 Suppose that assumptions (A1) and (A2) hold. If(SP) and (DSP) have the unique optimal solutionsˆx (t) and w (t), respectively. Then ˆx(n)(t) → ˆx (t) (weak∗) and
w(n)(t) → w (t) (weak∗) as n → ∞, where ˆx(n)(t) and w(n)(t) are defined as in (19) and
(28), respectively.
5 Numerical examples
Finally, for illustration purpose, we use two examples to implement the improved method and to show the quality of the proposed error bound.
Example 1 maximize 1 0 ln(t + 1/2)x(t)dt subject to 2x(t) − 7 t 0 x(s)ds ≤ et− 1, ∀t ∈ [0, 1] x(t) ∈ L∞+[0, 1].
Table 1 Approximate value Vn(S P) and error bound εnfor Examples 1 and 2
n Example 1 Example 2 Vn(S P) εn Vn(S P) εn 10 0.3505765 0.0568731 0.8917267 0.2040224 11 0.3518269 0.0285131 0.8958349 0.1025288 12 0.3524547 0.0142757 0.8978995 0.0513945 13 0.3527692 0.0071426 0.8989344 0.0257298 14 0.3529266 0.0035725 0.8994525 0.0128731 15 0.3530054 0.0017866 0.8997118 0.0064386 16 0.3530448 0.0008934 0.8998414 0.0032198 17 0.3530645 0.0004467 0.8999063 0.0016100 18 0.3530743 0.0002234 0.8999387 0.0008050 19 0.3530792 0.0001117 0.8999549 0.0004025 20 0.3530817 0.0000558 0.8999630 0.0002013
Example 2 maximize 1 0 (t − 1/2)x1(t) + (t2− 1/3)x2(t) dt subject to x1(t) + 3x2(t) − t 0 [4x1(s) + 2x2(s)] ds ≤ t, ∀ t ∈ [0, 1] x1(t), x2(t) ∈ L∞+[0, 1].
To illustrate the convergence, we select the partition number n from 10 to 20. Using MATLAB Version 7.0.1 on a PC for the experiment, the results obtained by running the program which implement the proposed algorithm are presented in Table1, where Vn(S P) is the objective
value of the approximate solution ˆx(n)(t) and εn is the proposed error bound defined as in (34).
From Table1, one can easily compute a range of the optimal value of(SP) for each n, and this range will approach to the optimal value of(SP) as n tends to infinite.
Acknowledgments The authors wish to thank the referees whose insightful comments and suggestions contributed significantly to an improved version of the paper.
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