Numerical Method for Solving the Robust Continuous-Time Linear Programming Problems

: A robust continuous-time linear programming problem is formulated and solved numerically in this paper. The data occurring in the continuous-time linear programming problem are assumed to be uncertain. In this paper, the uncertainty is treated by following the concept of robust optimization, which has been extensively studied recently. We introduce the robust counterpart of the continuous-time linear programming problem. In order to solve this robust counterpart, a discretization problem is formulated and solved to obtain the (cid:101) -optimal solution. The important contribution of this paper is to locate the error bound between the optimal solution and (cid:101) -optimal solution.


Introduction
The theory of continuous-time linear programming problem has received considerable attention for a long time.Tyndall [1,2] treated rigorously a continuous-time linear programming problem with the constant matrices, which was originated from the "bottleneck problem" proposed by Bellman [3].Levison [4] generalized the results of Tyndall by considering the time-dependent matrices in which the functions shown in the objective and constraints were assumed to be continuous on the time interval [0, T].In this paper, we are going to consider the continuous-time linear programming problem in which the data are assumed to be uncertain.The data here mean the real-valued functions or real numbers.Based on the assumption of uncertainty, we are going to propose and solve the so-called robust continuous-time linear programming problem.
Meidan and Perold [5], Papageorgiou [6] and Schechter [7] have obtained many interesting results of the continuous-time linear programming problem.Also, Anderson et al. [8][9][10], Fleischer and Sethuraman [11] and Pullan [12][13][14][15][16] investigated a subclass of the continuous-time linear programming problem, which is called the separated continuous-time linear programming problem and can be used to model the job-shop scheduling problems.Weiss [17] proposed a Simplex-like algorithm to solve the separated continuous-time linear programming problem.Also, Shindin and Weiss [17,18] studied a more generalized separated continuous-time linear programming problem.However, the error estimate was not studied in the above articles.One of the contributions of this paper is to obtain the error bound between the optimal solution and numerical optimal solution.
Wen and Wu [36][37][38] have developed different numerical methods to solve the continuous-time linear fractional programming problems.In order to solve the continuous-time problems, the discretized problems should be considered by dividing the time interval [0, T] into many subintervals.Since the functions considered in Wen and Wu [36][37][38] are assumed to be continuous on [0, T], we can take this advantage to equally divide the time interval [0, T].In other words, each subinterval has the same length.In Wu [39], the functions are assumed to be piecewise continuous on the time interval [0, T].In this case, the time interval cannot be equally divided.The reason is that, in order to develop the numerical technique, the functions should be continuous on each subinterval.Therefore a different methodology for not equally partitioning the time interval was proposed in Wu [39].In this paper, we shall solve a more general model that considers the uncertain data in continuous-time linear programming problem.In other words, we are going to solve the robust counterpart of the continuous-time linear programming problem.In this paper, we still consider the piecewise continuous functions on the time interval [0, T].Therefore, the time interval cannot be equally divided.
Addressing the uncertain data in optimization problems has become an attractive research topic.As early as the mid 1950s, Dantzig [40] introduced the stochastic optimization as an approach to model uncertain data by assuming scenarios for the data occurring with different probabilities.One of the difficulties is to present the exact distribution for the data.Therefore, the so-called robust optimization might be another choice for modeling the optimization problems with uncertain data.The basic idea of robust optimization is to assume that each uncertain data falls into a set.For example, the real-valued data can be assumed to fall into a closed interval for convenience.In order to address the optimization problems with uncertain data falling into the uncertainty sets, Ben-Tal and Nemirovski [41,42] and independently El Ghaoui [43,44] proposed to solve the so-called robust optimization problems.Now, this topic has increasingly received much attention.For example, the research articles contributed by Averbakh and Zhao [45], Ben-Tal et al. [46], Bertsimas et al. [47][48][49], Chen et al. [50], Erdo ǧan and Lyengar [51], Zhang [52] and the references therein can present the main stream of this topic.In this paper, we propose the robust counterpart of the continuous-time linear programming problem and design a practical algorithm to solve this problem.
This paper is organized as follows.In Section 2, a robust continuous-time linear programming problems is formulated.Under some algebraic calculation, it can be transformed into a traditional form of the continuous-time linear programming problem.In Section 3, we introduce the discretization problem of the transformed problem.Based on the solutions obtained from the discretization problem, we can construct the feasible solutions of the transformed problem.Under these settings, the error bound will be derived.In order to obtain the approximate solution, we also introduce the concept of -optimal solutions.In Section 4, we study the convergence of approximate solutions.In Section 5, based on the obtained results, the computational procedure is proposed and a numerical example is provided to demonstrate the usefulness of this practical algorithm.

Robust Continuous-Time Linear Programming Problems
In this section, a robust continuous-time linear programming problems will be formulated.We shall use some algebraic calculation to transform it into a traditional form of the continuous-time linear programming problem.Now, we consider the following continuous-time linear programming problem: for all t ∈ [0, T] and i = 1, • • • , p; z j ∈ L 2 [0, T] and z j (t) ≥ 0 for all j = 1, • • • , q and t ∈ [0, T], where B ij and K ij are nonnegative real numbers for i = 1, • • • , p and j = 1, • • • , q, and a j and c i are the real-valued functions.It is obvious that if the real-valued functions c i are assumed to be non-negative on [0, T] for i = 1, • • • , p, then the primal problem (CLP) is feasible with a trivial feasible solution z j (t) = 0 for all j = 1, • • • , q.
Suppose that some of the data B ij and K ij are uncertain such that they should fall into the uncertainty sets U B ij and U K ij , respectively.Given any fixed i ∈ {1, i , then B ij is uncertain, and if j ∈ I We also assume that functions a j and c i are pointwise-uncertain in the sense that, given each t ∈ [0, T], the uncertain data a j (t) and c i (t) should fall into the uncertainty sets V a j (t) and V c i (t), respectively.If function a j or c i is assumed to be certain, then each function value a j (t) or c i (t) is assumed to be certain for t ∈ [0, T].If function a j or c i is assumed to be uncertain, then the function value a j (t) or c i (t) may be certain for some t ∈ [0, T].We denote by I (a) and I (c) the sets of indices in which the functions a j and c i are assumed to be uncertain, respectively.In other words, if j ∈ I (a) , then the function a j is uncertain, and if i ∈ I (c) , then the function c i is uncertain.
The robust counterpart of problem (CLP) is assumed to take each data in the corresponding uncertainty sets, and it is formulated as follows: We can see that the robust counterpart (RCLP) is a continuous-time programming problem with infinitely many number of constraints.Therefore, it is difficult to solve.However, if we can determine the suitable uncertainty sets U B ij , U K ij , V a j (t) and V c i (t), then this semi-infinite problem can be transformed into a conventional continuous-time linear programming problem.
We assume that all the uncertain data fall into the closed intervals, which will be described below.
i , we assume that the uncertain data B ij and K ij should fall into the closed intervals respectively, where ij ≥ 0 are the known nominal data of B ij and K ij , respectively, and B ij ≥ 0 and K ij ≥ 0 are the uncertainties such that i , we use the notation B (0) ij to denote the certain data with uncertainty B ij = 0. Also, we use the notation K (0) ij to denote the certain data with uncertainty For a j with j ∈ I (a) and c i with i ∈ I (c) , we assume that where a (0) j (t) and c (0) i (t) are the known nominal data of a j (t) and c i (t), respectively, and a j (t) ≥ 0 and c i (t) ≥ 0 are the uncertainties of a j (t) and c i (t), respectively.For j ∈ I (a) , we use the notation a (0) j (t) to denote the certain function with uncertainties a j (t) = 0 for all t ∈ [0, T].Also, we use the notation c (0) i (t) to denote the certain function with uncertainties c j (t) = 0 for i ∈ I (c) and t ∈ [0, T].
In this case, the robust counterpart (RCLP) is written as follows: c) and for all t ∈ [0, T]; and for all t ∈ [0, T]; i ; for all a j (t) ∈ V a j (t) with j ∈ I (a) for all t ∈ [0, T]; Next, we are going to convert the above semi-infinite problem (RCLP) into a conventional continuous-time linear programming problem.
First of all, the problem (RCLP) can be rewritten as the following equivalent form and for all t ∈ [0, T]; c) and for all t ∈ [0, T];

Given any fixed
Similarly, for j ∈ I Using ( 1) and (2), we consider the following cases.
where the equality can be attained.

•
For i ∈ I (c) , we obtain where the equality can be attained.On the other hand, since a (0) j (t) − a j (t) ≤ a j (t) for j ∈ I (a) and t ∈ [0, T], we have where the equality can also be attained.Therefore, from (3), ( 4) and ( 5), we conclude that (φ, ) is a feasible solution of problem (RCLP1) if and only if it satisfies the following inequalities: This shows that problem (RCLP1) is equivalent to the following problem which can also be rewritten as the following continuous-time linear programming problem: for all t ∈ [0, T] and i ∈ I (c) ; According to the duality theory in continuous-time linear programming problem, the dual problem of (RCLP3) can be formulated as follows for all t ∈ [0, T] and j ∈ I (a) ; In the sequel, we are going to design a computational procedure to numerically solve the robust counterpart (RCLP3).

Discretization
In this section, we shall introduce the discretization problem.Based on the solutions obtained from the discretization problem, we can construct the feasible solutions of the transformed problem.Under these settings, the error bound can be derived.On the other hand, in order to obtain the approximate solution, we also introduce the concept of -optimal solutions.
In order to develop the efficient numerical method, we assume that the following conditions are satisfied.
i are piecewise continuous on [0, T].For j ∈ I (a) and i ∈ I (c) , the functions a j and c i are also piecewise continuous on [0, T], which says that the functions a (0) j − a j and c (0) i − c i are piecewise continuous on [0, T] for j ∈ I (a) and i ∈ I (c) .

•
For each j = 1, • • • , q, the following inequality is satisfied: • the following inequality is satisfied: In other words, Let A j denote the set of discontinuities of functions a (0) j − a j for j ∈ I (a) and a (0) j for j ∈ I (a) .Let C j denote the set of discontinuities of functions c (0) i − c i for i ∈ I (c) and c (0) i for i ∈ I (c) .Then, we see that A j and C i are finite subsets of [0, T].In order to determine the partition of the time interval [0, T], we consider the following set Then, D is a finite subset of [0, T] written by where, for convenience, we set d 0 = 0 and d r = T.It means that d 0 and d r may be the endpoint-continuities of functions āj and ci .Let P n be a partition of [0, T] such that D ⊆ P n , which means that each closed [d v , d v+1 ] is also divided into many closed subintervals.In this case, the time interval [0, T] is not necessarily equally divided into n closed subintervals.Let We also write In the limiting case, we shall assume that In this paper, we assume that there exists n * , n * ∈ N such that Therefore, in the limiting case, we assume that n * → ∞, which implies n → ∞.In the sequel, when we say that n → ∞, it implicitly means that n * → ∞.
For example, suppose that the length of closed interval In this case, the total subintervals are n = n * • r.We also see that n * = n * and Under the above construction for the partition P n , we see that the functions a and and the vectors Then, we see that we define the following linear programming problem: According to the duality theory of linear programming, the dual problem of (P n ) is given by

Now, let
Then, by dividing d on both sides of the constraints, the dual problem ( D n ) can be equivalently written by denotes the length of closed interval Ē(n) l .We also define Then, we have s For further discussion, we adopt the following notations: Now, we also have which say that It is obvious that τ for any n ∈ N and l = 1, • • • , n.
Proposition 1.The following statements hold true.
where σ is given in (7).We define , and consider the following vector , and consider the following vector .
Then, w(n) is a feasible solution of problem (D n ) satisfying the following inequalities li is non-negative and w (n) is an optimal solution of problem (D n ), then w(n) is also an optimal solution of problem (D n ).
Therefore, from the above two cases, we conclude that w(n) is indeed a feasible solution of problem (D n ).Since problem (D n ) is a minimization problem and it says that if w (n) is an optimal solution of problem (D n ), then w(n) is an optimal solution of problem (D n ).This completes the proof.
Proposition 2. Suppose that the primal problem (P n ) is feasible with a feasible solution Proof.By (6), for each j, there exists . If φ = 0, then K is a zero matrix .In this case, using the feasibility of z (n) , we have For the case of φ = 0, we want to show that We shall prove it by induction on l.Since z (n) is a feasible solution of problem (P n ), for l = 1, we have , Therefore, for each j, we obtain Suppose that Then, for each j, we have By the feasibility of z (n) , we have Therefore, by induction, we obtain , we complete the proof.
Then we have the following feasibility.
q ) defined in (29) is a feasible solution of problem (RCLP3).
Proof.Since z(n) is a feasible solution of problem (P n ), it follows that and Now, we consider the following two cases.
ij ≥ 0 for all i and j, and for i ∈ I (c) (by (11)) For l = 1, the desired inequality can be similarly obtained by (30).
• Suppose that t = T.Then, we can similarly show that for i ∈ I (c) (by ( 11)) Therefore, we conclude that ( z q ) is indeed a feasible solution of problem (RCLP3).This completes the proof.
Given any optimization problem (P), we denote by V(P) the optimal objective value of problem (P).For example, the optimal objective value of problem (RCLP3) is denoted by V(RCLP3).Now we assume that z(n) is an optimal solution of problem (P n ).Then we can also construct a feasible solution ( z q ) of problem (RCLP3) according to (29).Then, using (11), we have Therefore, we have T 0 a j (t) • z j (t)dt (by Proposition 3) ≥ V(P n ) (by (32)).
Using the weak duality theorem for the primal and dual pair of problems (DRCLP3) and (RCLP3), we see that Next, we want to show that Let l p ) be an optimal solution of problem (D n ).We define w(n is defined in (18), and consider the following vector Then, according to part (ii) of Proposition 1, we see that w(n) is an optimal solution of problem (D n ) satisfying the following inequalities where e (n) l is defined in (8). and Then, we have which says that and We first provide some useful lemmas.
Proof.According to the construction of partition P n , we see that a

Proof. It suffices to prove
as n → ∞.
From ( 9), since li is bounded according to (35), it follows that, given any fixed Now, for j ∈ I (a) , we have and, for j ∈ I (a) , we have Using Lemma 1, we complete the proof.
We define From (7), we see that 0 < σ ≤ σ * .Also, from (35), we see that the sequence { h(n) lj } ∞ n=1 of functions is uniformly bounded, which also says that {π n=1 is uniformly bounded.Therefore there exists a constant x such that π (n) l ≤ x for all n ∈ N and l = 1, • • • , n.Now, we define a real-valued function p (n) on [0, T] by The following lemma will be used for further discussion.
Lemma 3. We have a j (T) − a j (T) − a Moreover, the sequence of real-valued functions {f (n) } ∞ n=1 is uniformly bounded.
Proof.For t ∈ F (n) l , from (41), we have Since For t = e (n) n = T, we also have, For each j = 1, • • • , q and l = 1, • • • , n, we consider the following cases.44), we have l , by ( 43) and ( 40), we have Finally, it is obvious that the sequence of real-valued functions {f (n) } ∞ n=1 is uniformly bounded.This completes the proof.
For each i = 1, • • • , p, we define the step function w Remark 1.According to (35) and Lemma 3, we see that the family of vector-valued functions { w (n) } n∈N is uniformly bounded.
Proposition 4. For any n ∈ N, the vector-valued step function ( w (by the feasibility of w(n) for problem (D n ))    Therefore, we conclude that w (n) is indeed a feasible solution of problem (DRCLP3), and the proof is complete.
Proof.It is obvious that the following functions    are continuous a.e. on [0, T], i.e., they are Riemann-integrable on [0, T].In other words, their Riemann integral and Lebesgue integral are identical.From Lemma 1, we see that Since the set of functions { z bounded according to Proposition 2, using the Lebesgue bounded convergence theorem, we obtain (46).On the other hand, since set of functions { w(n) i } ∞ n=1 for i = 1, • • • , p is uniformly bounded according to Proposition 1, using the Lebesgue bounded convergence theorem, we can obtain (47).This completes the proof.Theorem 1.The following statements hold true. where satisfying ε n → 0 as n → ∞. (ii) (No Duality Gap).Suppose that the primal problem (P n ) is feasible.We have Proof.To prove part (i), we have (by Proposition 4) Since p (n) is continuous a.e. on [0, T], it follows that f (n) is also continuous a.e. on [0, T], which says that f (n) is Riemann-integrable on [0, T].In other words, the Riemann integral and Lebesgue integral of f (n) on [0, T] are identical.Since π (n) l → 0 as n → ∞ by Lemma 2, it follows that p (n) → 0 as n → ∞ a.e. on [0, T], which implies that f (n) → 0 as n → ∞ a.e. on [0, T].Applying the Lebesgue bounded convergence theorem for integrals, we obtain Using Lemma 4, we conclude that ε n → 0 as n → ∞.Also, from (49), we obtain It is easy to see that ε n can be written as (48), which proves part (i).
To prove part (ii), by part (i) and inequality (33), we obtain This completes the proof.
Proposition 5.The following statements hold true.
(i) Suppose that the primal problem (P n ) is feasible.Let z (n) j be defined in (29) for j = 1, • • • , q.Then, the error between V(RCLP3) and the objective value of ( z q ) is less than or equal to ε n defined in (48), i.e., be defined in (45) Then, the error between V(DRCLP3) and the objective value of ( w and (51) and part (ii) of Theorem 1) ≤ ε n (by part (i) of Theorem 1).
To prove part (ii), we have This completes the proof.
Definition 1.Given any > 0, we say that the feasible solution (z ( ) We say that the feasible solution (w ( ) Theorem 2. Given any > 0, the following statements hold true.
(i) The -optimal solution of problem (RCLP3) exists in the sense: There exists n ∈ N such that (z ( ) q ), where ( z q ) is obtained from Proposition 5 satisfying ε n < .(ii) The -optimal solution of problem (DRCLP3) exists in the sense: There exists n ∈ N such that (w ( ) p ), where ( w p ) is obtained from Proposition 5 satisfying ε n < .
Proof.Given any > 0, from Proposition 5, since ε n → 0 as n → ∞, there exists n ∈ N such that ε n < .Then, the result follows immediately.

Convergence of Approximate Solutions
In this section, we shall study the convergence of approximate solutions.We first provide some useful lemmas that can guarantee the feasibility of solutions.On the other hand, the strong duality theorem can also be established using the limits of approximate solutions.
In the sequel, by referring to ( 29) and ( 45), we shall present the convergent properties of the sequences { z (n) } ∞ n=1 and { w (n) } ∞ n=1 that are constructed from the optimal solutions z(n) of problem (P n ) and the optimal solution w(n) of problem (D n ), respectively.We first provide a useful lemma.Lemma 5. Let the real-valued function η be defined by on [0, T], and let (w p ) be a feasible solution of dual problem (DRCLP3).We define w (1) p ) is a feasible solution of dual problem (DRCLP3) satisfying w   Proof.By the feasibility of (w i (s)ds for j ∈ I (a) a (0) and w (1) i (s)ds − ∑ {i:j∈I i (s)ds for j ∈ I (a) a (0) i (s)ds − ∑ {i:j∈I i (s)ds For any fixed t ∈ [0, T], we define the index sets From (55), we obtain i (s)ds − ∑ {i:j∈I i (s)ds for j ∈ I (a) a (0) i (s)ds − ∑ {i:j∈I i (s)ds • Suppose that I > = ∅ and there exists i * ∈ I > such that B (0) i * by (7).Therefore, we obtain (57) From ( 52), we see that for all t ∈ [0, T].Using (66), (57) and the fact of w (1) i (s)ds − ∑ {i:j∈I i (s)ds for j ∈ I (a) a (0) i (s)ds − ∑ {i:j∈I i (s)ds for j ∈ I (a) .
Therefore, we conclude that (w (1) p ) is a feasible solution of (DRCLP3), and the proof is complete.
For further discussion, we need the following useful lemmas.Lemma 6. (Riesz and Sz.-Nagy [53] is uniformly bounded with respect to • 2 , then exists a subsequence { f k j } ∞ j=1 that weakly converges to some f 0 ∈ L 2 [0, T].In other words, for any g ∈ L 2 [0, T], we have p )} ∞ n=1 be the sequences that are constructed from the optimal solutions ( z (29) and (45), respectively.Then, the following statements hold true.
Proof.From Proposition 2, it follows that the sequence of functions {( z n=1 is uniformly bounded with respect to • 2 .Using Lemma 6, there exists a subsequence z Therefore we can construct the subsequences { z From Lemma 7, for each i, we have which says that w(0) i (t) ≥ a.e. in [0, T] for all i = 1, • • • , p.For j ∈ I (a) , by taking the limit inferior on both sides of (62), we obtain (by the weak convergence) For j ∈ I (a) , we can similarly obtain For each i = where the set N has measure zero.Then, for each i = • Suppose that t ∈ N .For j ∈ I (a) , from (63), we have w * i (s)ds.
For j ∈ I (a) , from (64), we can similarly obtain From ( 52), we see that for j ∈ I (a) and for all t ∈ [0, T].Therefore, for j ∈ I (a) , we obtain w * i (s)ds (by ( 65)).

Computational Procedure and Numerical Example
In this section, based on the above results, we can design a computational procedure.Also, a numerical example is provided to demonstrate the usefulness of this practical algorithm.The purpose is to provide the computational procedure to obtain the approximate solutions of the continuous-time linear programming problem (RCLP3).Of course, the approximate solutions will be the step functions.According to Proposition 5, it is possible to obtain the appropriate step functions so that the corresponding objective function value is close enough to the optimal objective function value when n is taken to be sufficiently large.
Recall that, from Theorem 1 and Proposition 5, the error upper bound between the approximate objective value and the optimal objective value is given by In order to further design the computational procedure, we need to assume that each α j is twice-differentiable on [0, T] for the purpose of applying the Newton's method, which also says that α j is twice-differentiable on the open interval E (76) The KKT condition says that if t * is an optimal solution of problem (76), then the following conditions should be satisfied:

) 0 =
0 and e (n) n = T.Then, the n closed subintervals are denoted by

jLemma 2 .
for j ∈ I (a) , a (0) j − a j for j ∈ I (a) , c (0) i for i ∈ I (c) and c (0) i − c i for i ∈ I (c) are continuous on the open interval E It is not hard to obtain the desired results.For each l = 1, • • • , n, we have
for all l = 1, • • • , n. From (75), we need to solve the following simple type of optimization problem max e 2, • • • , p}, we denote by I the set of indices in which B ij and K ij are assumed to be uncertain.In other words, if j ∈ I ≤ η(t) for each k and for all t ∈ [0, T].Let N 0 be the subset of [0, T] on which the inequalities (63) and (64) are violated, let N 1 be the subset of [0, T] on which ( 1, • • • , p, w * i (t) ≥ 0 for all t ∈ [0, T] and We also see that w * i ∈ L 2 [0, T].Now we are going to claim that ( w * 1 , • • • , w * p ) is a feasible solution of (DRCLP3).