Bilinear Modeling Solution Approach for Fixed Charged Network Flow Problems (Draft)

Size: px

Start display at page:

Download "Bilinear Modeling Solution Approach for Fixed Charged Network Flow Problems (Draft)"

Richard Hoover
5 years ago
Views:

1 Bilinear Modeling Solution Approach for Fixed Charged Network Flow Problems (Draft) Steffen Rebennack 1, Artyom Nahapetyan 2, Panos M. Pardalos 1 1 Department of Industrial & Systems Engineering, Center for Applied Optimization, University of Florida, Gainesville, FL 32611, USA, {steffen,pardalos}@ufl.edu 2 Innovative Scheduling Inc., Gainesville Technology Enterprise Center (GTEC), 2153 SE Hawthorne Road, Gainesville, FL 32641, artyom@innovativescheduling.com Summary. We present a continuous, bilinear formulation for the fixed charge network flow problem. This formulation is used to derive an exact algorithm for the fixed charge network flow problem; converging in a finite number of steps. A computational study is made to show the performance of the algorithm. Key words: Bilinear modeling; fixed charge network flow problem; exact formulation; concave minimization 1 Introduction [21] [13] [11] [10] [9] [8] [6] In Section 2, we discuss a linear and a bilinear semi-continuous formulation for the fixed charge network flow problem. A continuous bilinear model is then presented in Section 3. Computational feasibility tests are made in Section 4. We conclude with Section 5. 2 Two Mathematical Formulations of Fixed Charged Network Flow Problem In this section, we discuss two formulations of the fixed charge network flow problem over a continuous domain. The first formulation has a semicontinuous objective function with constrains describing network topology. The second formulation has the same feasible region but the objective is defined in terms of piecewise linear functions. In the section, we show that the second formulation is equivalent to the first one under certain conditions.

2 2 S. Rebennack, A. Nahapetyan, P. M. Pardalos Let G = (V,A) represent a directed network with node set V and arc set A V V. Let us assume that the network has n = V nodes and m = A arcs. With each arc a A, we associate a capacity λ a Z +, and λ Z A + denotes the corresponding vector of capacities. Let matrix B represent the n m (unique) node-arc incidence matrix of network G [5], i.e. 1, if node i is the head node of arc a: k V B ia = 1, if node i is the tail node of arc a: k V, 0, otherwise and b i denote the demand (b i 0) or the supply (b i 0) at node i V with i V b i = 0, [1]. In the fixed charge network flow problems, there is a variables cost of c a Z + and a fixed cost of s a Z +. This gives the following (concave) cost structure for each arc a A f a (x a ) = { sa + c a x a, if x a (0,λ a ] 0, if x a = 0. (1) Using above described notations, we can state the following formulation for the fixed-charged network flow problem [14]: FCNF : min x f(x) = a A s.t. Bx = b 0 x λ f a (x a ) Let us approximate the concave cost function f a in (1) by a piece-wise linear concave function φ ɛa a (x a ) = { sa + c a x a, if x a [ɛ a,λ a ] c ɛa a x a, if x a [0,ɛ a ) (2) with c ɛa a = c a + sa ɛ a, [14]. Recognize that φ ɛa a underestimates function f a for all ɛ a > 0 (see Figure??). I think you mean Figure 1 from [14]. I included the figure. I had problems compiling it. Do you have the source code of the figure? Maybe, we should also modify it a bit, as it has been published in this form. In analogon to formulation FCNF, we can define the continuous piecewise linear network flow problem for a given ɛ > 0 as CPLNF(ɛ) : min x φ ɛ (x) = a A s.t. Bx = b 0 x λ φ ɛa a (x a )

3 Biliner Model for FCNFP 3 In [14], authors proved that for a sufficiently small ɛ, CPLNF(ɛ) correctly models the fixed charge network flow problem. Below we briefly restate the theorem and for the proof we refer to [14]. Let V represent the set of vertices of the feasible region of FCNF. Then, define δ as the minimum among all positive components of all vertices x v in V; i.e. δ = min {x v a x v V,a A,x v a > 0}. Theorem 2.1 [14] Let x (x ɛ ) be an optimal solution to FCNF (CPLNF(ɛ)). Then, φ ɛ (x ɛ ) = f(x ) for all ɛ with ɛ a (0,δ] a A. 3 Bilinear Formulation In this section, we develop a continuous bilinear formulation for the fixed charge network flow problem, which is equivalent to CP LN F(ɛ) formulation. Based on the new formulation, we describe an algorithm that can be used to find an optimal solution to fixed charge network flow problem. Note that CPLNF(ɛ) has linear constrains and a piecewise linear continuous concave objective function. Next we will modify the objective function using binary variables, indicating in which one of the two parts the function lies. Let us define binary variables y a for all arcs a A y a = { 0, if xa [0,ɛ a ) 1, if x a (ɛ a,λ a ]. (3) Relaxing these variables allows us to formulate the following continuous bilinear network flow problem for any given ɛ > 0 CBLNF(ɛ) : ϕ(ɛ) = min x,y a A ( c ɛa a x a + (s a s ) a x a )y a ɛ a (4) s.t. Bx = b (5) 0 x λ (6) 0 y 1 (7) In the following theorem, we proof that CBLNF(ɛ) and CPLNF(ɛ) are equivalent. Theorem 3.1 (x,y ) is an optimal solution to CBLNF(ɛ), if and only if x is optimal for CPLNF(ɛ). Proof. To prove the theorem, below we show that any optimal solution of one of the problems corresponds to a feasible solution for the second problem with the same objective function value. = Let x denote an optimal solution of CPLNF(ɛ). Assign variable y values according to (3). Then, (x,y) is feasible to CBLNF(ɛ) and has the same objective function value.

4 4 S. Rebennack, A. Nahapetyan, P. M. Pardalos = Let (x,y) denote an optimal solution of CBLNF(ɛ). Obviously, x is also feasible for CPLNF(ɛ). Next, we need to prove that the objective function values are equal. Doing so, we need to show that if s a 0 and x a ɛ, then y respects its definition (3). As we are only considering the effect of variable y, we can ignore the term c ɛa a x a in the objective function (4). The remainder can be restated as ( y a s a s ) a x a ɛ a Case 1: Let x a < ɛ. Then s a sa ɛ a x a > 0. Since in CBLNF(ɛ) we minimize the objective function (4), at optimality y a has to be equal to 0. Case 2: Let x a > ɛ. Then s a sa ɛ a x a < 0. Similar to the previous case, at optimality y a has to be equal to 1. Let us adopt Theorem 2.1 to the new model CBLNF(ɛ). This will be used as a stopping criteria for the algorithm introduced below. Corollary 3.2 (Optimality Condition) Let (x,y ) be the optimal solution of problem CBLNF(ɛ). If. x a {0} [ɛ,λ] a A, (8) then x solves the fixed charge network flow problem optimally. Proof. It follows from Theorem 2.1, as ɛ 1 ɛ 2 implies that ϕ(ɛ 1 ) ϕ(ɛ 2 ). Let x denote an optimal solution to the FCNF and f = f(x ) represent its objective function value. We can make the following observations: The objective function of CBLNF(ɛ) is concave for any ɛ > 0, as its Hessian is negative (semi)definit. Hence, CBLN F(ɛ) is NP-hard to solve [16]. ϕ(ɛ) is a lower bound on f for all ɛ > 0. Any feasible solution ˆx to CBLNF(ɛ) is also feasible to the fixed charge network flow problem; however, the objective function value may differ. Hence, by evaluating f(ˆx), we obtain also an upper bound on f. With the optimality conditions (8) on hand, we can state an exact (continuous) algorithm for the fixed charge network flow problem; along the lines of Procedure 1 in [14]. This algorithm is given in Algorithm 3.1. Basically, a series of bilinear problems (Step 2) is solved to (global) optimality until the stopping criteria is met (Step 6). Corollary 3.3 Algorithm CBA solves the fixed charge network flow problem in a finite number of iterations.

5 Algorithm 3.1 Continuous BiLinear Algorithm (CBL) Input: Matrix B, vectors a,b, λ; parameters α, β (0, 1) Output: Optimal x and parameter ɛ 1: // Initialize y = 1, ɛ = α λ 2: // Solve continuous bilinear problem solve CBLNF(ɛ); let x be its optimal solution 3: // Check stopping criteria (8) if x a (0, ɛ) then 4: // Update ɛ a ɛ a β ɛ a 5: // Start from the beginning with an updated ɛ GoTo Step 2 else 6: // Optimal solution found return x and ɛ Biliner Model for FCNFP 5 Proof. The correctness of the algorithm follows from Theorem 3.1 together with Corollary 3.2. In the worst case, ɛ has to be updated in Step 4 of Algorithm CBA until ɛ a δ for each a A. Hence, for each arc a A, an upper bound L = l + 1 on the number of iterations is given by (the smallest) l Z + such that β l αλ δ; it is L + 1 as αλ might be already less than or equal to δ and one iterations is needed to check the stopping criteria. An upper bound on the number of iterations is then given by 1 + a A max { log β ( δ αλ a ) },0 Should we write more details why this is an upper bound of iterations? I added some explanations before the formula.. 4 Computational Tests In this section, we discuss some computational tests with Algorithm CBL, developed in Section 3. The aim of these computational tests is to validate the proposed algorithm and to see its performance with respect to the convergence rate. For this purposes, it is sufficient to use a general (global) nonlinear solver rather than a specialized global solver for minimizing concave linearly constrained quadratic programs. Let us briefly review some global algorithms for nonconvex quadratic programming. A finite branch-and-bound algorithms using semidefinite programming is given by Burer and Vandenbussche, [3]. A review of global opti-

6 6 S. Rebennack, A. Nahapetyan, P. M. Pardalos mization approaches for indefinite quadratic programming is given by Pardalos [15] whereas a general overview with applications is given by Floudas and Visweswaran [7]. Specialized algorithms for concave, linear constrained quadratic programming have been developed by Konno [12], Bomze and Danninger [2] as well as Chinchuluun et al. [4]. A parallel algorithm was introduced by Phillips and Rosen [17]. In order to reduce the number of iterations of the Continuous Bilinear Algorithm 3.1, in step 2 we use a heuristic called Adaptive Dynamic Cost Updating Procedure (ADCUP) described in [14]. Its basic idea is to iteratively fix y or x variables in formulation CBLNF(ɛ) and solve the resulting linear programs (LP) by dynamically updating the value of ɛ until a feasible solution is reached. ADCUP is a very fast converging algorithm and usually provides a feasible solution with objective function value very close to the optimal one. Next, we employ the feasible solution as a starting point and solve CBLN F(ɛ) via the Branch-And-Reduce Optimization Navigator (BARON) version of Sahinidis et al. [18, 20, 19]. Algorithm 3.1 is modeled in the General Algebraic Modeling System (GAMS) version The computational framework is Redhat version 5, running on a Dell power edge 2600 with two Pentium 4 3.2Ghz with 1MB cache processors and 6 GB of memory. For comparability purposes, only one processor is used for the tests. For the computational tests, the data for the network was uniformly random generated. For each arc a A, we have the variable cost c a [1,5] and fixed cost s a [50,100]. The supply and demand b i [30,50] for all nodes i V. All tested graphs have two supply and two demand nodes. The upper bound on the flow was set to (two times) the total demand; i.e. λ = i V b i. Parameters, α and β were set to 0.2 and 0.3, respectively. Why we are setting λ = i V b i? It seams that the upper bound will never be reached... Should we try to use some smaller numbers that λ plays a role? This is a very good comment. I actually do not remember why I decided to define λ in this way. However, I tried different values for λ (especially λ = minimum demand / supply). Unfortunately, the running time for the algorithm exploded totally. Much more time is spent to close the gaps for the bilinear programs. But you are right that CPLEX spent also more time on it. For one instance with 15 node and 100 arcs, we have the following result for CBA. The values for the large λ are given in brackets. CPU time: (116.53) seconds, innerloopcnt = 41 (43), outerloopcnt = 7 (2). CPLEX: 0.33 (0.28) seconds; but it needs much more iterations. Hence, I suggest leaving it the way we have it right now. For each network size, 10 randomly generated instances were solved and computational results are presented in Table 1. The number of inner iterations (# inner iter.) is the number of LP s solved by ACUP heuristic procedure while the number of outer iterations (# outer iter.) reports how often CBLNF(ɛ) is solved until the global optimal is reached. The number of outer iterations is crucial for the performance of the algorithm as the computation of the global

7 Biliner Model for FCNFP 7 Table 1. Computational results for different sizes of the networks. Shown are the results for 10 randomly generated instances per problem size G # inner iter. # outer iter. n m min max average σ min max average σ instance is infeasible 2 4 instances are infeasible optimum of CBLNF(ɛ) is the bottleneck of CBA algorithm. The average number of outer iterations was quite low, ranging from 1.0 to 3.8. There were only 8 instances out of 100 tested cases where the number of outer iterations was larger than 4. Typically, the heuristics is called only once or twice after the first bilinear model has been solved. Hence, the largest fraction of the inner iterations take the first call of the heuristics. (I did not understand the last sentence...i added one sentence before. ) Typically, after CBLN F(ɛ) is solved and an updated ɛ is passed to the heuristics along with the currently best solution, only two iterations are needed during the heuristics, i.e. two times y variables have to be updated after the first LP has been solved. In order to check if the global optimum was reached after termination of the algorithm, we solved a mixed-integer programming formulation for the fixed charged network flow problem with ILOG CPLEX. The objective function values (and even the solutions) are the same for all tested instances. However, we have to mention that CPLEX requires only a very small fraction of the CPU time used by algorithm CBL to solve the tested instances. Algorithm CBL has two parameters α and β to tune. Parameter α is not very crucial for the convergence, as it effects only the initial value of ɛ. However, parameter β has a significant impact on the inner as well as the outer number of iterations. This effect is shown in Figure 1. As expected, if β is closer to one, then more inner as well as outer iterations are needed; for value 0.99, 1023 LP s and 205 bilinear models have to be solved. In contrary, if the value is closer to 0, then less iterations are required. However, for the tested instances, there was no difference in the number of inner as well as outer iterations for the values 0.01, 0.001, and This is also expected as at some point, each ɛ a is updated only once during the

8 8 S. Rebennack, A. Nahapetyan, P. M. Pardalos algorithm. Nevertheless, choosing parameter β too small leads to numerical stability problems, as the cost in model CBLNF(ɛ) depends on 1 ɛ # inner iter. # outer iter β Fig. 1. Computational tests for different values of parameter β for a graph with 15 nodes and 200 arcs 5 Conclusion Acknowledgement. We would like to thank... References 1. Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Immanuel M. Bomze and Gabriele Danninger. A Global Optimization Algorithm for Concave Quadratic Programming Problems. SIAM Journal on Optimization, 3(4): , 1993.

9 Biliner Model for FCNFP 9 3. Samuel Burer and Dieter Vandenbussche. A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Mathematical Programming, 113: , Altannar Chinchuluun, Panos M. Pardalos, and Rentsen Enkhbat. Global minimization algorithms for concave quadratic programming problems. Optimization, 54(6): , Reinhard Diestel. Graph Theory. Springer, New-York, Electronic Edition B. Eksioglu, S. Duni-Eksioglu, and Panos. M. Pardalos. Equilibrium Problems and Variational Models, chapter Solving Large Scale Fixed Charge Network Flow Problems. Kluwer Academic Publishers, C. Floudas and V. Visweswaran. Handbook of Global Optimization, chapter Quadratic optimization, pages Kluwer Academic Publishers, R. Horst and P. Pardalos (eds.). 8. G. M. Guisewite and P. M. Pardalos. Minimum Concave-Cost Network Flow Problems: Applications, Complexity, and Algorithms. Annals of Operations Research, 25:75 100, D. Kim and P. M. Pardalos. Dynamic Slope Scaling and Trust Interval Techniques for Solving Concave Piecewise Linear Network Flow Problems. Networks, 35(3):216222, D. Kim and P.M. Pardalos. A solution approach to the fixed charge network flow problem using a dynamic slope scaling procedure. Operations Research Letters, 24: , Dukwon Kim and Panos M. Pardalos Xinyan Pan. An Enhanced Dynamic Slope Scaling Procedure with Tabu Scheme for Fixed Charge Network Flow Problems. Computational Economics, 2 3: , Hiroshi Konno. A cutting plane algorithm for solving bilinear programs. Mathematical Programming, 11:14 27, Artyom Nahapetyan and Panos M. Pardalos. A Bilinear Relaxation Based Algorithm for Concave Piecewise Linear Network Flow Problem. Journal of Industrial and Management Optimization, 3:71 85, Artyom Nahapetyan and Panos M. Pardalos. Adaptive Dynamic Cost Updating Procedure for Solving Fixed Charge Network Flow Problems. Computational Optimization and Applications, 39:37 50, Panos M. Pardalos. Global optimization algorithms for linearly constrained indefinite quadratic problems. Computers & Mathematics with Applications, 21(6-7):87 97, Panos M. Pardalos and Stephen A. Vavasis. Quadratic programming with one negative eigenvalue is np-hard. Journal of Global Optimization, 1:15 22, A. T. Phillips and J. B. Rosen. A parallel algorithm for constrained concave quadratic global minimization. Mathematical Programming, 42(1-4): , Hong S. Ryoo and Nikolaos V. Sahinidis. Global optimization of non-convex NLPs and MINLPs with application in process design. Computers & Chemical Engineering, 19(5): , Hong S. Ryoo and Nikolaos V. Sahinidis. A branch-and-reduce approach to global optimization. Journal of Global Optimization, 8(2): , March Nikolaos V. Sahinidis. BARON: A general purpose global optimization software package. Journal of Global Optimization, 8(2): , March 1996.

10 10 S. Rebennack, A. Nahapetyan, P. M. Pardalos 21. Hoang Tuy. Strong Polynomial Time Solvability of Minimum Concave Cost Network Flow Problem. ACTA MATHEMATICA VIETNAMICA, 25(2): , 2000.

Bilinear Programming

Bilinear Programming Artyom G. Nahapetyan Center for Applied Optimization Industrial and Systems Engineering Department University of Florida Gainesville, Florida 32611-6595 Email address: artyom@ufl.edu