Uncapacitated Facility Location Models with Concave Operating Costs
3. Algorithms for UFLPCOC
Our exact solution approach is mainly based on obtaining efficient lower and upper bounds for U F LP COC. In Section 3.1, we develop an MIP, for which the objective function value of its optimal solution provides a lower bound forU F LP COC. In Section 3.2, we present a heuristic, which is based on a neighborhood search over the location set from the solution to the MIP. It is used to find an upper bound forU F LP COC. The exact approach presented in 3.3 is based on successive lower and upper bound improvements forU F LP COC.
3.1 A lower bound for
U F LP COCIn order to determine a lower bound forU F LP COC, the concave functionF(Λj)is replaced by a linear functionFL(Λj)such thatFL(Λj) ≤ F(Λj)forΛminj ≤ Λj ≤ Λmaxj ∀j ∈M. Since F(Λj)is concave, every chord lies below the graph ofF. Thus, the condition required above is satisfied if a chord is taken asFL.
Thus, being FL(Λj) = ajΛj +bj a chord of F(Λj), j = 1, . . . , M and by replacingF(Λj) with FL(Λj) inU F LP COC will result the following mixed integer program, which we call LBM IP:
151
minX
j∈M
(fj+bj)xj+X
i∈N
X
j∈M
λi(aj+cij)yij s.t.
(2)−(3)
xj ∈ {0, 1}, yij ≥0, ∀i∈N, j ∈M.
3.2 An upper bound for
U F LP COCWe note that any feasible location vectorxincluding the one produced by solvingLBM IP generates a feasible solution toU F LP COC. This is achieved by first defining the assignment vectory(x)using the assumption that customers are assigned to open facilities level by level in an increasing order of shipping cost. Then, the resulting value of the objective function (1), provides an upper bound forU F LP COC.
Denote bySxthe set of facility locations, which are open given location vectorx. To find an improved upper bound, the heuristic uses a descent approach in a neighborhood search for Sx−the location set produced by solvingLBM IP. The distance-kneighborhood ofS ⊆M is defined as
Nk(S) ={S′⊆M :|S−S′|+|S′−S| ≤k}
i.e.,S′ is in the distance-kneighborhood ofS if the number of non-overlapping elements in the two sets does not exceedk.
Once the neighborhood is well defined, the descent algorithm is straightforward: use the solution toLBM IP as a starting subsetSx; evaluate the change in the value of the objective function (1) for all the subsets in the neighborhood; if an improved subset exists in the neigh-borhood, move the search to the best vector in the neighborhood. Repeat the process with the new subset until no improved vector exists in the neighborhood. The last subset is the solution.
Denote bySx(the set of facility locations under vectorx) the solution subset to the descent approach, and lety(x)be the assignment vector where customers are assigned to least access cost open facilities atSx. The resulting value of the objective function (1) is our new upper bound and the solution to the descent approachxandy(x)is the solution to our heuristic.
3.3 An exact approach for
U F LP COCThe exact approach presented is based on successive improvements on lower and upper bounds on U F LP COC in each step of the algorithm. In this approach, we first find initial lower and upper bounds forU F LP COC by solving the heuristic proposed in Section 3.2. In the next step, we find an improved lower bound by solving an improvedLBM IP. An im-provedLBM IP isLBM IP with additional cuts, which exclude the pre-examined location vectors from the feasible region (at the first step LBM IP is solved without any cuts), and with a tighten objective function, which is found by solving2M integer programs. After find-ing an improved lower bound, the location set produced by solvfind-ing the improvedLBM IP is used as a starting location set in a neighborhood search to find an improved upper bound using the descent approach named in Section 3.2. The procedure continues until the lower bound is greater than the upper bound, so that it is evident that the unexamined location sets are unable to improve the current upper bound.
4. Conclusions
An efficient approach for solving uncapacitated facility location models with concave operat-ing costs is presented. Preliminary computational results are beoperat-ing carried out on a PC Intelr
152 Robert Aboolian, Emilio Carrizosa and Vanesa Guerrero
CoreTMi7-2600K, 16GB of RAM. We use the optimization engine CPLEX v12.4 (CPLEX 2012) for solving all optimization problems.
Acknowledgments
This research is funded in part by projects MTM2012-36163 (Ministerio de Economía y Com-petitividad, Spain), P11-FQM-7603 (Junta de Andalucía), both supported by EU ERD funds, and by Fundación Cámara.
References
[1] R. Aboolian, T. Cui and Z.J.M. Shen.An efficient Approach for Solving Reliable Facility Location Models. Addison Wesley, Massachusetts, INFORMS Journal on Computing, 25(4), 720-729, 2012.
[2] G. Cornuéjols, G.L. Nemhauser and L.A. Wolsey The uncapacitated facility location problem, in: Discrete Location Theory, eds. P:B: Mirchandani and R.L. Francis, Wiley, New York, 119-171, 1990.
Proceedings of MAGO 2014, pp. 153 – 156.
An Introduction to Lipschitz Global Optimization
∗Yaroslav D. Sergeyev1,2
1DIMES, University of Calabria, Via P. Bucci, 42C – 87036, Rende (CS), Italy 2Software Department, N. I. Lobachevsky State University, Nizhni Novgorod, Russia yaro@si.dimes.unical.it
Abstract This lecture deals with the global optimization problems where the objective function can be "black box", multiextremal, and possibly non-differentiable. It is also assumed that evaluation of the ob-jective function at a point is a time-consuming operation. Two statements of the problem are taken into consideration: (i) the objective function satisfies the Lipschitz condition; (ii) the gradient of the objective function satisfies the Lipschitz condition. Two cases are considered for both problems: the Lipschitz constant is either known a priori or unknown (in this case it should be estimated). Local tuning on the behavior of the objective function and a new technique, namedlocal improvement, are used in order to accelerate the search. Convergence condition are given and extensive numerical experiments are presented.
Keywords: Lipschitz global optimization, Numerical methods, Partition strategies, Peano-Hilbert space-filling curves
1. Introduction
Global optimization is a thriving branch of applied mathematics and an extensive literature has been dedicated to this field (see, e. g., [1–27]). In this lecture, the global optimization prob-lem of a multidimensional function satisfying the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant is considered:
f∗ =f(x∗) = min
x∈D f(x), (1)
|f(x′)−f(x′′)| ≤Lkx′−x′′k, x′, x′′∈D, (2) whereL,0< L <∞,is called the Lipschitz constant,
D= [a, b] ={x∈RN :a(j)≤x(j)≤b(j)}, (3) andk · kdenotes, usually, the Euclidean norm (however, other norms can be also used). It is supposed that the objective function can be "black box", multiextremal, and non-differentiable.
It is also assumed that evaluation of the objective function at a point is a time-consuming operation. Two statements are considered: the Lipschitz constantLis either known a priori or unknown (in this case it should be estimated).
A particular class of the Lipschitz global optimization problems is also discussed in this lec-ture, namely, the class of problems with differentiable objective functions having the Lipschitz gradientf′(x), i.e.,
kf′(x′)−f′(x′′)k ≤Kkx′−x′′k, x′, x′′∈D, 0< K <∞. (4) Again, similarly to the situation regarding the constantL, the constantKcan be either known a priori or unknown and, therefore, should be estimated in a way.
∗This research was partially supported by the INdAM–GNCS 2014 Research Project of the Italian National Group for Scientific Computation of the National Institute for Advanced Mathematics “F. Severi”.
154 Yaroslav D. Sergeyev