Uncapacitated Facility Location Models with Concave Operating Costs
3. Multi-dimensional generalizations
154 Yaroslav D. Sergeyev
An Introduction to Lipschitz Global Optimization 155
-5 -4 -3 -2 -1 0 1 2 3 4 5
DLT-LI DGE-LI DKC-LI DLT DGE DKC LT-LI GE-LI PKC-LI LT GE PKC 0 2 4
421 163
41 43
37 33
78 26 132
35 31 19
Figure 1: Graph of the function number 38 from [15] and trial points generated by the 12 methods tested.
in solving applied problems. In these algorithms, the search hyperinterval is adaptively parti-tioned into smaller hyperintervals and the objective function is evaluated only at two vertices corresponding to the main diagonal of the generated hyperintervals. It is demonstrated that the traditional diagonal partition strategies do not fulfil the requirements of computational efficiency because of executing many redundant evaluations of the objective function.
A new adaptive diagonal partition strategy that allows one to avoid such computational redundancy is described. Some powerful multidimensional global optimization algorithms based on the new strategy are introduced. Results of extensive numerical experiments per-formed on the GKLS-generator (see [2]) to test the proposed methods demonstrate their ad-vantages with respect to traditional diagonal algorithms in terms of both number of trials of the objective function and qualitative analysis of the search domain, which is characterized by the number of generated hyperintervals. A number of directions of possible developments is discussed briefly. Among them we can mention problems with multiextremal partially gen-erated constraints, the usage of parallel non-redundant computations, and theoretical results on the possible speed-up.
156 Yaroslav D. Sergeyev
References
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[2] M. Gaviano, D. E. Kvasov, D. Lera, and Ya. D. Sergeyev, “Algorithm 829: Software for generation of classes of test functions with known local and global minima for global optimization”,ACM Transactions on Mathematical Software,29(4):469–480, 2003.
[3] S. Yu. Gorodetsky, “Paraboloid triangulation methods in solving multiextremal optimization problems with constraints for a class of functions with Lipschitz directional derivatives”,Vestnik of Lobachevsky State Univer-sity of Nizhni Novgorod,1(1):144–155, 2012. In Russian.
[4] D. R. Jones, C. D. Perttunen, and B. E. Stuckman, “Lipschitzian optimization without the Lipschitz constant”, Journal of Optimization Theory and Applications,79: 157–181, 1993.
[5] R. Horst and P. M. Pardalos,Handbook of Global Optimization, Kluwer, Dordrecht, 1995.
[6] D. E. Kvasov, D. Menniti, A. Pinnarelli, Ya. D. Sergeyev, and N. Sorrentino, “Tuning fuzzy power-system sta-bilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions”, Electric Power Systems Research,78(7):1217–1229, 2008.
[7] D. E. Kvasov and Ya. D. Sergeyev, “A univariate global search working with a set of Lipschitz constants for the first derivative”,Optimization Letters,3(2):303–318, 2009.
[8] D. E. Kvasov and Ya. D. Sergeyev, “Lipschitz gradients for global optimization in a one-point-based partition-ing scheme”,Journal of Computational and Applied Mathematics,236(16):4042–4054, 2012.
[9] D. E. Kvasov and Ya. D. Sergeyev, “Univariate geometric Lipschitz global optimization algorithms”,Numerical Algebra, Control and Optimization,2(1):69–90, 2012.
[10] D. Lera and Ya. D. Sergeyev, “Lipschitz and Holder global optimization using space-filling curves”,Applied Numerical Mathematics,60(1-2):115–129, 2010.
[11] D. Lera and Ya. D. Sergeyev, “An information global minimization algorithm using the local improvement technique”,Journal of Global Optimization,48(1):99–112, 2010.
[12] D. Lera and Ya. D. Sergeyev, “Acceleration of univariate global optimization algorithms working with Lips-chitz functions and LipsLips-chitz first derivatives”,SIAM Journal on Optimization,23(1):508–529, 2013.
[13] R. Paulaviˇcius and J. Žilinskas,Simplicial Global Optimization, Springer, New York, 2014.
[14] J. D. Pintér,Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications, Kluwer, Dordrecht, 1996.
[15] J. D. Pintér, Global optimization: software, test problems, and applications, In P. M. Pardalos and H. E.
Romeijn, editors,Handbook of Global Optimization, volume 2, pages 515–569. Kluwer Academic Publishers, Dordrecht, 2002.
[16] Ya. D. Sergeyev, “An information global optimization algorithm with local tuning”, SIAM Journal on Opti-mization,5(4):858–870, 1995.
[17] Ya. D. Sergeyev, “Global one-dimensional optimization using smooth auxiliary functions”,Mathematical Pro-gramming,81(1):127–146, 1998.
[18] Ya. D. Sergeyev, P. Pugliese, D. Famularo, “Index information algorithm with local tuning for solving mul-tidimensional global optimization problems with multiextremal constraints”, Mathematical Programming, 96(3):489–512, 2003.
[19] Ya. D. Sergeyev and D. E. Kvasov, “Global search based on efficient diagonal partitions and a set of Lipschitz constants”,SIAM Journal on Optimization,16(3):910–937, 2006.
[20] Ya. D. Sergeyev and D. E. Kvasov,Diagonal Global Optimization Methods, FizMatLit, Moscow, 2008. In Russian.
[21] Ya. D. Sergeyev and D. E. Kvasov, “Lipschitz global optimization”, in J. J. Cochran et al., (Eds.),Wiley Encyclo-pedia of Operations Research and Management Science (in 8 volumes), John Wiley & Sons, New York,4:2812–2828, 2011.
[22] Ya. D. Sergeyev, R. G. Strongin, and D. Lera,Introduction to Global Optimization Exploiting Space-Filling Curves, Springer, New York, 2013.
[23] R. G. Strongin, Numerical Methods in Multi-Extremal Problems (Information-Statistical Algorithms), Nauka, Moscow, 1978. In Russian.
[24] R. G. Strongin and Ya. D. Sergeyev,Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms, Kluwer, Dordrecht, 2000.
[25] A. A. Zhigljavsky,Theory of Global Random Search, Kluwer, Dordrecht, 1991.
[26] A. A. Zhigljavsky and A. Žilinskas,Stochastic Global Optimization, Springer, New York, 2008.
Proceedings of MAGO 2014, pp. 157 – 160.
Solving a Huff-like Stackelberg problem on networks
∗Kristóf Kovács and Boglárka G.-Tóth
Budapest University of Technology and Economics, Hungary, kkovacs@math.bme.hu, bog@math.bme.hu
Abstract This work deals with a Huff-like Stackelberg problem, where the leader facility wants to decide its location so that its profit is maximal after the competitor (the follower) also built its facility. It is assumed that the follower makes a rational decision, maximizing their profit. The inelastic demand is aggregated into the vertices of a graph, and facilities can be located along the edges.
This Stackelberg model is a bi-level problem that makes global solvability extremely hard. Even though the problem is tackled by a Branch and Bound method, so that global optimality is provided.
Keywords: Branch&Bound, Stackelberg problem, facility location, Interval Analysis, DC decomposition, global optimization, bi-level problem
1. Introduction
In competitive facility location the general aim is to locate one or more new facilities for an existing or a newcomer chain maximizing its market share or profit. When competitors are likely to react with their own expansion, the owner has to take that into account. This leads to a bi-level optimization problem, where the optimal location of the first player, theleader, has to be determined depending on the location of the second player, the follower, who de-cides its location with the knowledge of the location of the leader. This problem is called the Stackelberg problem, or the(r, p)-centroid problem whenrleader andpfollower facilities are located.
The underlying location problem depends on many factors starting from the decision space, through properties of the demand till costumer’s choices. In this work static competition with inelastic demand is considered. Demand is concentrated in a discrete set of points, called demand points. Costumers are assumed to follow the probabilistic choice for the facilities, i.e.
they split their demand proportionally to the attraction they feel to the facilities. Attraction of a facility determined by its quality and the distances to it, through a gravitational or logit type model. The objective function to be maximized is the profit obtained by the chain, to be understood as the income due to the market share captured by the chain minus its operational costs. The location space in our model is a network, with the vertices being demand points and the facilities located on its edges.
Many papers dealing with Stackelberg problems assume binary costumer choice, that al-lows to narrow the solution candidates to a discrete set of points, transforming it a combi-natorial optimization problem [5], or already offering only a discrete set for the locations [2].
Continous problems on the plane with analogous objectives has been addressed by [1, 3] of-fering heuristic methods. In [4] a similar problem was proposed and solved reliably, although on a planar space for the maximization of the market share.
∗This research has been supported by the Ministry of Economy and Competitiveness of Spain under the research project ECO2011-24927, in part financed by the European Regional Development Fund (ERDF), and the Fundación Séneca (The Agency of Science and Technology of the Region of Murcia) under the research project 15254/PI/10 and by Junta de Andalucía (P11-TIC7176.)
158 Kristóf Kovács and Boglárka G.-Tóth
2. Problem Formulation
Let us now introduce formally the problem under consideration. Let us given a network N = (V, E), where each eij ∈ E refers to the edge with end pointsai andaj ∈ V denoting its length by leij. It allows us to talk about points in an edge: edgeeij is identified with the interval[0, leij],and we thus denote anyx ∈[0, leij]by the point in the edgeeij at distancex ofai and distanceleij−xofaj.
The demand is concentrated at the vertices of N, where each a ∈ V has associated its buying powerωa. The functionda(x)gives the distance between demand pointaand facility x. Assuming thatxis located on edgeeij, it is calculated as follows
da(x) = min{x+d(ai, a), leij−x+d(aj, a)} whered(ai, a)is the length of the shortest path from demand pointaitoa.
In a competitive environment it is usual to assume that both firms have preexisting facilities.
Considering m existing facilities, we refer to the leader’s facilities as xi i = 1. . . k and to the follower owed ones as xj j = k+ 1. . . m. Every facility is given its fixed qualityqi for i = 1, . . . m. The new facility of the leader and the follower is denoted by the indexlandf, respectively, thus the location of the leader’s new facility is denoted by xl its quality by ql, similarly we havexf andqf for the follower.
The market share captured by the leader (with new facility atxl) after the follower locates atxf is
Ml(xl, xf) =X
a∈V
ωa ql/ϕa(da(xl)) +Pk
j=1qj/ϕa(da(xj)) ql/ϕa(da(xl)) +qf/ϕa(da(xf)) +Pm
j=1qj/ϕa(da(xj)), while the market share captured by the follower is
Mf(xl, xf) =X
a∈V
ωa qf/ϕa(da(xf)) +Pm
j=k+1qj/ϕa(da(xj)) ql/ϕa(da(xl)) +qf/ϕa(da(xf)) +Pm
j=1qj/ϕa(da(xj)).
The functionϕis a positive nondecreasing function on non negative values. The usual choice is ϕ(t) = tλ, where λ = 2 gives the so called gravitational model. Both firms are assumed to have renting and/or operational costs depending on the facility’s proximity to demand points. Locations near highly populated areas likely to be more expensive, therefore
Gl(xl) =X
a∈V
ωa ql
ψa(da(xl)), Gf(xf) =X
a∈V
ωa qf ψa(da(xf))
are considered, whereψ is a similar function toϕ, though the two should not be the same.
Thus the profit of the two firms are
Fl(xl, xf) =Ml(xl, xf)−Gl(xl), Ff(xl, xf) =Mf(xl, xf)−Gf(xl).
Using the previous functions we can formulate the objective of the leader problem as maxxl∈E Fl(xl, x∗f)
s.t. x∗f = argmax
xf∈E
Ff(xl, xf) Naturally the objective of the follower problem for a givenxlis
xmaxf∈EFf(xl, xf).
Solving a Huff-like Stackelberg problem on networks 159