Relevant bibliographies by topics / Bilevel program

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Bilevel program'

Author: Grafiati
Published: 14 March 2023

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Journal articles on the topic "Bilevel program"

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Xu, Yi, and Lili Han. "Quadratic Program on a Structured Nonconvex Set." Mathematical Problems in Engineering 2020 (March 12, 2020): 1–6. http://dx.doi.org/10.1155/2020/4318186.

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In this paper, we focus on a special nonconvex quadratic program whose feasible set is a structured nonconvex set. To find an effective method to solve this nonconvex program, we construct a bilevel program, where the low-level program is a convex program while the upper-level program is a small-scale nonconvex program. Utilizing some properties of the bilevel program, we propose a new algorithm to solve this special quadratic program. Finally, numerical results show that our new method is effective and efficient.
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Mitsos, Alexander, Panayiotis Lemonidis, and Paul I. Barton. "Global solution of bilevel programs with a nonconvex inner program." Journal of Global Optimization 42, no. 4 (December 5, 2007): 475–513. http://dx.doi.org/10.1007/s10898-007-9260-z.

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Aboussoror, Abdelmalek, Hicham Babahadda, and Abdelatif Mansouri. "Bilevel programs with extremal value function: global optimality." International Journal of Mathematics and Mathematical Sciences 2005, no. 3 (2005): 419–35. http://dx.doi.org/10.1155/ijmms.2005.419.

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For a bilevel program with extremal value function, a necessary and sufficient condition for global optimality is given, which reduces the bilevel program to amax-minproblem with linked constraints. Also, for the case where the extremal value function is polyhedral, this optimality condition gives the possibility of a resolution via a maximization problem of a polyhedral convex function over a convex set. Finally, this case is completed by an algorithm.
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Liao, Jiagen, and Zhongping Wan. "On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problems on Riemannian manifolds." Filomat 36, no. 11 (2022): 3609–24. http://dx.doi.org/10.2298/fil2211609l.

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Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints (MPCC). First, we present KKT reformulation of the bilevel optimization problems on Riemannian manifolds. Moreover, we show that global optimal solutions of theMPCCcorrespond to global optimal solutions of the bilevel problem on the Riemannian manifolds provided the lower level convex problem satisfies the Slater?s constraint qualification. But the relationship between the local solutions of the bilevel problem and its corresponding MPCC is incomplete equivalent. We then also show by examples that these correspondences can fail if the Slater?s constraint qualification fails to hold at lower-level convex problem. In addition,M- and C-type optimality conditions for the bilevel problem on Riemannian manifolds are given.
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Lin, Gui-Hua, Mengwei Xu, and Jane J. Ye. "On solving simple bilevel programs with a nonconvex lower level program." Mathematical Programming 144, no. 1-2 (January 30, 2013): 277–305. http://dx.doi.org/10.1007/s10107-013-0633-4.

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Li, Hecheng, and Zhicang Wang. "An Evolutionary Algorithm Using Parameter Space Searching for Interval Linear Fractional Bilevel Programming Problems." International Journal of Pattern Recognition and Artificial Intelligence 30, no. 04 (April 12, 2016): 1659011. http://dx.doi.org/10.1142/s0218001416590114.

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Interval programming is one of main approaches treating imprecise and uncertain elements involved in optimization problems. In this paper, an interval linear fractional bilevel program is considered, which is characterized in that both objective coefficients and the right-hand side vector are interval numbers, and an evolutionary algorithm (EA) is proposed to solve the problem. First, the interval parameter space of the follower’s problem is taken as the search space of the proposed EA. For each individual, one can evaluate it by dealing with a simplified interval bilevel program in which only the leader’s objective involves interval parameters. In addition, the optimality conditions of linear fractional programs are applied to convert and solve the simplified problem. Finally, some computational examples were solved and the results show that the proposed algorithm is efficient and robust.
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Li, Hecheng, and Lei Fang. "An Evolutionary Algorithm Using Duality-Base-Enumerating Scheme for Interval Linear Bilevel Programming Problems." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/737515.

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Interval bilevel programming problem is hard to solve due to its hierarchical structure as well as the uncertainty of coefficients. This paper is focused on a class of interval linear bilevel programming problems, and an evolutionary algorithm based on duality bases is proposed. Firstly, the objective coefficients of the lower level and the right-hand-side vector are uniformly encoded as individuals, and the relative intervals are taken as the search space. Secondly, for each encoded individual, based on the duality theorem, the original problem is transformed into a single level program simply involving one nonlinear equality constraint. Further, by enumerating duality bases, this nonlinear equality is deleted, and the single level program is converted into several linear programs. Finally, each individual can be evaluated by solving these linear programs. The computational results of 7 examples show that the algorithm is feasible and robust.
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ALVIANO, MARIO, and RAFAEL PEÑALOZA. "Fuzzy answer sets approximations." Theory and Practice of Logic Programming 13, no. 4-5 (July 2013): 753–67. http://dx.doi.org/10.1017/s1471068413000471.

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AbstractFuzzy answer set programming (FASP) is a recent formalism for knowledge representation that enriches the declarativity of answer set programming by allowing propositions to be graded. To now, no implementations of FASP solvers are available and all current proposals are based on compilations of logic programs into different paradigms, like mixed integer programs or bilevel programs. These approaches introduce many auxiliary variables which might affect the performance of a solver negatively. To limit this downside, operators for approximating fuzzy answer sets can be introduced: Given a FASP program, these operators compute lower and upper bounds for all atoms in the program such that all answer sets are between these bounds. This paper analyzes several operators of this kind which are based on linear programming, fuzzy unfounded sets and source pointers. Furthermore, the paper reports on a prototypical implementation, also describing strategies for avoiding computations of these operators when they are guaranteed to not improve current bounds. The operators and their implementation can be used to obtain more constrained mixed integer or bilevel programs, or even for providing a basis for implementing a native FASP solver. Interestingly, the semantics of relevant classes of programs with unique answer sets, like positive programs and programs with stratified negation, can be already computed by the prototype without the need for an external tool.
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Naebi Toutounchi, A., SJ Seyed Shenava, SS Taheri, and H. Shayeghi. "MPEC approach for solving preventive maintenance scheduling of power units in a market environment." Transactions of the Institute of Measurement and Control 40, no. 2 (July 27, 2016): 436–45. http://dx.doi.org/10.1177/0142331216659336.

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Yearly preventive maintenance scheduling of generating units in a restructured power system is one of the most important problems that have to be solved in modern power systems. In this paper, a bilevel approach is used for modelling of the preventive maintenance scheduling problem. The upper level of this bilevel problem represents the revenue function of power units owned by a generation company (GENCO), whereas the lower-level problem represents the market-clearing process and is usually called the independent system operator (ISO) level. This bilevel problem is then formulated as a mathematical program with equilibrium constraints (MPEC) using the primal–dual theorem, which converts the problem into a single-level mixed-integer non-linear optimization problem that can be solved using programming software. Various case studies are conducted using the IEEE reliability test system (RTS) and the obtained results are compared.
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Scaparra, Maria P., and Richard L. Church. "A bilevel mixed-integer program for critical infrastructure protection planning." Computers & Operations Research 35, no. 6 (June 2008): 1905–23. http://dx.doi.org/10.1016/j.cor.2006.09.019.

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Dissertations / Theses on the topic "Bilevel program"

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savelli, iacopo. "Towards the Integration of Electricity Markets: System-wide and Local Solutions." Doctoral thesis, Università di Siena, 2019. http://hdl.handle.net/11365/1068717.

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The creation of an efficient, sustainable, and resilient energy system is of paramount importance in the European agenda. To reach this goal, the integration of the existing structures, and the proposal of new design paradigms are key factors. The contribution of the thesis goes along this line, by proposing two solutions to foster the electricity market integration. At system-wide level, a novel market clearing approach is proposed, to deal with some of the main issues of the European electricity market. At local level, a novel community microgrid market model is developed, where people can pool their resources, trade in a local electricity market, and provide ancillary services. The European market clearing problem is characterized by a set of heterogeneous orders and rules that force the implementation of heuristic and iterative solving methods. In particular, curtailable block orders and the uniform purchase price pose serious difficulties. A block order spans over multiple hours, and can be either fully accepted or fully rejected. The uniform purchase price prescribes that all consumers pay a common price in all the zones, while producers receive zonal prices, which can differ from one zone to another. The uniform purchase price scheme leads to a non-linear optimization problem involving both primal and dual variables, whereas block orders introduce multi-temporal constraints and binary variables into the problem. The market clearing problem in the presence of both the uniform purchase price and block orders is still an open issue in the European context. To deal with this integration problem, a novel, non-iterative, and heuristic-free approach is proposed, which results in a mixed-integer linear program, built starting from a non-linear integer bilevel program. At local level, the increasing share of renewable energy sources, and the availability of storage systems in distribution grids, pave the way to new market designs that favor the local usage of energy. Community microgrids fit in this context. A community microgrid is a collection of entities that pool their resources to achieve an efficient use of their assets. To foster the integration of entities at local level, a novel market model for community microgrid is proposed. By using the community, participants can match their demand and supply through an internal local market with a significant reduction of the exchanges with the grid. As a consequence, each participant can benefit from a reduction of its energy costs, a drop of the energy peak, and can effectively provide ancillary services to the grid. The proposed community model ensures that no entity is penalized by participating in the community. This requirement is termed Pareto superior condition. The proposed approach is structured as a bilevel model, which is then recast as a single mathematical program by using primal and dual relations. Numerical results and sensitivity analyses are reported to show the effectiveness of the two proposed approaches.
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Claus, Matthias [Verfasser], and Rüdiger [Akademischer Betreuer] Schultz. "Advancing stability analysis of mean-risk stochastic programs : Bilevel and two-stage models / Matthias Claus ; Betreuer: Rüdiger Schultz." Duisburg, 2016. http://d-nb.info/1119705525/34.

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Henkel, Charlotte [Verfasser], Rüdiger [Akademischer Betreuer] Schultz, and René [Akademischer Betreuer] Henrion. "An algorithm for the global resolution of linear stochastic bilevel programs / Charlotte Henkel. Gutachter: René Henrion. Betreuer: Rüdiger Schultz." Duisburg, 2014. http://d-nb.info/1055906975/34.

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Heß, Maximilian [Verfasser], and Simone [Akademischer Betreuer] Göttlich. "An enumerative method for convex programs with linear complementarity constraints and application to the bilevel problem of a forecast model for high complexity products / Maximilian Heß ; Betreuer: Simone Göttlich." Mannheim : Universitätsbibliothek Mannheim, 2017. http://d-nb.info/1153339021/34.

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Hellman, Fredrik. "Towards the Solution of Large-Scale and Stochastic Traffic Network Design Problems." Thesis, Uppsala University, Department of Information Technology, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-130013.

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This thesis investigates the second-best toll pricing and capacity expansion problems when stated as mathematical programs with equilibrium constraints (MPEC). Three main questions are rised: First, whether conventional descent methods give sufficiently good solutions, or whether global solution methods are to prefer. Second, how the performance of the considered solution methods scale with network size. Third, how a discretized stochastic mathematical program with equilibrium constraints (SMPEC) formulation of a stochastic network design problem can be practically solved. An attempt to answer these questions is done through a series ofnumerical experiments.

The traffic system is modeled using the Wardrop’s principle for user behavior, separable cost functions of BPR- and TU71-type. Also elastic demand is considered for some problem instances.

Two already developed method approaches are considered: implicit programming and a cutting constraint algorithm. For the implicit programming approach, several methods—both local and global—are applied and for the traffic assignment problem an implementation of the disaggregate simplicial decomposition (DSD) method is used. Regarding the first question concerning local and global methods, our results don’t give a clear answer.

The results from numerical experiments of both approaches on networks of different sizes shows that the implicit programming approach has potential to solve large-scale problems, while the cutting constraint algorithm scales worse with network size.

Also for the stochastic extension of the network design problem, the numerical experiments indicate that implicit programming is a good approach to the problem.

Further, a number of theorems providing sufficient conditions for strong regularity of the traffic assignment solution mapping for OD connectors and BPR cost functions are given.

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Bai, Kuang. "Directional constraint qualifications and optimality conditions with application to bilevel programs." Thesis, 2020. http://hdl.handle.net/1828/11939.

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The main purpose of this dissertation is to investigate directional constraint qualifications and necessary optimality conditions for nonsmooth set-constrained mathematical programs. First, we study sufficient conditions for metric subregularity of the set-constrained system. We introduce the directional version of the quasi-/pseudo-normality as a sufficient condition for metric subregularity, which is weaker than the classical quasi-/pseudo-normality, respectively. Then we apply our results to complementarity and Karush-Kuhn-Tucker systems. Secondly, we study directional optimality conditions of bilevel programs. It is well-known that the value function reformulation of bilevel programs provides equivalent single-level optimization problems， which are nonsmooth and never satisfy the usual constraint qualifications such as the Mangasarian-Fromovitz constraint qualification (MFCQ). We show that even the first-order sufficient condition for metric subregularity (which is generally weaker than MFCQ) fails at each feasible point of bilevel programs. We introduce the directional Clarke calmness condition and show that under the directional Clarke calmness condition, the directional necessary optimality condition holds. We perform directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional Clarke calmness.
Graduate
2021-07-07
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Shie, Huei Jiun, and 謝慧君. "Existence Theorems of Quasi-Variational Inclusion With Applications to Bilevel Problems and Mathematical Programs With Equilibrium Constraint." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/48088538893121645315.

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碩士
國立彰化師範大學
數學系所
93
In this paper, we establish existence theorems of quasi-variational inclusion, from which we establish existence theorems of mathematical programs with quasi-variational inclusion constraint, bilevel problems, mathematical programs with equilibrium constraint and semi-infinite problems.
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Mucavele, Custodio Estevao. "The economics of smallholder rice producers in Bilene-Macia District, southern Mozambique." Diss., 2001. http://hdl.handle.net/2263/29117.

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Research carried out in several countries has concluded that high yielding varieties (HYV) of rice have tremendous potential compared to traditional rice varieties in alleviating poverty. On the other hand, it is argued that despite increase in yields, high yielding varieties do not benefit the poor farmers because of high costs associated with input acquisition and other operational costs. To further understand the issues related to production and economics of HYV, this study was carried out with its primary objective to assess the profitability of smallholder rice production in Mangol, Bilene district in southern Mozambique. Out of a total population of eighty farmers, sixty farmers participated in this research. Two groups of farmers, one using traditional rice varieties and the other using high yielding varieties were surveyed. The study consisted of field observations; yield measurements and interviews of the farmers using a semi-structured questionnaire. Gross margins were calculated for both traditional input users (TIU) and modem input users (MIU). Results show that an average farmer using high yielding varieties produces 4.4 tons of rice per ha, corresponding to a gross margin of 4 238 000,00 MZM. The highest yield obtained with high yielding varieties was 7.3 tons of rice per ha, producing a gross margin of 10 038 000,00 MZM. On the other hand, an average TIU farmer produces 2.5 tons per ha, realising a gross margin of 3 483 125, 00 MZM. The maximum yield attainable using traditional inputs is 2.9 tons per ha, producing a gross margin of 4 283 125, 00 MZM. The results also show that the lowest yield obtained by farmers using modern inputs was 3.1 tons per ha, producing gross margins of 1 638 000,00 MZM while the lowest attainable when using traditional inputs is 1.7 tons producing gross margins of 1 913 125,00 MZM. The main conclusion of this study was that HYV are more profitable than traditional varieties, both in financial terms, to individual farmer and in economic terms, to the society as a whole. However, due to high production costs, farmers producing HYV must attain yield levels not less than 3.4 tons per ha as yields below that level make little financial incentives and does not motivate farmers to shift from traditional varieties to modern varieties. In order to ensure high yields, farmers must be assisted to carefully implement the recommended agronomic practices because the adoption of high yielding varieties does not ensure profitable yields if the other agronomic and management practices are not observed during the production process. The study also assessed the contribution of agricultural support services such as input supply, credit and extension, on the adoption of new technologies and it was concluded that when agricultural support services are available, farmers can successfully adopt recommended technologies.
Dissertation (M Inst Agrar (Agricultural Economics))--University of Pretoria, 2006.
Agricultural Economics, Extension and Rural Development
unrestricted
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Books on the topic "Bilevel program"

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Ben-Ayed, Omar. Construction of a real world bilevel linear program of the highway network design problem. [Urbana, Ill.]: College of Commerce and Business Administration, University of Illinois at Urbana-Champaign, 1988.

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Book chapters on the topic "Bilevel program"

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Calvete, Herminia I., and Carmen Galé. "A Multiobjective Bilevel Program for Production-Distribution Planning in a Supply Chain." In Lecture Notes in Economics and Mathematical Systems, 155–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-04045-0_13.

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Dempe, Stephan, Vyacheslav Kalashnikov, Gerardo A. Pérez-Valdés, and Nataliya Kalashnykova. "Convex Bilevel Programs." In Energy Systems, 117–31. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-45827-3_4.

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Labbé, Martine, Patrice Marcotte, and Gilles Savard. "On a class of bilevel programs." In Applied Optimization, 183–206. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/978-1-4757-3226-9_10.

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Tuy, Hoang, and Saied Ghannadan. "A New Branch and Bound Method for Bilevel Linear Programs." In Multilevel Optimization: Algorithms and Applications, 231–49. Boston, MA: Springer US, 1998. http://dx.doi.org/10.1007/978-1-4613-0307-7_10.

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DeNegre, S. T., and T. K. Ralphs. "A Branch-and-cut Algorithm for Integer Bilevel Linear Programs." In Operations Research and Cyber-Infrastructure, 65–78. Boston, MA: Springer US, 2009. http://dx.doi.org/10.1007/978-0-387-88843-9_4.

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Gaar, Elisabeth, Jon Lee, Ivana Ljubić, Markus Sinnl, and Kübra Tanınmış. "SOCP-Based Disjunctive Cuts for a Class of Integer Nonlinear Bilevel Programs." In Integer Programming and Combinatorial Optimization, 262–76. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-06901-7_20.

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Visweswaran, V., C. A. Floudas, M. G. Ierapetritou, and E. N. Pistikopoulos. "A Decomposition-Based Global Optimization Approach for Solving Bilevel Linear and Quadratic Programs." In Nonconvex Optimization and Its Applications, 139–62. Boston, MA: Springer US, 1996. http://dx.doi.org/10.1007/978-1-4613-3437-8_10.

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Vicente, Luis N., and Paul H. Calamai. "Geometry and Local Optimality Conditions for Bilevel Programs with Quadratic Strictly Convex Lower Levels." In Nonconvex Optimization and Its Applications, 141–51. Boston, MA: Springer US, 1995. http://dx.doi.org/10.1007/978-1-4613-3557-3_10.

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Ban, Xuegang (Jeff), Michael Ferris, and Henry X. Liu. "Numerical Studies on Reformulation Techniques for Continuous Network Design with Asymmetric User Equilibria." In Innovations in Information Systems for Business Functionality and Operations Management, 138–57. IGI Global, 2012. http://dx.doi.org/10.4018/978-1-4666-0933-4.ch008.

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In this article, we aim to find the most effective reformulation techniques to solve the MPCC (mathematical program with complementarity constraints) model that we proposed recently for continuous network design problems under asymmetric user equilibria. The MPCC model is based on a link-node nonlinear complementarity formulation for asymmetric user equilibria. By applying various reformulation techniques for the lower level nonlinear complementarity, the original bilevel formulation can be converted to a single level nonlinear programming problem. We show that certain reformulations are more effective than others to solve the proposed MPCC model. Recommendations are thus provided on how to choose a reformulation of the continuous network design problem that can be solved effectively and/or efficiently.
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Martires, Joanne S., Reuben Ram, and Jeanne Wallace. "Introduction to Sleep-Disordered Breathing and Treatment." In Integrative Sleep Medicine, edited by Valerie Cacho and Esther Lum, 509–28. Oxford University Press, 2021. http://dx.doi.org/10.1093/med/9780190885403.003.0031.

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Sleep-related breathing disorders encompass a wide range of problems that occur during sleep. The most common sleep-related breathing disorder is obstructive sleep apnea syndrome (OSAS). OSAS is increasingly prevalent, underdiagnosed and can lead to significant daytime sleepiness and disability. Continuous positive airway pressure (CPAP) is the treatment of choice for OSAS but lack of adherence to CPAP is rampant. Supportive care, education, and motivational enhancement programs do show promise in improving compliance. Alternatives to CPAP such as oral appliances, hypoglossal nerve stimulation, and myofunctional therapy have not been proved as efficacious but may provide some benefit. In the future, personalized treatment may be the best way to approach this condition. Central sleep apnea and hypoventilation are other forms of sleep-related breathing disorders. Treatment largely depends on the underlying condition but may require the use of advanced modes of noninvasive ventilation such as adaptive servo-ventilation, bilevel positive airway pressure, and volume-assured pressure support. Treatment of sleep-related breathing disorders is complex and includes seeking patient input, understanding barriers to treatment, and reevaluating patients after treatment is initiated. Successful treatment of sleep-disordered breathing can lead to improvements in daytime function, quality of life, and overall health.
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Conference papers on the topic "Bilevel program"

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Ouattara, Aurelien, and Anil Aswani. "Duality Approach to Bilevel Programs with a Convex Lower Level." In 2018 Annual American Control Conference (ACC). IEEE, 2018. http://dx.doi.org/10.23919/acc.2018.8431802.

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Gupta, Abhishek, and Yew-Soon Ong. "An evolutionary algorithm with adaptive scalarization for multiobjective bilevel programs." In 2015 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2015. http://dx.doi.org/10.1109/cec.2015.7257083.

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Li, Hong, Li Zhang, and Hecheng Li. "Modified NSGA-II Based Interactive Algorithm for Linear Multiobjective Bilevel Programs." In 2019 15th International Conference on Computational Intelligence and Security (CIS). IEEE, 2019. http://dx.doi.org/10.1109/cis.2019.00095.

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Sinha, Ankur, Pekka Malo, and Kalyanmoy Deb. "Solving optimistic bilevel programs by iteratively approximating lower level optimal value function." In 2016 IEEE Congress on Evolutionary Computation (CEC). IEEE, 2016. http://dx.doi.org/10.1109/cec.2016.7744017.

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Hawthorne, Bryant D., and Jitesh H. Panchal. "Policy Design for Sustainable Energy Systems Considering Multiple Objectives and Incomplete Preferences." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70426.

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The focus of this paper is on policy design problems related to large scale complex systems such as the decentralized energy infrastructure. In such systems, the policy affects the technical decisions made by stakeholders (e.g., energy producers), and the stakeholders are coordinated by market mechanisms. The decentralized decisions of the stakeholders affect the sustainability of the overall system. Hence, appropriate design of policies is an important aspect of achieving sustainability. The state-of-the-art computational approach to policy design problem is to model them as bilevel programs, specifically mathematical programs with equilibrium constraints. However, this approach is limited to single-objective policy design problems and is based on the assumption that the policy designer has complete information of the stakeholders’ preferences. In this paper, we take a step towards addressing these two limitations. We present a formulation based on the integration of multi-objective mathematical programs with equilibrium constraints with games with vector payoffs, and Nash equilibra of games with incomplete preferences. The formulation, along with a simple solution approach, is presented using an illustrative example from the design of feed-in-tariff (FIT) policy with two stakeholders. The contributions of this paper include a mathematical formulation of the FIT policy, the extension of computational policy design problems to multiple objectives, and the consideration of incomplete preferences of stakeholders.
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Reports on the topic "Bilevel program"

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Hart, William Eugene, Jean-Paul Watson, John Daniel Siirola, and Richard Li-Yang Chen. Modeling Bilevel Programs in Pyomo. Office of Scientific and Technical Information (OSTI), April 2016. http://dx.doi.org/10.2172/1561200.

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