Integer Programming on the Junction Tree Polytope for Influence Diagrams

Influence diagrams (ID) and limited memory influence diagrams (LIMID) are flexible tools to represent discrete stochastic optimization problems, with the Markov decision process (MDP) and partially observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In an ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types: chance, decision, and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the “single policy update,” the default algorithm for addressing IDs.

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Multirow Intersection Cuts Based on the Infinity Norm

INFORMS Journal on Computing ◽

10.1287/ijoc.2020.1027 ◽

2021 ◽

Author(s):

Álinson S. Xavier ◽

Ricardo Fukasawa ◽

Laurent Poirrier

Keyword(s):

Optimization Problems ◽

Valid Inequalities ◽

Linear Optimization ◽

Computational Cost ◽

General Purpose ◽

Mixed Integer ◽

Infinity Norm ◽

Intersection Cuts ◽

Linear Optimization Problems ◽

Mixed Integer Linear Optimization

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

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A Cluster Graph Approach to Land Cover Classification Boosting

Data ◽

10.3390/data4010010 ◽

2019 ◽

Vol 4 (1) ◽

pp. 10

Author(s):

Lloyd Hughes ◽

Simon Streicher ◽

Ekaterina Chuprikova ◽

Johan Du Preez

Keyword(s):

Land Cover ◽

Graphical Model ◽

Land Cover Classification ◽

Probabilistic Graphical Model ◽

Observation Data ◽

Computationally Efficient ◽

Efficient Manner ◽

Uncertainty Estimates ◽

Spatio Temporal ◽

Cluster Graph

When it comes to land cover classification, the process of deriving the land classes is complex due to possible errors in algorithms, spatio-temporal heterogeneity of the Earth observation data, variation in availability and quality of reference data, or a combination of these. This article proposes a probabilistic graphical model approach, in the form of a cluster graph, to boost geospatial classifications and produce a more accurate and robust classification and uncertainty product. Cluster graphs can be characterized as a means of reasoning about geospatial data such as land cover classifications by considering the effects of spatial distribution, and inter-class dependencies in a computationally efficient manner. To assess the capabilities of our proposed cluster graph boosting approach, we apply it to the field of land cover classification. We make use of existing land cover products (GlobeLand30, CORINE Land Cover) along with data from Volunteered Geographic Information (VGI), namely OpenStreetMap (OSM), to generate a boosted land cover classification and the respective uncertainty estimates. Our approach combines qualitative and quantitative components through the application of our probabilistic graphical model and subjective expert judgments. Evaluating our approach on a test region in Garmisch-Partenkirchen, Germany, our approach was able to boost the overall land cover classification accuracy by 1.4% when compared to an independent reference land cover dataset. Our approach was shown to be robust and was able to produce a diverse, feasible and spatially consistent land cover classification in areas of incomplete and conflicting evidence. On an independent validation scene, we demonstrated that our cluster graph boosting approach was generalizable even when initialized with poor prior assumptions.

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Mathematical models for the periodic vehicle routing problem with time windows and time spread constraints

An International Journal of Optimization and Control Theories & Applications (IJOCTA) ◽

10.11121/ijocta.01.2021.00899 ◽

2020 ◽

Vol 11 (1) ◽

pp. 10-23

Author(s):

Hande Öztop ◽

Damla Kizilay ◽

Zeynel Abidin Çil

Keyword(s):

Vehicle Routing ◽

Vehicle Routing Problem ◽

Optimization Problems ◽

Valid Inequalities ◽

Time Windows ◽

Mixed Integer ◽

Routing Problem ◽

Periodic Vehicle Routing Problem ◽

Time Spread ◽

Periodic Vehicle Routing

The periodic vehicle routing problem (PVRP) is an extension of the well-known vehicle routing problem. In this paper, the PVRP with time windows and time spread constraints (PVRP-TWTS) is addressed, which arises in the high-value shipment transportation area. In the PVRP-TWTS, period-specific demands of the customers must be delivered by a fleet of heterogeneous capacitated vehicles over the several planning periods. Additionally, the arrival times to a customer should be irregular within its time window over the planning periods, and the waiting time is not allowed for the vehicles due to the security concerns. This study, proposes novel mixed-integer linear programming (MILP) and constraint programming (CP) models for the PVRP-TWTS. Furthermore, we develop several valid inequalities to strengthen the proposed MILP and CP models as well as a lower bound. Even though CP has successful applications for various optimization problems, it is still not as well-known as MILP in the operations research field. This study aims to utilize the effectiveness of CP in solving the PVRP-TWTS. This study presents a CP model for PVRP-TWTS for the first time in the literature to the best of our knowledge. Having a comparison of the CP and MILP models can help in providing a baseline for the problem. We evaluate the performance of the proposed MILP and CP models by modifying the well-known benchmark set from the literature. The extensive computational results show that the CP model performs much better than the MILP model in terms of the solution quality.

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Robust Mixed 0-1 Programming and Submodularity

INFORMS Journal on Optimization ◽

10.1287/ijoo.2019.0042 ◽

2021 ◽

pp. ijoo.2019.0042

Author(s):

Seulgi Joung ◽

Sungsoo Park

Keyword(s):

Optimization Problems ◽

Valid Inequalities ◽

Continuous Variable ◽

Integer Program ◽

Data Uncertainty ◽

Mixed Integer ◽

Mixed Integer Program ◽

Robust Solution ◽

Polyhedral Study ◽

Set Function

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.

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Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

Optimization Letters ◽

10.1007/s11590-020-01660-6 ◽

2020 ◽

Author(s):

Thomas Kleinert ◽

Martine Labbé ◽

Fränk Plein ◽

Martin Schmidt

Keyword(s):

Branch And Bound ◽

Optimization Problems ◽

Valid Inequalities ◽

Bilevel Optimization ◽

Branch And Cut ◽

Mixed Integer ◽

Valid Inequality ◽

Single Level ◽

Primal Dual ◽

Level Problem

Abstract Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

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A Probabilistic-Graphical-Model Based Approach for Representing Lineages in Uncertain Data

Chinese Journal of Computers ◽

10.3724/sp.j.1016.2011.01897 ◽

2011 ◽

Vol 34 (10) ◽

pp. 1897-1906 ◽

Cited By ~ 2

Author(s):

Kun YUE ◽

Wei-Yi LIU ◽

Yun-Lei ZHU ◽

Wei ZHANG

Keyword(s):

Graphical Model ◽

Uncertain Data ◽

Probabilistic Graphical Model ◽

Model Based

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A Multi-Objective Approach toward Optimal Design of Sustainable Integrated Biodiesel/Diesel Supply Chain Based on First- and Second-Generation Feedstock with Solid Waste Use

Energies ◽

10.3390/en14082261 ◽

2021 ◽

Vol 14 (8) ◽

pp. 2261

Author(s):

Evgeniy Ganev ◽

Boyan Ivanov ◽

Natasha Vaklieva-Bancheva ◽

Elisaveta Kirilova ◽

Yunzile Dzhelil

Keyword(s):

Supply Chain ◽

Optimal Design ◽

Solid Waste ◽

Second Generation ◽

Agricultural Land ◽

Optimization Problems ◽

Biodiesel Production ◽

Mixed Integer ◽

Economic Criterion ◽

Multi Objective

This study proposes a multi-objective approach for the optimal design of a sustainable Integrated Biodiesel/Diesel Supply Chain (IBDSC) based on first- (sunflower and rapeseed) and second-generation (waste cooking oil and animal fat) feedstocks with solid waste use. It includes mixed-integer linear programming (MILP) models of the economic, environmental and social impact of IBDSC, and respective criteria defined in terms of costs. The purpose is to obtain the optimal number, sizes and locations of bio-refineries and solid waste plants; the areas and amounts of feedstocks needed for biodiesel production; and the transportation mode. The approach is applied on a real case study in which the territory of Bulgaria with its 27 districts is considered. Optimization problems are formulated for a 5-year period using either environmental or economic criteria and the remainder are defined as constraints. The obtained results show that in the case of the economic criterion, 14% of the agricultural land should be used for sunflower and 2% for rapeseed cultivation, while for the environmental case, 12% should be used for rapeseed and 3% for sunflower. In this case, the price of biodiesel is 14% higher, and the generated pollutants are 6.6% lower. The optimal transport for both cases is rail.

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