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Journal articles on the topic 'Mixed-Integer Linear Programs'

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Published: 6 September 2023

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1

Guieu, Olivier, and John W. Chinneck. "Analyzing Infeasible Mixed-Integer and Integer Linear Programs." INFORMS Journal on Computing 11, no. 1 (February 1999): 63–77. http://dx.doi.org/10.1287/ijoc.11.1.63.

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2

Cornuéjols, Gérard. "Valid inequalities for mixed integer linear programs." Mathematical Programming 112, no. 1 (January 24, 2007): 3–44. http://dx.doi.org/10.1007/s10107-006-0086-0.

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3

Berthold, Timo, and Gregor Hendel. "Learning To Scale Mixed-Integer Programs." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 5 (May 18, 2021): 3661–68. http://dx.doi.org/10.1609/aaai.v35i5.16482.

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Many practical applications require the solution of numerically challenging linear programs (LPs) and mixed integer programs (MIPs). Scaling is a widely used preconditioning technique that aims at reducing the error propagation of the involved linear systems, thereby improving the numerical behavior of the dual simplex algorithm and, consequently, LP-based branch-and-bound. A reliable scaling method often makes the difference whether these problems can be solved correctly or not. In this paper, we investigate the use of machine learning to choose at the beginning of the solution process between two common scaling methods: Standard scaling and Curtis-Reid scaling. The latter often, but not always, leads to a more robust solution process, but may suffer from longer solution times. Rather than training for overall solution time, we propose to use the attention level of a MIP solution process as a learning label. We evaluate the predictive power of a random forest approach and a linear regressor that learns the (square-root of the) difference in attention level. It turns out that the resulting classification not only reduces various types of numerical errors by large margins, but it also improves the performance of the dual simplex algorithm. The learned model has been implemented within the FICO Xpress MIP solver and it is used by default since release 8.9, May 2020, to determine the scaling algorithm Xpress applies before solving an LP or a MIP.

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4

Laporte, Gilbert, and Frédéric Semet. "An optimality cut for mixed integer linear programs." European Journal of Operational Research 119, no. 3 (December 1999): 671–77. http://dx.doi.org/10.1016/s0377-2217(98)00357-9.

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5

Proll, L. G. "Stronger formulations of mixed integer linear programs: an example." International Journal of Mathematical Education in Science and Technology 28, no. 5 (September 1997): 707–12. http://dx.doi.org/10.1080/0020739970280507.

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6

Klotz, Ed, and Alexandra M. Newman. "Practical guidelines for solving difficult mixed integer linear programs." Surveys in Operations Research and Management Science 18, no. 1-2 (October 2013): 18–32. http://dx.doi.org/10.1016/j.sorms.2012.12.001.

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7

Bonami, Pierre, Gérard Cornuéjols, Sanjeeb Dash, Matteo Fischetti, and Andrea Lodi. "Projected Chvátal–Gomory cuts for mixed integer linear programs." Mathematical Programming 113, no. 2 (December 8, 2006): 241–57. http://dx.doi.org/10.1007/s10107-006-0051-y.

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8

Pryor, Jennifer, and John W. Chinneck. "Faster integer-feasibility in mixed-integer linear programs by branching to force change." Computers & Operations Research 38, no. 8 (August 2011): 1143–52. http://dx.doi.org/10.1016/j.cor.2010.10.025.

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9

Chen, Binyuan, Simge Küçükyavuz, and Suvrajeet Sen. "Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs." Operations Research 59, no. 1 (February 2011): 202–10. http://dx.doi.org/10.1287/opre.1100.0882.

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10

Li, Xiaobo, Karthik Natarajan, Chung-Piaw Teo, and Zhichao Zheng. "Distributionally robust mixed integer linear programs: Persistency models with applications." European Journal of Operational Research 233, no. 3 (March 2014): 459–73. http://dx.doi.org/10.1016/j.ejor.2013.07.009.

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11

Grimstad, Bjarne, and Henrik Andersson. "ReLU networks as surrogate models in mixed-integer linear programs." Computers & Chemical Engineering 131 (December 2019): 106580. http://dx.doi.org/10.1016/j.compchemeng.2019.106580.

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12

Warwicker, John Alasdair, and Steffen Rebennack. "A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting." INFORMS Journal on Computing 34, no. 2 (March 2022): 1042–47. http://dx.doi.org/10.1287/ijoc.2021.1114.

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The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

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13

Fischetti, Matteo, Ivana Ljubić, Michele Monaci, and Markus Sinnl. "A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs." Operations Research 65, no. 6 (December 2017): 1615–37. http://dx.doi.org/10.1287/opre.2017.1650.

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14

Belotti, Pietro, Banu Soylu, and Margaret M. Wiecek. "Fathoming rules for biobjective mixed integer linear programs: Review and extensions." Discrete Optimization 22 (November 2016): 341–63. http://dx.doi.org/10.1016/j.disopt.2016.09.003.

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15

Carvajal, R., S. Ahmed, G. Nemhauser, K. Furman, V. Goel, and Y. Shao. "Using diversification, communication and parallelism to solve mixed-integer linear programs." Operations Research Letters 42, no. 2 (March 2014): 186–89. http://dx.doi.org/10.1016/j.orl.2013.12.012.

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16

Guastaroba, G., M. Savelsbergh, and M. G. Speranza. "Adaptive Kernel Search: A heuristic for solving Mixed Integer linear Programs." European Journal of Operational Research 263, no. 3 (December 2017): 789–804. http://dx.doi.org/10.1016/j.ejor.2017.06.005.

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17

SANNOMIYA, NOBUO, and KATSUHIKO OKAMOTO. "A method for decomposing mixed-integer linear programs with staircase structure." International Journal of Systems Science 16, no. 1 (January 1985): 99–111. http://dx.doi.org/10.1080/00207728508926658.

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18

Owen, Jonathan H., and Sanjay Mehrotra. "A disjunctive cutting plane procedure for general mixed-integer linear programs." Mathematical Programming 89, no. 3 (February 2001): 437–48. http://dx.doi.org/10.1007/pl00011407.

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19

Cheung, Kevin K. H., and Babak Moazzez. "Certificates of Optimality for Mixed Integer Linear Programming Using Generalized Subadditive Generator Functions." Advances in Operations Research 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/5017369.

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We introduce generalized subadditive generator functions for mixed integer linear programs. Our results extend Klabjan’s work from pure integer programs with nonnegative entries to general MILPs. These functions suffice to achieve strong subadditive duality. Several properties of the functions are shown. We then use this class of functions to generate certificates of optimality for MILPs. We have performed a computational test study on knapsack problems to investigate the efficiency of the certificates.

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20

Brand, Cornelius, Martin Koutecký, and Sebastian Ordyniak. "Parameterized Algorithms for MILPs with Small Treedepth." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 14 (May 18, 2021): 12249–57. http://dx.doi.org/10.1609/aaai.v35i14.17454.

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Solving (mixed) integer (linear) programs, (M)I(L)Ps for short, is a fundamental optimisation task with a wide range of applications in artificial intelligence and computer science in general. While hard in general, recent years have brought about vast progress for solving structurally restricted, (non-mixed) ILPs: n-fold, tree-fold, 2-stage stochastic and multi-stage stochastic programs admit efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth. In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d)poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth, and n is the number of variables. This is enabled by proving bounds on the denominators (fractionality) of the vertices of bounded-treedepth (non-integer) linear programs. We do so by carefully analysing the inverses of invertible sub-matrices of the constraint matrix. This allows us to afford scaling up the mixed program to the integer grid, and applying the known methods for integer programs. We then trace the limiting boundary of our "bounded fractionality" approach both in terms of going beyond MILP (by allowing non-linear objectives) as well as its usefulness for generalising other important known tractable classes of ILP. On the positive side, we show that our result can be generalised from MILP to MIP with piece-wise linear separable convex objectives with integer breakpoints. On the negative side, we show that going even slightly beyond such objectives or considering other natural related tractable classes of ILP leads to unbounded fractionality. Finally, we show that restricting the structure of only the integral variables in the constraint matrix does not yield tractable special cases.

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21

Owen, Jonathan H., and Sanjay Mehrotra. "On the Value of Binary Expansions for General Mixed-Integer Linear Programs." Operations Research 50, no. 5 (October 2002): 810–19. http://dx.doi.org/10.1287/opre.50.5.810.370.

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22

Patel, Jagat, and John W. Chinneck. "Active-constraint variable ordering for faster feasibility of mixed integer linear programs." Mathematical Programming 110, no. 3 (July 11, 2006): 445–74. http://dx.doi.org/10.1007/s10107-006-0009-0.

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23

Zare, M. Hosein, Osman Y. Özaltın, and Oleg A. Prokopyev. "On a class of bilevel linear mixed-integer programs in adversarial settings." Journal of Global Optimization 71, no. 1 (August 21, 2017): 91–113. http://dx.doi.org/10.1007/s10898-017-0549-2.

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24

Newby, Eric, and M. M. Ali. "Linear transformation based solution methods for non-convex mixed integer quadratic programs." Optimization Letters 11, no. 5 (December 26, 2015): 967–81. http://dx.doi.org/10.1007/s11590-015-0988-y.

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25

Ferber, Aaron, Bryan Wilder, Bistra Dilkina, and Milind Tambe. "MIPaaL: Mixed Integer Program as a Layer." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 02 (April 3, 2020): 1504–11. http://dx.doi.org/10.1609/aaai.v34i02.5509.

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Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.

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26

Gondzio, Jacek, and E. Alper Yıldırım. "Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations." Journal of Global Optimization 81, no. 2 (April 20, 2021): 293–321. http://dx.doi.org/10.1007/s10898-021-01017-y.

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AbstractA standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

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27

Douaioui, Kaoutar, Mouhsene Fri, Charif Mabrouki, and El Alami Semma. "A multi-objective integrated procurement, production, and distribution problem of supply chain network under fuzziness uncertainties." Pomorstvo 35, no. 2 (December 22, 2021): 191–206. http://dx.doi.org/10.31217/p.35.2.1.

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In this paper, we devoted a design under uncertainty of a four-echelon supply chain network including multiple suppliers, multiple plants, multiple distributors and multiple customers. The proposed model is a bi-objective mixed integer linear programming which considers several constraints and aims to minimize the total costs including the procurement, production, storage and distribution costs as well as to maximize on-time deliveries (OTD). To bring the model closer to real-world planning problems, the objective function coefficients (e.g. procurement cost, production cost, inventory holding and transport costs) and other parameters (e.g., demand, production capacity and safety stock level), are all considered triangular fuzzy numbers. Besides, a hybrid mathematical model-based on credibility approach is constructed for the problem, i.e., expected value and chance constrained models. Moreover, to build the crisp equivalent model, we use different property of the credibility measure. The resulted crisp equivalent model is a bi-objective mixed integer linear programs (BOMILP). To transform this crisp BOMILP into a single objective mixed integer linear programs (MILP) model, we apply three different aggregation functions. Finally, numerical results are reported for a real case study to demonstrate the efficiency and applicability of the proposed model.

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28

Gollmer, Ralf, Frederike Neise, and Rüdiger Schultz. "Stochastic Programs with First-Order Dominance Constraints Induced by Mixed-Integer Linear Recourse." SIAM Journal on Optimization 19, no. 2 (January 2008): 552–71. http://dx.doi.org/10.1137/060678051.

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29

Mehmanchi, Erfan, Andrés Gómez, and Oleg A. Prokopyev. "Fractional 0–1 programs: links between mixed-integer linear and conic quadratic formulations." Journal of Global Optimization 75, no. 2 (September 6, 2019): 273–339. http://dx.doi.org/10.1007/s10898-019-00817-7.

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30

Alves, Maria João, and João Paulo Costa. "Graphical exploration of the weight space in three-objective mixed integer linear programs." European Journal of Operational Research 248, no. 1 (January 2016): 72–83. http://dx.doi.org/10.1016/j.ejor.2015.06.072.

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31

Mahmoodian, Vahid, Hadi Charkhgard, and Yu Zhang. "Multi-objective optimization based algorithms for solving mixed integer linear minimum multiplicative programs." Computers & Operations Research 128 (April 2021): 105178. http://dx.doi.org/10.1016/j.cor.2020.105178.

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32

Huang, Kuo-Ling, and Sanjay Mehrotra. "An empirical evaluation of walk-and-round heuristics for mixed integer linear programs." Computational Optimization and Applications 55, no. 3 (February 16, 2013): 545–70. http://dx.doi.org/10.1007/s10589-013-9540-0.

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33

Adelgren, Nathan, and Akshay Gupte. "Branch-and-Bound for Biobjective Mixed-Integer Linear Programming." INFORMS Journal on Computing 34, no. 2 (March 2022): 909–33. http://dx.doi.org/10.1287/ijoc.2021.1092.

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We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, checking node fathoming, presolve, and duality gap measurement. Our branch-and-bound is predominantly a decision space search method because the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and on a new set of instances that we generate using a benchmark library (MIPLIB2017) for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm. Summary of Contribution: Biobjective mixed-integer optimization problems have two linear objectives and a mixed-integer feasible region. Such problems have many applications in operations research, because many real-world optimization problems naturally comprise two conflicting objectives to optimize or can be approximated in such a manner and are even harder than single objective mixed-integer programs. Solving them exactly requires the computation of all the nondominated solutions in the objective space, whereas some applications may also require finding at least one solution in the decision space corresponding to each nondominated solution. This paper provides an exact algorithm for solving these problems using the branch-and-bound method, which works predominantly in the decision space. Of the many ingredients of this algorithm, some parts are direct extensions of the single-objective version, but the main parts are newly designed algorithms to handle the distinct challenges of optimizing over two objectives. The goal of this study is to improve solution quality and speed and show that decision-space algorithms perform comparably to, and sometimes better than, algorithms that work mainly in the objective-space.

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34

Letchford, Adam N., and Daniel J. Grainger. "A note on representations of linear inequalities in non-convex mixed-integer quadratic programs." Operations Research Letters 45, no. 6 (November 2017): 631–34. http://dx.doi.org/10.1016/j.orl.2017.10.007.

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35

Vielma, Juan Pablo, Shabbir Ahmed, and George L. Nemhauser. "A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs." INFORMS Journal on Computing 20, no. 3 (August 2008): 438–50. http://dx.doi.org/10.1287/ijoc.1070.0256.

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36

Escudero, Laureano F., María Araceli Garín, María Merino, and Gloria Pérez. "On time stochastic dominance induced by mixed integer-linear recourse in multistage stochastic programs." European Journal of Operational Research 249, no. 1 (February 2016): 164–76. http://dx.doi.org/10.1016/j.ejor.2015.03.050.

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37

Ogbe, Emmanuel, and Xiang Li. "Extended cross decomposition for mixed-integer linear programs with strong and weak linking constraints." Computers & Chemical Engineering 119 (November 2018): 237–57. http://dx.doi.org/10.1016/j.compchemeng.2018.09.011.

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38

Sherali, Hanif D., and Danny C. Myers. "Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs." Discrete Applied Mathematics 20, no. 1 (May 1988): 51–68. http://dx.doi.org/10.1016/0166-218x(88)90041-8.

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39

Britz, Wolfgang. "Automated Calibration of Farm-Sale Mixed Linear Programming Models using Bi-Level Programming." German Journal of Agricultural Economics 70, no. 3 (September 1, 2021): 165–81. http://dx.doi.org/10.30430/70.2021.3.165-181.

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We calibrate Linear and Mixed Integer Programs with a bi-level estimator, minimizing under First-order-conditions (FOC) conditions a penalty function considering the calibration fit and deviations from given parameters. To deal with non-convexity, a heuristic generates restart points from current best-fit parameters and their means. Monte-Carlo analysis assesses the approach by drawing parameters for a model optimizing acreages under maximal crop shares, a land balance and annual plus intra-annual labour constraints; a variant comprises integer based investments. Resulting optimal solutions perturbed by white noise provide calibration targets. The approach recovers the true parameters and thus allows for systematic and automated calibration.

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40

Shi, Tong Ju, and Xin Shun Ma. "Development of Web-Based Experiment for Integer Programming Using Java Applet." Advanced Materials Research 756-759 (September 2013): 1998–2002. http://dx.doi.org/10.4028/www.scientific.net/amr.756-759.1998.

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Integer (Mixed integer) programming is one of important mathematical programs for solving many practical problems, such as economic management, optimization control and supply chain. The integer linear programming problem and its solution algorithm are studied in this paper, and Web-based mathematical experiments of Branch-and-bound algorithm developed using Java applets. The system of the experiment can not only demonstrate the basic principle and iterating procedure of the algorithm by online example, but also solve any new integer programming inputted by user. The developed system provides users an efficient assisted learning platform with an interactive mode.

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41

Chen, Binyuan, Simge Küçükyavuz, and Suvrajeet Sen. "A computational study of the cutting plane tree algorithm for general mixed-integer linear programs." Operations Research Letters 40, no. 1 (January 2012): 15–19. http://dx.doi.org/10.1016/j.orl.2011.10.009.

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42

Shimizu, Yoshiaki. "Process Systems Engineering. Parametric Problems of PROLP. Application to solution of mixed-integer linear programs." KAGAKU KOGAKU RONBUNSHU 22, no. 5 (1996): 1046–54. http://dx.doi.org/10.1252/kakoronbunshu.22.1046.

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43

Razavyan, S. "A Method for Generating a Well-Distributed Pareto Set in Multiple Objective Mixed Integer Linear Programs Based on the Decision Maker’s Initial Aspiration Level." Asia-Pacific Journal of Operational Research 33, no. 04 (August 2016): 1650031. http://dx.doi.org/10.1142/s0217595916500317.

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This paper attempt to generate a representative subset of the Pareto optimal set for multiple objective mixed integer linear programming problem using the weighted L1 norm distance. The procedure presented in this paper is somewhat similar to the one used in the ideal-point methods and its aim is to generate at each iteration the closest-points to the ideal vector corresponding to the decision maker’s initial aspiration level for a new tradeoff parameter. Unlike most of the known algorithms for generating a discrete representation of the Pareto optimal set, the procedure generates at each iteration a nondominated point by solving only one mixed integer linear programming problem. The obtained solution minimizes the weighted L1 norm distance to the ideal vector with respect to the distance between the ideal vector and previously found vectors. More generally, this approach is able to generate all Pareto optimal solutions, where all of the decision variables are restricted to be integer. In order to explain the presented details, several illustrative examples are provided.

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44

Louaqad, Saad, and Oulaid Kamach. "Mixed Integer Linear Programs for Blocking and No Wait Job Shop Scheduling Problems in Robotic cells." International Journal of Computer Applications 153, no. 10 (November 17, 2016): 1–7. http://dx.doi.org/10.5120/ijca2016912158.

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45

Rinaldi, Giovanni, Ulrich Voigt, and Gerhard J. Woeginger. "The mathematics of playing golf, or: a new class of difficult non-linear mixed integer programs." Mathematical Programming 93, no. 1 (June 1, 2002): 77–86. http://dx.doi.org/10.1007/s101070200298.

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46

Solanki, Rajendra. "Generating the noninferior set in mixed integer biobjective linear programs: An application to a location problem." Computers & Operations Research 18, no. 1 (January 1991): 1–15. http://dx.doi.org/10.1016/0305-0548(91)90037-r.

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47

Zeile, Clemens, Tobias Weber, and Sebastian Sager. "Combinatorial Integral Approximation Decompositions for Mixed-Integer Optimal Control." Algorithms 15, no. 4 (March 31, 2022): 121. http://dx.doi.org/10.3390/a15040121.

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Solving mixed-integer nonlinear programs (MINLPs) is hard from both a theoretical and practical perspective. Decomposing the nonlinear and the integer part is promising from a computational point of view. In general, however, no bounds on the objective value gap can be established and iterative procedures with potentially many subproblems are necessary. The situation is different for mixed-integer optimal control problems with binary variables that switch over time. Here, a priori bounds were derived for a decomposition into one continuous nonlinear control problem and one mixed-integer linear program, the combinatorial integral approximation (CIA) problem. In this article, we generalize and extend the decomposition idea. First, we derive different decompositions and analyze the implied a priori bounds. Second, we propose several strategies to recombine promising candidate solutions for the binary control functions in the original problem. We present the extensions for ordinary differential equations-constrained problems. These extensions are transferable in a straightforward way, though, to recently suggested variants for certain partial differential equations, for algebraic equations, for additional combinatorial constraints, and for discrete time problems. We implemented all algorithms and subproblems in AMPL for a proof-of-concept study. Numerical results show the improvement compared to the standard CIA decomposition with respect to objective function value and compared to general-purpose MINLP solvers with respect to runtime.

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48

Torres, Juan J., Can Li, Robert M. Apap, and Ignacio E. Grossmann. "A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software." Algorithms 15, no. 4 (March 22, 2022): 103. http://dx.doi.org/10.3390/a15040103.

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This paper presents a tutorial on the state-of-the-art software for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose. The methodologies are classified according to the decomposition alternatives and the types of the variables in the problem. We review the fundamentals of Benders decomposition, dual decomposition and progressive hedging, as well as possible improvements and variants. We also present extensive numerical results to underline the properties and performance of each algorithm using software implementations, including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the strengths and weaknesses of each methodology and propose future research directions.

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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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Debeer, Dries, Peter W. van Rijn, and Usama S. Ali. "Multidimensional Test Assembly Using Mixed-Integer Linear Programming: An Application of Kullback–Leibler Information." Applied Psychological Measurement 44, no. 1 (February 25, 2019): 17–32. http://dx.doi.org/10.1177/0146621619827586.

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Many educational testing programs require different test forms with minimal or no item overlap. At the same time, the test forms should be parallel in terms of their statistical and content-related properties. A well-established method to assemble parallel test forms is to apply combinatorial optimization using mixed-integer linear programming (MILP). Using this approach, in the unidimensional case, Fisher information (FI) is commonly used as the statistical target to obtain parallelism. In the multidimensional case, however, FI is a multidimensional matrix, which complicates its use as a statistical target. Previous research addressing this problem focused on item selection criteria for multidimensional computerized adaptive testing (MCAT). Yet these selection criteria are not directly transferable to the assembly of linear parallel test forms. To bridge this gap the authors derive different statistical targets, based on either FI or the Kullback–Leibler (KL) divergence, that can be applied in MILP models to assemble multidimensional parallel test forms. Using simulated item pools and an item pool based on empirical items, the proposed statistical targets are compared and evaluated. Promising results with respect to the KL-based statistical targets are presented and discussed.

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