Tamás Terlaky · 2013
Abstract : Fundamental Disjunctive Conic Cut (DCC) methodology for Mixed-Integer Conic Linear Optimization (MICO) was developed. To describe the convex hull of the intersection of a convex set E and a linear disjunction is the fundamental problem, and that served as the core of solution techniques for MICO. It was proved that if there exists a cone K that has the same intersection with the boundary of the disjunction as the convex set E, then the convex hull of the disjunction is the intersection of E with K. While uniqueness of a DCC is proved for general MICO, the existence of such a cone is difficult to prove for the general case. Thorough analysis of a parametric family of quadrics allows to prove the existence and uniqueness of a second order cone, when E is the intersection of an affine space and a second order cone. An efficiently computable method was developed for finding that cone, which provided novel and powerful DCCs for Mixed Integer Second Order Cone Optimization (MISOCO), which can be used in branch-and-cut algorithms when solving MISOCO problems. All special and degenerate cases are carefully analyzed and easy to compute criteria are developed to compute a DCC for all cases. Limited, but rigorous computational experiments gave strong indication of the power of the DCCs.
Cited by 2
Štěpán Škoda · Digital Repository (National Repository of Grey Literature) · 2011
In the present work we study the problems of integer linear optimi- zation, at first from the theoretical point of view (Part I) and subsequently on the basis of empirical data (Part II). First section explains with what this field deals with and where is applied. Other sections contain annotated mathematical formulation of problems, definitions and theorems needed to understand the general methods for solving integer linear programs. In the last section of Part I, we introduce the two best known groups of algorithms that are used by commercial software. The second part provides more details on an Internet library that contains some practical problems, that has been needed to be solved in the past. Furthermore, there are sections dealing with solvers and the advanced options of GAMS. The last section presents data obtained in the course of solving problems using several codes (solvers) of software. 1
Cited by 0
MANSINI, Renata, SPERANZA, Maria Grazia · Institutional Research Information System (Università degli Studi di Brescia) · 2000
Cited by 0
Margot, Francois · KiltHub Repository · 2009
In this paper, a methodology for testing the accuracy and strength of cut generators for mixed-integer linear programming is presented. The procedure amounts to random diving towards a feasible solution, recording several kinds of failures. This allows for a ranking of the accuracy of the generators. Then, for generators deemed to have similar accuracy, statistical tests are performed to compare their relative strength. An application on eight Gomory cut generators and six Reduce-and-Split cut generators is given. The problem of constructing benchmark instances for which feasible solutions can be obtained is also addressed.
Cited by 0Open access
Rutgers University Community Repository (Rutgers University) · 2015
In this work we focus on various cutting-plane methods for Mixed-integer Linear Programming (MILP) problems. It is well-known that MILP is a fundamental hard problem and many famous combinatorial optimization problems can be modeled using MILP formulations. It is also well-known that MILP formulations are very useful in many real life applications. Our first, rather theoretical, contribution is a new family of superadditive valid inequalities that are obtained from value functions of special surrogate optimization problems. Superadditive functions hold particular interest in MILP as they are fundamental in building integer programming duality, and all ``deepest valid inequalities'' are known to arise from superadditive functions. We propose a new family of superadditive functions that generate inequalities that are at least as strong as Chvatal-Gomory (CG-) inequalities. A special subfamily provides a new characterization of CG-cuts. Value functions of optimization problems are known to be super additive. We look at special surrogate optimization problems, and measure their complexity in terms of the number of integer variables in them. It turns out that the lowest possible nontrivial complexity class here includes all CG-cuts, and provides some stronger ones, as well. Our next contribution is a practically efficient, polynomial time method to produce ``deepest'' cuts form multiple simplex rows for the so called corner polyhedra. These inequalities have been receiving considerable attention lately. We provide a polynomial time column-generation algorithm to obtain such inequalities, based on an arbitrary (fixed) number of rows of the simplex tableau. We provide numerical evidence that these inequalities improve the CPLEX integrality gap at the root node on a well-known set of MILP instances, MIPLIB. In the last chapter, we consider a particular MILP instance, Optimal Resilient Distribution Grid Design Problem (ORDGDP). This is a problem of critical importance to infrastructure security and recently attracted a lot of attention from various government agencies (e.g. Presidential Policy Directive of Critical Infrastructure Security 2013). We formulate this problem as a MILP and propose various efficient solution methods blending together well-known decomposition ideas to overcome the numerical intractability encountered using commercial MILP software such as CPLEX.
Cited by 0
Bagus Sartono, Schoen, Eric, Goos, Peter · Figshare · 2015
We present a mixed integer linear programming approach to orthogonally block two-level, multilevel, and mixed-level orthogonal designs. The approach involves an exact optimization technique which guarantees an optimal solution. It can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used. By means of 54-run and 64-run examples, we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two-factor interaction contrasts and in terms of the D-efficiency for models with main effects and some two-factor interaction contrasts. We demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength-4 designs. The approach can also be applied to strength-2 designs, but, for these cases, alternative methods described in the literature may perform equally well. Supplementary materials for this article are available online.
Cited by 0Open access
Luke Fina, Christopher Petersen · 2026
In this work, we introduce mixed-integer linear programming model predictive control (MILP-MPC) to model decision making problems for spacecraft. The term mixed-integer denotes the optimization decision variables are both continuous and integer. The minimum thrust propulsion and multi-mode propulsion problems are formulated as MILP-MPCs. MILPs in the literature are treated as intractable for real time systems due being theoretically NP-Hard. MILP solver advancements over the past decades alleviate these computational issues. Computational feasibility and empirical stability are verified in simulation for the proposed mixed-integer MPC formulations.
Cited by 0
Kemal Sahin, Korhan Gürsoy, Amy Ciric · Encyclopedia of Optimization · 2001
Cited by 0
Hamoud Bin Obaid, Theodore B. Trafali̇s, Mastoor M. Abushaega et al. · Mathematics · 2024
This study presents a new approach to optimize the dynamic evacuation process through a dynamic traffic assignment model formulated using mixed-integer linear programming (MILP). The model approximates the travel time for evacuee groups with a piecewise linear function that accounts for variations in travel time due to load-dependent factors. Significant delays are transferred to subsequent groups to simulate delay propagation. The primary objective is to minimize the network clearance time—the total time required for the last group of evacuees to reach safety from the start of the evacuation. Given the model’s computational intensity, a simplified version is introduced for comparison. Both the original and simplified models are tested on small networks and benchmarked against the Cell Transmission Model, a well-regarded method in dynamic traffic assignment literature. Additional objectives, including average travel time and average evacuation time, are explored. A sensitivity analysis is conducted to assess how varying the number of evacuee groups impacts model outcomes.
Cited by 1Open access
Stojan Kravanja · Civil Engineering ' ... · 2018
The paper discusses the Mixed-Integer Non-linear Programming (MINLP) of problems in civil engineering. The MINLP enables the optimization of continuous parameters simultaneously with discrete alternatives. While continuous parameters are in structural optimization structural costs, masses, loads, stresses, resistances and deflections, as well as the discrete alternatives are in most cases defined as different topologies, standard sizes and materials. The continuous parameters are in the models expressed by continuous variables, whilst the discrete alternatives by discrete (0-1) variables. The MINLP optimization of a structure is usually a comprehensive and highly non-linear calculation process. The MINLP approach requires that a structure is generated as an MINLP superstructure including a number of structure alternatives. One of them is the optimal one. For each optimization problem/structure, an MINLP optimization model of the structure must be developed, where the cost or mass objective function of the structure is subjected to structural analysis and dimensioning equality/inequality constraints. The Modified Outer-Approximation/Equality-Relaxation algorithm and a threephase MINLP strategy are applied. Three numerical examples, i.e. the MINLP optimization of a cantilever beam, composite floor and high-pressure penstock are presented at the end of the paper.
Cited by 0Open access
Bagus Sartono, Eric D. Schoen, Peter Goos · Figshare · 2015
We present a mixed integer linear programming approach to orthogonally block two-level, multilevel, and mixed-level orthogonal designs. The approach involves an exact optimization technique which guarantees an optimal solution. It can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used. By means of 54-run and 64-run examples, we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two-factor interaction contrasts and in terms of the D-efficiency for models with main effects and some two-factor interaction contrasts. We demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength-4 designs. The approach can also be applied to strength-2 designs, but, for these cases, alternative methods described in the literature may perform equally well. Supplementary materials for this article are available online.
Cited by 0Open access
Bagus Sartono, Schoen, Eric, Goos, Peter · Figshare · 2015
Cited by 0Open access
Bagus Sartono, Eric D. Schoen, Peter Goos · Figshare · 2015
We present a mixed integer linear programming approach to orthogonally block two-level, multilevel, and mixed-level orthogonal designs. The approach involves an exact optimization technique which guarantees an optimal solution. It can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used. By means of 54-run and 64-run examples, we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two-factor interaction contrasts and in terms of the D-efficiency for models with main effects and some two-factor interaction contrasts. We demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength-4 designs. The approach can also be applied to strength-2 designs, but, for these cases, alternative methods described in the literature may perform equally well. Supplementary materials for this article are available online.
Cited by 0Open access
Mark Turner, Thorsten Koch, Felipe Serrano et al. · Open Journal of Mathematical Optimization · 2023
Cutting plane selection is a subroutine used in all modern mixed-integer linear programming solvers with the goal of selecting a subset of generated cuts that induce optimal solver performance. These solvers have millions of parameter combinations, and so are excellent candidates for parameter tuning. Cut selection scoring rules are usually weighted sums of different measurements, where the weights are parameters. We present a parametric family of mixed-integer linear programs together with infinitely many family-wide valid cuts. Some of these cuts can induce integer optimal solutions directly after being applied, while others fail to do so even if an infinite amount are applied. We show for a specific cut selection rule, that any finite grid search of the parameter space will always miss all parameter values, which select integer optimal inducing cuts in an infinite amount of our problems. We propose a variation on the design of existing graph convolutional neural networks, adapting them to learn cut selection rule parameters. We present a reinforcement learning framework for selecting cuts, and train our design using said framework over MIPLIB 2017 and a neural network verification data set. Our framework and design show that adaptive cut selection does substantially improve performance over a diverse set of instances, but that finding a single function describing such a rule is difficult. Code for reproducing all experiments is available at https://github.com/Opt-Mucca/Adaptive-Cutsel-MILP .
Cited by 18Open access
Steffen, Marthe Carine, Honnemyr, Ida Kristine · Duo Research Archive (University of Oslo) · 2023
Personnel planning is an important component of business management. Improved\npersonnel planning can lead to more satisfied employees and decreased personnel\ncosts. This thesis explores how mixed integer linear programming (MILP) can assist\nthe police´s operations centre with personnel planning, by allocating employees to\npredefined shifts following legally required work regulations. Further, the effects of\nrecommended health-promoting shift patterns and employee satisfaction are explored.\n\nThe research finds that the optimisation model considering only the legally required\nregulations, can provide complete schedules within a satisfactory time limit. However, when implementing recommended health-promoting shift patterns and personnel\npreferences, the complexity of the model increases significantly. Key findings reflect\nthat the running time and complexity of the rostering problem increases significantly\nwhen considering all regulations, preferences, and health-promoting shift patterns.\nHence, for the purpose of generating high quality solutions that considers all kinds of\npreferences and regulations, also the less important ones, heuristics approaches should\nbe considered to obtain improved solutions more efficiently.
Cited by 0Open access
Emre Yamangil · Rutgers University Community Repository (Rutgers University) · 2015
In this work we focus on various cutting-plane methods for Mixed-integer Linear Programming (MILP) problems. It is well-known that MILP is a fundamental hard problem and many famous combinatorial optimization problems can be modeled using MILP formulations. It is also well-known that MILP formulations are very useful in many real life applications. Our first, rather theoretical, contribution is a new family of superadditive valid inequalities that are obtained from value functions of special surrogate optimization problems. Superadditive functions hold particular interest in MILP as they are fundamental in building integer programming duality, and all ``deepest valid inequalities'' are known to arise from superadditive functions. We propose a new family of superadditive functions that generate inequalities that are at least as strong as Chvatal-Gomory (CG-) inequalities. A special subfamily provides a new characterization of CG-cuts. Value functions of optimization problems are known to be super additive. We look at special surrogate optimization problems, and measure their complexity in terms of the number of integer variables in them. It turns out that the lowest possible nontrivial complexity class here includes all CG-cuts, and provides some stronger ones, as well. Our next contribution is a practically efficient, polynomial time method to produce ``deepest'' cuts form multiple simplex rows for the so called corner polyhedra. These inequalities have been receiving considerable attention lately. We provide a polynomial time column-generation algorithm to obtain such inequalities, based on an arbitrary (fixed) number of rows of the simplex tableau. We provide numerical evidence that these inequalities improve the CPLEX integrality gap at the root node on a well-known set of MILP instances, MIPLIB. In the last chapter, we consider a particular MILP instance, Optimal Resilient Distribution Grid Design Problem (ORDGDP). This is a problem of critical importance to infrastructure security and recently attracted a lot of attention from various government agencies (e.g. Presidential Policy Directive of Critical Infrastructure Security 2013). We formulate this problem as a MILP and propose various efficient solution methods blending together well-known decomposition ideas to overcome the numerical intractability encountered using commercial MILP software such as CPLEX.
Cited by 2Open access
MANSINI, Renata, SPERANZA, Maria Grazia, W. OGRYCZAK · Institutional Research Information System (Università degli Studi di Brescia) · 2012
Cited by 0
Renata Mansini, Wlodzimierz Ogryczak, M. Grazia Speranza · 2012
Cited by 0
Marc De Leenheer, Lagrange Relaxation, Linear Constraints · 2013
Cited by 0
Eric Newby · University of the Witwatersrand, Johannesburg Institutional Repository on DSpace (University of the Witwatersrand, Johannesburg) · 2013
A dissertation submitted to the Faculty of Science School of Computational and Applied Mathematics, University of the Witwatersrand, \nJohannesburg. April 27, 2013.
Cited by 5Open access