scholarly journals Inverse Mixed Integer Optimization: Polyhedral Insights and Trust Region Methods

2022 ◽  
Author(s):  
Merve Bodur ◽  
Timothy C. Y. Chan ◽  
Ian Yihang Zhu
Keyword(s):  
Optimization Problems ◽  
Linear Optimization ◽  
Trust Region ◽  
Inverse Optimization ◽  
Mixed Integer ◽  
Integer Optimization ◽  
Solution Algorithms ◽  
Linear Optimization Problems ◽  
Mixed Integer Linear Optimization ◽  
Plane Solution

Inverse optimization—determining parameters of an optimization problem that render a given solution optimal—has received increasing attention in recent years. Although significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality conditions is established and leveraged to design new cutting plane solution algorithms. Through an extensive set of computational experiments, we show that our methods provide substantial improvements over existing methods in solving the largest and most difficult instances to date.

Multirow Intersection Cuts Based on the Infinity Norm

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.

scholarly journals Mixed integer nonlinear programming (MINLP)-based bandwidth utility function on internet pricing scheme with monitoring and marginal cost

2019 ◽  
Vol 9 (2) ◽  
pp. 1240
Author(s):  
Robinson Sitepu ◽  
Fitri Maya Puspita ◽  
Elika Kurniadi ◽  
Yunita Yunita ◽  
Shintya Apriliyani
Keyword(s):  
Utility Function ◽  
Marginal Cost ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Monitoring Cost ◽  
Internet Pricing ◽  
Pricing Scheme ◽  
Linear Optimization Problems

<span>The development of the internet in this era of globalization has increased fast. The need for internet becomes unlimited. Utility functions as one of measurements in internet usage, were usually associated with a level of satisfaction of users for the use of information services used. There are three internet pricing schemes used, that are flat fee, usage based and two-part tariff schemes by using one of the utility function which is Bandwidth Diminished with Increasing Bandwidth with monitoring cost and marginal cost. Internet pricing scheme will be solved by LINGO 13.0 in form of non-linear optimization problems to get optimal solution. The optimal solution is obtained using the either usage-based pricing scheme model or two-part tariff pricing scheme model for each services offered, if the comparison is with flat-fee pricing scheme. It is the best way for provider to offer network based on usage based scheme. The results show that by applying two part tariff scheme, the providers can maximize its revenue either for homogeneous or heterogeneous consumers.</span>

Optimal Test Assembly in Practice

2013 ◽  
Vol 221 (3) ◽  
pp. 190-200 ◽  
Author(s):  
Jörg-Tobias Kuhn ◽  
Thomas Kiefer
Keyword(s):  
Particle Physics ◽  
Large Scale ◽  
Block Design ◽  
Linear Optimization ◽  
Mixed Integer ◽  
Optimal Test ◽  
Automated Test Assembly ◽  
Test Assembly ◽  
Block Level ◽  
Mixed Integer Linear Optimization

Several techniques have been developed in recent years to generate optimal large-scale assessments (LSAs) of student achievement. These techniques often represent a blend of procedures from such diverse fields as experimental design, combinatorial optimization, particle physics, or neural networks. However, despite the theoretical advances in the field, there still exists a surprising scarcity of well-documented test designs in which all factors that have guided design decisions are explicitly and clearly communicated. This paper therefore has two goals. First, a brief summary of relevant key terms, as well as experimental designs and automated test assembly routines in LSA, is given. Second, conceptual and methodological steps in designing the assessment of the Austrian educational standards in mathematics are described in detail. The test design was generated using a two-step procedure, starting at the item block level and continuing at the item level. Initially, a partially balanced incomplete item block design was generated using simulated annealing, whereas in a second step, items were assigned to the item blocks using mixed-integer linear optimization in combination with a shadow-test approach.

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