scholarly journals Outer approximation for global optimization of mixed-integer quadratic bilevel problems

2021 ◽  
Author(s):  
Thomas Kleinert ◽  
Veronika Grimm ◽  
Martin Schmidt
Keyword(s):  
Optimization Problems ◽  
Strong Duality ◽  
Bilevel Optimization ◽  
Outer Approximation ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Solution Algorithms ◽  
Bilevel Problems ◽  
Upper Level ◽  
Theoretical Developments

AbstractBilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP bilevel problems, i.e., models with a mixed-integer convex-quadratic upper level and a continuous convex-quadratic lower level. This setting allows for a strong-duality-based transformation of the lower level which yields, in general, an equivalent nonconvex single-level reformulation of the original bilevel problem. Under reasonable assumptions, we can derive both a multi- and a single-tree outer-approximation-based cutting-plane algorithm. We show finite termination and correctness of both methods and present extensive numerical results that illustrate the applicability of the approaches. It turns out that the proposed methods are capable of solving bilevel instances with several thousand variables and constraints and significantly outperform classical solution approaches.

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

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.

Uncertainty Handling in Bilevel Optimization for Robust and Reliable Solutions

2018 ◽  
Vol 26 (Suppl. 2) ◽  
pp. 1-24 ◽  
Author(s):  
Zhichao Lu ◽  
Kalyanmoy Deb ◽  
Ankur Sinha
Keyword(s):  
Systematic Study ◽  
Optimization Problems ◽  
Bilevel Optimization ◽  
Test Cases ◽  
Recent Past ◽  
Uncertainty Handling ◽  
Bilevel Problems ◽  
Upper Level ◽  
Definition Of ◽  
First Time

Uncertainties in variables and parameters cause optimization problems to move away from globally-optimal and uncertain solutions. Practitioners resort to finding robust and reliable solutions in such situations. Bilevel optimization problems involving a hierarchy of two nested optimization problems have received a growing attention in the recent past due to their relevance in practice. While a number of studies on bilevel solution methodologies and applications are available for a deterministic setup, but studies on uncertainties in bilevel optimization are rare. In this paper, we suggest methodologies for handling uncertainty in both lower and upper level variables that may occur from different practicalities. For the first time, we perform a systematic study demonstrating the effect of uncertainties in each level along with the definition of robustness and reliability in the context of bilevel optimization. The issues and complexities introduced due to such uncertainties are then studied through a number of test cases, for brevity, we only show results on three test cases. Finally, two real-world bilevel problems involving uncertainties in their variables are solved. The study provides foundations and demon- strates viable directions for further research in uncertainty-based bilevel optimization problems.

scholarly journals Partially distributed outer approximation

2021 ◽  
Author(s):  
Alexander Murray ◽  
Timm Faulwasser ◽  
Veit Hagenmeyer ◽  
Mario E. Villanueva ◽  
Boris Houska
Keyword(s):  
Large Scale ◽  
Optimization Problems ◽  
Outer Approximation ◽  
Mixed Integer ◽  
Global Optimality ◽  
Integer Optimization ◽  
Global Minimizers ◽  
Outer Approximation Method ◽  
Coupling Constraints ◽  
Outer Approximation Algorithm

AbstractThis paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

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.

scholarly journals Optimistic Variants of Single-Objective Bilevel Optimization for Evolutionary Algorithms

2020 ◽  
Vol 19 (03) ◽  
pp. 2050020
Author(s):  
Anuraganand Sharma
Keyword(s):  
Optimization Problems ◽  
Bilevel Optimization ◽  
Benchmark Problems ◽  
New Variant ◽  
Real World Applications ◽  
Bilevel Problems ◽  
Specialized Form ◽  
Comparative Statistical Analysis ◽  
Constraint Optimization Problems ◽  
Single Objective

Single-objective bilevel optimization is a specialized form of constraint optimization problems where one of the constraints is an optimization problem itself. These problems are typically non-convex and strongly NP-Hard. Recently, there has been an increased interest from the evolutionary computation community to model bilevel problems due to its applicability in real-world applications for decision-making problems. In this work, a partial nested evolutionary approach with a local heuristic search has been proposed to solve the benchmark problems and have outstanding results. This approach relies on the concept of intermarriage-crossover in search of feasible regions by exploiting information from the constraints. A new variant has also been proposed to the commonly used convergence approaches, i.e., optimistic and pessimistic. It is called an extreme optimistic approach. The experimental results demonstrate the algorithm converges differently to known optimum solutions with the optimistic variants. Optimistic approach also outperforms pessimistic approach. Comparative statistical analysis of our approach with other recently published partial to complete evolutionary approaches demonstrates very competitive results.

scholarly journals Exit lane allocation with the lane-based signal optimization method at isolated intersections

2020 ◽  
Vol 47 (12) ◽  
pp. 1345-1358
Author(s):  
Qinrui Tang ◽  
Alexander Sohr
Keyword(s):  
Optimization Problems ◽  
Signal Sequence ◽  
Optimization Method ◽  
Signalized Intersections ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Auxiliary Variables ◽  
Signal Optimization ◽  
Pavement Markings ◽  
Lane Based

In signal optimization problems, incompatible movements are usually in either of two states: predecessor or successor. However, if the exit lane is well allocated, the incompatible movements merging at the same destination arm can exist in parallel. The corresponding longer green duration is expected to increase the capacity of intersections. This paper aims to solve the exit lane allocation problem with the lane-based method by applying the three states among incompatible movements at conventional signalized intersections. After introducing auxiliary variables, the problem is formulated as a mixed integer programming and can be solved using a standard branch-and-cut algorithm. In addition to the exit lane allocation results, this proposed method can also determine the cycle length, green duration, start of green, and signal sequence. The results show that the proposed method can obtain a higher capacity than that without the exit lane allocation. The pavement markings are further suggested for safety.

Computing Feasible Points of Bilevel Problems with a Penalty Alternating Direction Method

2020 ◽  
Author(s):  
Thomas Kleinert ◽  
Martin Schmidt
Keyword(s):  
Optimization Problems ◽  
Mathematical Optimization ◽  
Alternating Direction Method ◽  
Convergence Theory ◽  
Mixed Integer ◽  
Solution Quality ◽  
Alternating Direction ◽  
Bilevel Problems ◽  
Primal Heuristics ◽  
Feasible Points

Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, and so on. Often these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization, one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with a mixed-integer linear or quadratic upper level and a linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2,800 instances—which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method in terms of both running times and solution quality. This renders the method a suitable subroutine in global bilevel solvers as well as a reasonable standalone approach. Summary of Contribution: Bilevel optimization problems form a very important class of optimization problems in the field of operations research, which is mainly due to their capability of modeling hierarchical decision processes. However, real-world bilevel problems are usually very hard to solve—especially in the case in which additional mixed-integer aspects are included in the modeling. Hence, the development of fast and reliable primal heuristics for this class of problems is very important. This paper presents such a method.

Modeling Defender-Attacker Problems as Robust Linear Programs with Mixed-Integer Uncertainty Sets

2021 ◽  
Author(s):  
Juan S. Borrero ◽  
Leonardo Lozano
Keyword(s):  
Robust Optimization ◽  
Convex Sets ◽  
Optimization Problems ◽  
Computational Study ◽  
Mixed Integer ◽  
Computational Time ◽  
Point Problem ◽  
Solution Algorithms ◽  
Uncertainty Sets ◽  
Uncertainty Set

We study a class of sequential defender-attacker optimization problems where the defender’s objective is uncertain and depends on the operations of the attacker, which are represented by a mixed-integer uncertainty set. The defender seeks to hedge against the worst possible data realization, resulting in a robust optimization problem with a mixed-integer uncertainty set, which requires the solution of a challenging mixed-integer problem, which can be seen as a saddle-point problem over a nonconvex domain. We study two exact solution algorithms and present two feature applications for which the uncertainty is naturally modeled as a mixed-integer set. Our computational experiments show that the considered algorithms greatly outperform standard algorithms both in terms of computational time and solution quality. Moreover, our results show that modeling uncertainty with mixed-integer sets, instead of approximating the data using convex sets, results in less conservative solutions, which translates to a lower cost for the defender to protect from uncertainty. Summary of Contribution: We consider a class of defender-attacker problems where the defender has to make operational decisions that depend on uncertain actions from an adversarial attacker. Due to the type of information available to the defender, neither probabilistic modeling, nor robust optimization methods with convex uncertainty sets, are well suited to address the defender’s decision-making problem. Consequently, we frame the defender’s problem as a class of robust optimization problems with a mixed-integer uncertainty sets, and devise two exact algorithms that solve this class of problems. A comprehensive computational study shows that for the considered applications, our algorithms improves the performance of existing robust optimization approaches that can be adapted to solve this class of problems. Moreover, we show how mixed-integer uncertainty sets can reduce the level of over-conservatism that is a known issue of robust optimization approaches.

scholarly journals SOLVING THE NONLINEAR DISCRETE TRANSPORTATION PROBLEM BY MINLP OPTIMIZATION

Transport ◽  
2013 ◽  
Vol 29 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Uroš Klanšek
Keyword(s):  
Transportation Problem ◽  
Optimization Problems ◽  
Relaxation Method ◽  
Optimization Method ◽  
Critical Issue ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Cut Method ◽  
Optimum Solution ◽  
Selection Of

The Nonlinear Discrete Transportation Problem (NDTP) belongs to the class of the optimization problems that are generally difficult to solve. The selection of a suitable optimization method by which a specific NDTP can be appropriately solved is frequently a critical issue in obtaining valuable results. The aim of this paper is to present the suitability of five different Mixed-Integer Nonlinear Programming (MINLP) methods, specifically for the exact optimum solution of the NDTP. The evaluated MINLP methods include the extended cutting plane method, the branch and reduce method, the augmented penalty/outer-approximation/equality-relaxation method, the branch and cut method, and the simple branch and bound method. The MINLP methods were tested on a set of NDTPs from the literature. The gained solutions were compared and a correlative evaluation of the considered MINLP methods is shown to demonstrate their suitability for solving the NDTPs.

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