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.

scholarly journals An Effective Branch-and-cut algorithm in Order to Solve the Mixed Integer Bi-level Programming

2017 ◽  
Vol 5 (1) ◽  
pp. 1 ◽  
Author(s):  
Arsalan Rahmani ◽  
Majid Yousefikhoshbakht
Keyword(s):  
Linear Programming ◽  
Branch And Bound ◽  
Historical Perspective ◽  
Preference Function ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Branch And Bound Method ◽  
Computational Results ◽  
High Quality ◽  
Level Problem

<p>In this paper, a new branch-and-cut algorithm for mixed integer bi-level programming is proposed. For achieving this purpose, a historical perspective of the development of enumeration methods in the field of bi-level linear programming is considered. Then, we present some obstacles for using branch and bound method based on them, and an algorithm is developed to solve for mixed integer bi-level problem. Finally, we use a preference function to determine the choice of branching and specialized cuts in a branch and cut tree. Computational results are reported and compared favorably to those of previous methods and then implications discussed. The results show that not only the proposed algorithm can find high quality solutions for solving a number of the problems, but also it is competitive with other famous published algorithms.</p>

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.

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.

A Criterion Space Branch-and-Cut Algorithm for Mixed Integer Bilinear Maximum Multiplicative Programs

2022 ◽  
Author(s):  
Vahid Mahmoodian ◽  
Iman Dayarian ◽  
Payman Ghasemi Saghand ◽  
Yu Zhang ◽  
Hadi Charkhgard
Keyword(s):  
Objective Function ◽  
Branch And Bound ◽  
Capacity Allocation ◽  
Branch And Cut ◽  
Nash Bargaining ◽  
Mixed Integer ◽  
Bargaining Solution ◽  
Reliability Optimization ◽  
Criterion Space ◽  
Multiplicative Programs

This study introduces a branch-and-bound algorithm to solve mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). This class of optimization problems arises in many applications, such as finding a Nash bargaining solution (Nash social welfare optimization), capacity allocation markets, reliability optimization, etc. The proposed algorithm applies multiobjective optimization principles to solve MIBL-MMPs exploiting a special characteristic in these problems. That is, taking each multiplicative term in the objective function as a dummy objective function, the projection of an optimal solution of MIBL-MMPs is a nondominated point in the space of dummy objectives. Moreover, several enhancements are applied and adjusted to tighten the bounds and improve the performance of the algorithm. The performance of the algorithm is investigated by 400 randomly generated sample instances of MIBL-MMPs. The obtained result is compared against the outputs of the mixed-integer second order cone programming (SOCP) solver in CPLEX and a state-of-the-art algorithm in the literature for this problem. Our analysis on this comparison shows that the proposed algorithm outperforms the fastest existing method, that is, the SOCP solver, by a factor of 6.54 on average. Summary of Contribution: The scope of this paper is defined over a class of mixed-integer programs, the so-called mixed-integer bilinear maximum multiplicative programs (MIBL-MMPs). The importance of MIBL-MMPs is highlighted by the fact that they are encountered in applications, such as Nash bargaining, capacity allocation markets, reliability optimization, etc. The mission of the paper is to introduce a novel and effective criterion space branch-and-cut algorithm to solve MIBL-MMPs by solving a finite number of single-objective mixed-integer linear programs. Starting with an initial set of primal and dual bounds, our proposed approach explores the efficient set of the multiobjective problem counterpart of the MIBL-MMP through a criterion space–based branch-and-cut paradigm and iteratively improves the bounds using a branch-and-bound scheme. The bounds are obtained using novel operations developed based on Chebyshev distance and piecewise McCormick envelopes. An extensive computational study demonstrates the efficacy of the proposed algorithm.

A New Formulation for the Dial-a-Ride Problem

2021 ◽  
Author(s):  
Yannik Rist ◽  
Michael A. Forbes
Keyword(s):  
Time Window ◽  
Valid Inequalities ◽  
Route Planning ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Planning Problem ◽  
Current State ◽  
New Formulation ◽  
First Time ◽  
Delivery Location

This paper proposes a new mixed integer programming formulation and branch and cut (BC) algorithm to solve the dial-a-ride problem (DARP). The DARP is a route-planning problem where several vehicles must serve a set of customers, each of which has a pickup and delivery location, and includes time window and ride time constraints. We develop “restricted fragments,” which are select segments of routes that can represent any DARP route. We show how to enumerate these restricted fragments and prove results on domination between them. The formulation we propose is solved with a BC algorithm, which includes new valid inequalities specific to our restricted fragment formulation. The algorithm is benchmarked on existing and new instances, solving nine existing instances to optimality for the first time. In comparison with current state-of-the-art methods, run times are reduced between one and two orders of magnitude on large instances.

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.

scholarly journals Integer Programming on the Junction Tree Polytope for Influence Diagrams

2020 ◽  
Vol 2 (3) ◽  
pp. 209-228
Author(s):  
Axel Parmentier ◽  
Victor Cohen ◽  
Vincent Leclère ◽  
Guillaume Obozinski ◽  
Joseph Salmon
Keyword(s):  
Graphical Model ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Random Variables ◽  
Mixed Integer ◽  
Influence Diagrams ◽  
Probabilistic Graphical Model ◽  
Computationally Efficient ◽  
Junction Tree ◽  
Acyclic Digraph

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.

Team Orienteering with Time-Varying Profit

2021 ◽  
Author(s):  
Qinxiao Yu ◽  
Yossiri Adulyasak ◽  
Louis-Martin Rousseau ◽  
Ning Zhu ◽  
Shoufeng Ma
Keyword(s):  
Valid Inequalities ◽  
Iterated Local Search ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Orienteering Problem ◽  
Time Varying ◽  
High Quality ◽  
Hybrid Heuristic ◽  
Team Orienteering Problem ◽  
Team Orienteering

This paper studies the team orienteering problem, where the arrival time and service time affect the collection of profits. Such interactions result in a nonconcave profit function. This problem integrates the aspect of time scheduling into the routing decision, which can be applied in humanitarian search and rescue operations where the survival rate declines rapidly. Rescue teams are needed to help trapped people in multiple affected sites, whereas the number of people who could be saved depends as well on how long a rescue team spends at each site. Efficient allocation and scheduling of rescue teams is critical to ensure a high survival rate. To solve the problem, we formulate a mixed-integer nonconcave programming model and propose a Benders branch-and-cut algorithm, along with valid inequalities for tightening the upper bound. To solve it more effectively, we introduce a hybrid heuristic that integrates a modified coordinate search (MCS) into an iterated local search. Computational results show that valid inequalities significantly reduce the optimality gap, and the proposed exact method is capable of solving instances where the mixed-integer nonlinear programming solver SCIP fails in finding an optimal solution. In addition, the proposed MCS algorithm is highly efficient compared with other benchmark approaches, whereas the hybrid heuristic is proven to be effective in finding high-quality solutions within short computing times. We also demonstrate the performance of the heuristic with the MCS using instances with up to 100 customers. Summary of Contribution: Motivated by search and rescue (SAR) operations, we consider a generalization of the well-known team orienteering problem (TOP) to incorporate a nonlinear time-varying profit function in conjunction with routing and scheduling decisions. This paper expands the envelope of operations research and computing in several ways. To address the scalability issue of this highly complex combinatorial problem in an exact manner, we propose a Benders branch-and-cut (BBC) algorithm, which allows us to efficiently deal with the nonconcave component. This BBC algorithm is computationally enhanced through valid inequalities used to strengthen the bounds of the BBC. In addition, we propose a highly efficient hybrid heuristic that integrates a modified coordinate search into an iterated local search. It can quickly produce high-quality solutions to this complex problem. The performance of our solution algorithms is demonstrated through a series of computational experiments.

Sign in / Sign up

Export Citation Format

Share Document