A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems

2019 ◽  
Vol 75 (2) ◽  
pp. 493-513 ◽  
Author(s):  
Ken Kobayashi ◽  
Yuich Takano
Keyword(s):  
Optimization Problems ◽  
Branch And Cut ◽  
Semidefinite Optimization ◽  
Mixed Integer

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 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 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.

Branch-and-cut for combinatorial optimization problems without auxiliary binary variables

2001 ◽  
Vol 16 (1) ◽  
pp. 25-39 ◽  
Author(s):  
I. R. DE FARIAS ◽  
E. L. JOHNSON ◽  
G. L. NEMHAUSER
Keyword(s):  
Production Scheduling ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Polyhedral Combinatorics ◽  
Branch And Cut ◽  
Combinatorial Structure ◽  
Mixed Integer ◽  
Continuous Variables ◽  
Binary Variables ◽  
Combinatorial Optimization Problems

Many optimisation problems involve combinatorial constraints on continuous variables. An example of a combinatorial constraint is that at most one variable in a group of nonnegative variables may be positive. Traditionally, in the mathematical programming community, such problems have been modeled as mixed-integer programs by introducing auxiliary binary variables and additional constraints. Because the number of variables and constraints becomes larger and the combinatorial structure is not used to advantage, these mixed-integer programming models may not be solved satisfactorily, except for small instances. Traditionally, constraint programming approaches to such problems keep and use the combinatorial structure, but do not use linear programming bounds in the search for an optimal solution. Here we present a branch-and-cut approach that considers the combinatorial constraints without the introduction of binary variables. We review the development of this approach and show how strong constraints can be derived using ideas from polyhedral combinatorics. To illustrate the ideas, we present a production scheduling model that arises in the manufacture of fibre optic cables.

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 A Multi-Objective Approach toward Optimal Design of Sustainable Integrated Biodiesel/Diesel Supply Chain Based on First- and Second-Generation Feedstock with Solid Waste Use

Energies ◽  
2021 ◽  
Vol 14 (8) ◽  
pp. 2261
Author(s):  
Evgeniy Ganev ◽  
Boyan Ivanov ◽  
Natasha Vaklieva-Bancheva ◽  
Elisaveta Kirilova ◽  
Yunzile Dzhelil
Keyword(s):  
Supply Chain ◽  
Optimal Design ◽  
Solid Waste ◽  
Second Generation ◽  
Agricultural Land ◽  
Optimization Problems ◽  
Biodiesel Production ◽  
Mixed Integer ◽  
Economic Criterion ◽  
Multi Objective

This study proposes a multi-objective approach for the optimal design of a sustainable Integrated Biodiesel/Diesel Supply Chain (IBDSC) based on first- (sunflower and rapeseed) and second-generation (waste cooking oil and animal fat) feedstocks with solid waste use. It includes mixed-integer linear programming (MILP) models of the economic, environmental and social impact of IBDSC, and respective criteria defined in terms of costs. The purpose is to obtain the optimal number, sizes and locations of bio-refineries and solid waste plants; the areas and amounts of feedstocks needed for biodiesel production; and the transportation mode. The approach is applied on a real case study in which the territory of Bulgaria with its 27 districts is considered. Optimization problems are formulated for a 5-year period using either environmental or economic criteria and the remainder are defined as constraints. The obtained results show that in the case of the economic criterion, 14% of the agricultural land should be used for sunflower and 2% for rapeseed cultivation, while for the environmental case, 12% should be used for rapeseed and 3% for sunflower. In this case, the price of biodiesel is 14% higher, and the generated pollutants are 6.6% lower. The optimal transport for both cases is rail.

Optimization of Real Power Loss and Voltage Stability Index for Distribution Systems with Distributed Generation

2017 ◽  
Vol 33 ◽  
pp. 100-114 ◽  
Author(s):  
Mostafa Elshahed ◽  
Mahmoud Dawod ◽  
Zeinab H. Osman
Keyword(s):  
Genetic Algorithm ◽  
Distributed Generation ◽  
Distribution Systems ◽  
Voltage Stability ◽  
Reactive Power ◽  
Optimization Problems ◽  
Stability Index ◽  
Mixed Integer ◽  
Voltage Profile ◽  
Voltage Stability Index

Integrating Distributed Generation (DG) units into distribution systems can have an impact on the voltage profile, power flow, power losses, and voltage stability. In this paper, a new methodology for DG location and sizing are developed to minimize system losses and maximize voltage stability index (VSI). A proper allocation of DG has to be determined using the fuzzy ranking method to verify best compromised solutions and achieve maximum benefits. Synchronous machines are utilized and its power factor is optimally determined via genetic optimization to inject reactive power to decrease system losses and improve voltage profile and VSI. The Augmented Lagrangian Genetic Algorithm with nonlinear mixed-integer variables and Non-dominated Sorting Genetic Algorithm have been implemented to solve both single/multi-objective function optimization problems. For proposed methodology effectiveness verification, it is tested on 33-bus and 69-bus radial distribution systems then compared with previous works.

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