A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

Resolution Method for Mixed Integer Linear Multiplicative-Linear Bilevel Problems Based on Decomposition Technique

I In this paper, we propose an algorithm base on decomposition technique for solving the mixed integer linear multiplicative-linear bilevel problems. In fact, this algorithm is an application of the algorithm given by G. K. Saharidis et al for the case in which the first level objective function is linear multiplicative. We use properties of quasi-concave of bilevel programming problems and decompose the initial problem into two subproblems named RM P and SP . The lower and upper bound provided from the RM P and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is less than an arbitrary tolerance. In conclusion, some numerical examples are presented in order to show the efficiency of algorithm

Designing a clothing supply chain network considering pricing and demand sensitivity to discounts and advertisement

none3These days, clothing companies are becoming more and more developed around the world. Due to the rapid development of these companies, designing an efficient clothing supply chain network can be highly beneficial, especially with the remarkable increase in demand and uncertainties in both supply and demand. In this study, a bi-objective stochastic mixed-integer linear programming model is proposed for designing the supply chain of the clothing industry. The first objective function maximizes total profit and the second one minimizes downside risk. In the presented network, the initial demand and price are uncertain and are incorporated into the model through a set of scenarios. To solve the bi-objective model, weighted normalized goal programming is applied. Besides, a real case study for the clothing industry in Iran is proposed to validate the presented model and developed method. The obtained results showed the validity and efficiency of the current study. Also, sensitivity analyses are conducted to evaluate the effect of several important parameters, such as discount and advertisement, on the supply chain. The results indicate that considering the optimal amount for discount parameter can conceivably enhance total profit by about 20% compared to the time without this discount scheme. When the optimized parameter is taken into account for advertisement, 12% is obtained as total profit.openPaydar M.M.; Olfati M.; Triki C.Paydar, M. M.; Olfati, M.; Triki, C

Meta-heuristic global optimization algorithms for aircraft engines modelling and controller design; A review, research challenges, and exploring the future

Utilizing meta-heuristic global optimization algorithms in gas turbine aero-engines modelling and control problems is proposed over the past two decades as a methodological approach. The purpose of the review is to establish evident shortcomings of these approaches and to identify the remaining research challenges. These challenges need to be addressed to enable the novel, cost-effective techniques to be adopted by aero-engine designers. First, the benefits of global optimization algorithms are stated in terms of philosophy and the nature of different types of these methods. Then, a historical coverage is given for the applications of different optimization techniques applied in different aspects of gas turbine modelling, controller design, and tuning fields. The main challenges for the application of meta-heuristic global optimization algorithms in new advanced engine designs are presented. To deal with these challenges, two efficient optimization algorithms, Competent Genetic Algorithm in single objective feature and aggregative gradient-based algorithm in multi-objective feature are proposed and applied in a turbojet engine controller gain-tuning problem as a case study. A comparison with the publicly available results show that optimization time and convergence indices will be enhanced noticeably. Based on this comparison and analysis, the potential solutions for the remaining research challenges for application to aerospace engineering problems in the future include the implementation of enhanced and modified optimization algorithms and hybrid optimization algorithms in order to achieve optimal results for the advanced engine modelling and controller design procedure with affordable computational effort

The Watermelon Algorithm for The Bilevel Integer Linear Programming Problem

This paper presents an exact algorithm for the bilevel integer linear programming (BILP) problem. The proposed algorithm, which we call the watermelon algorithm, uses a multiway disjunction cut to remove bilevel infeasible solutions from the search space, which was motivated by how watermelon seeds can be carved out by a scoop. Serving as the scoop, a polyhedron is designed to enclose as many bilevel infeasible solutions as possible, and then the complement of this polyhedron is applied to the search space as a multiway disjunction cut in a branch-and-bound framework. We have proved that the watermelon algorithm is able to solve all BILP instances finitely and correctly, providing either a global optimal solution or a certificate of infeasibility or unboundedness. Computational experiment results on two sets of small- to medium-sized instances suggest that the watermelon algorithm could be significantly more efficient than previous branch-and-bound based BILP algorithms

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

International audienceBilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

Three essays on bilevel optimization algorithms and applications

This thesis consists of three journal papers I have worked on during the past three years of my PhD studies. In the first paper, we presented a multi-objective integer programming model for the gene stacking problem. Although the gene stacking problem is proved to be NP-hard, we have been able to obtain Pareto frontiers for smaller sized instances within one minute using the state-of-the-art commercial computer solvers in our computational experiments. In the second paper, we presented an exact algorithm for the bilevel mixed integer linear programming (BMILP) problem under three simplifying assumptions. Compared to these existing ones, our new algorithm relies on weaker assumptions, explicitly considers infinite optimal, infeasible, and unbounded cases, and is proved to terminate infinitely with the correct output. We report results of our computational experiments on a small library of BMILP test instances, which we have created and made publicly available online. In the third paper, we presented the watermelon algorithm for the bilevel integer linear programming (BILP) problem. To our best knowledge, it is the first exact algorithm which promises to solve all possible BILPs, including finitely optimal, infeasible, and unbounded cases. What is more, our algorithm does not rely on any simplifying condition, allowing even the case of unboundedness for the high point problem. We prove that the watermelon algorithm must finitely terminate with the correct output. Computational experiments are also reported showing the efficiency of our algorithm