An Evolutionary Algorithm for Solving Bilevel Programming Problems Using Duality Conditions

Bilevel programming is characterized by two optimization problems located at different levels, in which the constraint region of the upper level problem is implicitly determined by the lower level problem. This paper is focused on a class of bilevel programming with a linear lower level problem and presents a new algorithm for solving this kind of problems by combining an evolutionary algorithm with the duality principle. First, by using the prime-dual conditions of the lower level problem, the original problem is transformed into a single-level nonlinear programming problem. In addition, for the dual problem of the lower level, the feasible bases are taken as individuals in population. For each individual, the values of dual variables can be obtained by taking the dual problem into account, thus simplifying the single-level problem. Finally, the simplified problem is solved, and the objective value is taken as the fitness of the individual. Besides, when nonconvex functions are involved in the upper level, a coevolutionary scheme is incorporated to obtain global optima. In the computational experiment, 10 problems, smaller or larger-scale, are solved, and the results show that the proposed algorithm is efficient and robust.

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Non-Probabilistic Based Structural Design Optimization Under External Load Uncertainty With Eigenvalue-Superposition of Convex Models

Volume 3: 38th Design Automation Conference, Parts A and B ◽

10.1115/detc2012-71316 ◽

2012 ◽

Author(s):

Xike Zhao ◽

Hae Chang Gea ◽

Limei Xu

Keyword(s):

Design Optimization ◽

Structural Design ◽

External Load ◽

Optimization Problems ◽

Global Optimum ◽

Global Optimality ◽

Lower Level Problem ◽

Structural Design Optimization ◽

Level Problem ◽

Upper Level

The non-probabilistic-based structural design optimization problems with external load uncertainties are often solved through a two-level approach. However there are several challenges in this method. Firstly, to assure the reliability of the design, the lower level problem must be solved to its global optimality. Secondly, the sensitivity of the upper level problem cannot be analytically derived. To overcome these challenges, a new method based on the Eigenvalue-Superposition of Convex Models (ESCM) is proposed in this paper. The ESCM method replaces the global optimum of the lower level problem by a confidence bound, namely the ESCM bound, and with which the two-level problem can be formulated into a single level problem. The advantages of the ESCM method in efficiency and stability are demonstrated through numerical examples.

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On Bilevel Optimization with Inexact Follower

Decision Analysis ◽

10.1287/deca.2019.0392 ◽

2020 ◽

Vol 17 (1) ◽

pp. 74-95 ◽

Cited By ~ 1

Author(s):

M. Hosein Zare ◽

Oleg A. Prokopyev ◽

Denis Sauré

Keyword(s):

Optimization Problems ◽

Broad Class ◽

Bilevel Optimization ◽

Modeling Framework ◽

Optimization Framework ◽

Lower Level Problem ◽

Level Problem ◽

Potential Impact ◽

Upper Level ◽

Optimal Reaction

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

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A New Method To Solve Bi-Level Quadratic Linear Fractional Programming Problems

International Game Theory Review ◽

10.1142/s0219198915400174 ◽

2015 ◽

Vol 17 (02) ◽

pp. 1540017

Author(s):

Sanjeet Singh ◽

Nivedita Haldar

Keyword(s):

Objective Function ◽

Fractional Programming ◽

Optimal Solution ◽

Linear Constraints ◽

New Method ◽

Linear Fractional Programming ◽

Single Level ◽

Lower Level Problem ◽

Level Problem ◽

Upper Level

In this paper, we have developed a new method to solve bi-level quadratic linear fractional programming (BLQLFP) problems in which the upper-level objective function is quadratic and the lower-level objective function is linear fractional. In this method a BLQLFP problem is transformed into an equivalent single-level quadratic programming (QP) problem with linear constraints by forcing the duality gap of the lower-level problem to zero. Then by obtaining all vertices of the constraint region of the dual of the lower-level problem, which is a convex polyhedron, the single-level QP problem is converted into a series of finite number of QP problems with linear constraints which can be solved by any standard method for solving a QP. The best among the optimal solutions gives the desired optimal solution for the original bi-level programming (BLP) problem. Theoretical results have been illustrated with the help of a numerical example.

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Multi-Objective Robust Optimization Under Interval Uncertainty Using Online Approximation and Constraint Cuts

Journal of Mechanical Design ◽

10.1115/1.4003918 ◽

2011 ◽

Vol 133 (6) ◽

Cited By ~ 26

Author(s):

W. Hu ◽

M. Li ◽

S. Azarm ◽

A. Almansoori

Keyword(s):

Optimization Problem ◽

Optimization Problems ◽

Computational Effort ◽

Interval Uncertainty ◽

Test Results ◽

Robust Solution ◽

Multi Objective ◽

Function Calls ◽

Level Problem ◽

Upper Level

Many engineering optimization problems are multi-objective, constrained and have uncertainty in their inputs. For such problems it is desirable to obtain solutions that are multi-objectively optimum and robust. A robust solution is one that as a result of input uncertainty has variations in its objective and constraint functions which are within an acceptable range. This paper presents a new approximation-assisted MORO (AA-MORO) technique with interval uncertainty. The technique is a significant improvement, in terms of computational effort, over previously reported MORO techniques. AA-MORO includes an upper-level problem that solves a multi-objective optimization problem whose feasible domain is iteratively restricted by constraint cuts determined by a lower-level optimization problem. AA-MORO also includes an online approximation wherein optimal solutions from the upper- and lower-level optimization problems are used to iteratively improve an approximation to the objective and constraint functions. Several examples are used to test the proposed technique. The test results show that the proposed AA-MORO reasonably approximates solutions obtained from previous MORO approaches while its computational effort, in terms of the number of function calls, is significantly reduced compared to the previous approaches.

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Reduction of Dimension of the Upper Level Problem in a Bilevel Programming Model Part 2

Intelligent Decision Technologies - Smart Innovation, Systems and Technologies ◽

10.1007/978-3-642-22194-1_27 ◽

2011 ◽

pp. 265-272 ◽

Cited By ~ 2

Author(s):

Vyacheslav V. Kalashnikov ◽

Stephan Dempe ◽

Gerardo A. Pérez-Valdés ◽

Nataliya I. Kalashnykova

Keyword(s):

Bilevel Programming ◽

Programming Model ◽

Level Problem ◽

Reduction Of Dimension ◽

Upper Level ◽

Bilevel Programming Model

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Solving Signal Control Problems with Second-Order Sensitivity Information of Equilibrium Network Flows

Journal of Applied Mathematics ◽

10.1155/2014/947190 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Hsun-Jung Cho ◽

You-Heng Huang

Keyword(s):

Control Problem ◽

Network Flows ◽

Stackelberg Game ◽

Nonlinear Approximation ◽

Second Order ◽

Signal Control ◽

Reaction Function ◽

Lower Level Problem ◽

Level Problem ◽

Upper Level

The equilibrium network signal control problem is represented as a Stackelberg game. Due to the characteristics of a Stackelberg game, solving the upper-level problem and lower-level problem iteratively cannot be expected to converge to the solution. The reaction function of the lower-level problem is the key information to solve a Stackelberg game. Usually, the reaction function is approximated by the network sensitivity information. This paper firstly presents the general form of the second-order sensitivity formula for equilibrium network flows. The second-order sensitivity information can be applied to the second-order reaction function to solve the network signal control problem efficiently. Finally, this paper also demonstrates two numerical examples that show the computation of second-order sensitivity and the speed of convergence of the nonlinear approximation algorithm.

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Minimization of cost and congestion management using interline power flow controller

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-07-2015-0255 ◽

2016 ◽

Vol 35 (5) ◽

pp. 1495-1512

Author(s):

Uma Velayutham ◽

Lakshmi Ponnusamy ◽

Gomathi Venugopal

Keyword(s):

Social Welfare ◽

Power Flow ◽

Optimal Power Flow ◽

Real Power ◽

Content Type ◽

Optimal Power ◽

Lower Level Problem ◽

Level Problem ◽

Upper Level ◽

Power Flow Controller

Purpose The purpose of this paper is to optimally locate and size the FACTS device, namely, interline power flow controller in order to minimize the total cost and relieve congestion in a power system. This security analysis helps independent system operator (ISO) to have a better planning and market clearing criteria during any operating state of the system. Design/methodology/approach A multi-objective optimization problem has been developed including real power performance index (RPPI) and expected security cost (ESC). A security constrained optimal power flow has been developed as expected security cost optimal power flow problem which gives the probabilities of operating the system in all possible pre-contingency and post-contingency states subjected to various equality and inequality constraints. Maximizing social welfare is the objective function considered for normal state, while minimizing compensations for generations rescheduling and maximizing social welfare are the objectives in case of contingency states. The proposed work is viewed as a two level problem wherein the upper-level problem is to optimally locate IPFC using RPPI and the lower-level problem is to minimize the ESC subjected to various system constraints. Both upper-level and lower-level problem are solved using particle swarm optimization and The performance of the proposed algorithm is tested under severe line outages and has been validated using IEEE 30 bus system. Findings The proposed methodology shows that IPFC controls the power flows in the network without generation rescheduling or topological changes and thus improves the performance of the system. It is found that the benefit achieved in the ESC due to the installation of IPFC is greater than the annual investment cost of the device. ISO cannot achieve minimum total system cost by merely rescheduling generators. Instead of rescheduling, FACTS devices can be used for compensation by achieving minimum cost. IPFC can be used to compensate the congested lines and transfer cheaper power from generators to consumers. Originality/value Operational reliability, financial profitability and efficient utilization of the existing transmission system infrastructure has been achieved using single FACTS device. Instead of using multiple FATCS devices, if a single FACTS device like IPFC which itself can compensate several transmission lines is used, then in addition to the facility for independently controlled reactive (series) compensation of each individual line, it provides a capability to directly transfer real power between the compensated lines. Hence an attempt has been made in this paper to incorporate IPFC for relieving congestion in a deregulated environment. However, no previous researches have considered incorporating compensation of multi-transmission line using single IPFC in minimizing ESC. Thus, in this paper, the authors indicate how much the ESC is reduced by installing IPFC.

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An Evolutionary Algorithm Using Duality-Base-Enumerating Scheme for Interval Linear Bilevel Programming Problems

Mathematical Problems in Engineering ◽

10.1155/2014/737515 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Hecheng Li ◽

Lei Fang

Keyword(s):

Evolutionary Algorithm ◽

Bilevel Programming ◽

Search Space ◽

Linear Programs ◽

Single Level ◽

Linear Bilevel Programming ◽

Interval Linear ◽

Bilevel Programming Problems ◽

The Right ◽

Nonlinear Equality

Interval bilevel programming problem is hard to solve due to its hierarchical structure as well as the uncertainty of coefficients. This paper is focused on a class of interval linear bilevel programming problems, and an evolutionary algorithm based on duality bases is proposed. Firstly, the objective coefficients of the lower level and the right-hand-side vector are uniformly encoded as individuals, and the relative intervals are taken as the search space. Secondly, for each encoded individual, based on the duality theorem, the original problem is transformed into a single level program simply involving one nonlinear equality constraint. Further, by enumerating duality bases, this nonlinear equality is deleted, and the single level program is converted into several linear programs. Finally, each individual can be evaluated by solving these linear programs. The computational results of 7 examples show that the algorithm is feasible and robust.

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The Use of the Duality Principle to Solve Optimization Problems

International Journal of Recent Contributions from Engineering Science & IT (iJES) ◽

10.3991/ijes.v6i1.8224 ◽

2018 ◽

Vol 6 (1) ◽

pp. 33

Author(s):

Rowland Jerry Okechukwu Ekeocha ◽

Chukwunedum Uzor ◽

Clement Anetor

Keyword(s):

Optimization Problem ◽

Dual Problem ◽

Optimization Problems ◽

Linear Program ◽

Duality Gap ◽

Duality Principle ◽

Primal Problem ◽

Optimal Values ◽

Primal And Dual Problems ◽

Primal And Dual

The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem. However the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. In other words given any linear program, there is another related linear program called the dual. In this paper, an understanding of the dual linear program will be developed. This understanding will give important insights into the algorithm and solution of optimization problem in linear programming. Thus the main concepts of duality will be explored by the solution of simple optimization problem.

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