On Bilevel Optimization with Inexact Follower

2020 ◽  
Vol 17 (1) ◽  
pp. 74-95 ◽  
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.

scholarly journals Inexact Derivative-Free Optimization for Bilevel Learning

2021 ◽  
Author(s):  
Matthias J. Ehrhardt ◽  
Lindon Roberts
Keyword(s):  
Approximate Solutions ◽  
Bilevel Optimization ◽  
Worst Case ◽  
Lower Level Problem ◽  
Derivative Free Optimization ◽  
Derivative Free ◽  
Common Strategy ◽  
Regularization Techniques ◽  
Level Problem ◽  
Upper Level

AbstractVariational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by-now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require exact lower-level problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worst-case complexity bound for our approach. We test our proposed framework on ROF denoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as high-accuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).

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

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.

scholarly journals Proximal Gradient Method for Solving Bilevel Optimization Problems

2020 ◽  
Vol 25 (4) ◽  
pp. 66
Author(s):  
Seifu Endris Yimer ◽  
Poom Kumam ◽  
Anteneh Getachew Gebrie
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Gradient Methods ◽  
Split Feasibility Problem ◽  
Bilevel Optimization ◽  
Feasibility Problem ◽  
Proximal Gradient Method ◽  
Level Problem ◽  
Upper Level ◽  
The Split Feasibility Problem

In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly convergent iterative algorithm with an inertia effect solving the bilevel optimization problem under our consideration. Furthermore, we present a numerical example of our algorithm to illustrate its applicability.

scholarly journals An Evolutionary Algorithm for Solving Bilevel Programming Problems Using Duality Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hecheng Li ◽  
Lei Fang
Keyword(s):  
Evolutionary Algorithm ◽  
Bilevel Programming ◽  
Dual Problem ◽  
Optimization Problems ◽  
Duality Principle ◽  
Single Level ◽  
Nonconvex Functions ◽  
Lower Level Problem ◽  
Level Problem ◽  
Upper Level

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.

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

2011 ◽  
Vol 133 (6) ◽  
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.

Reduction of the Dimension of the Upper Level Problem in a Bilevel Optimization Model

2015 ◽  
pp. 289-308
Author(s):  
Stephan Dempe ◽  
Vyacheslav Kalashnikov ◽  
Gerardo A. Pérez-Valdés ◽  
Nataliya Kalashnykova
Keyword(s):  
Optimization Model ◽  
Bilevel Optimization ◽  
Level Problem ◽  
Upper Level

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

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.

Minimization of cost and congestion management using interline power flow controller

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.

Path Planning and Control of Redundant Manipulators Using Bilevel Optimization

2020 ◽  
Vol 142 (4) ◽  
Author(s):  
Uriel Nusbaum ◽  
Miri Weiss Cohen ◽  
Yoram Halevi
Keyword(s):  
Optimal Control ◽  
Dynamic Systems ◽  
Bilevel Optimization ◽  
Complex Tasks ◽  
Planning And Control ◽  
Optimal Inputs ◽  
Saturation Constraints ◽  
Level Problem ◽  
Upper Level ◽  
And Control

Abstract Redundancy is a useful feature in dynamic systems which can be exploited to enhance performance in various tasks. In this work, redundancy will be utilized to minimize the energy consumption of a linear manipulator, while in some cases an additional task of end-effector tracking will also be required and achieved. Optimal control theory has been extensively used for the optimization of dynamic systems; however, complex tasks and redundancy make these problems computationally expensive, numerically difficult to solve, and in many cases, ill-defined. In this paper, evolutionary bilevel optimization for the problem is presented. This is done by setting up an upper level optimization problem for a set of decision variables and a lower level one that actually calculates the optimal inputs and trajectories. The upper level problem is solved by a genetic algorithm (GA), whereas the lower level problem uses classical optimal control. As a result, the proposed algorithm allows the optimization of complex tasks that usually cannot be solved in practice using standard optimal control tools. In addition, despite the use of penalty functions to enforce saturation constraints, the algorithm leads to global energy minimization. Illustrative examples of a redundant x-y robotic manipulator with complex overall tasks will be presented, solved, and discussed.

scholarly journals Particle Swarm Optimization Based on Smoothing Approach for Solving a Class of Bi-Level Multiobjective Programming Problem

2017 ◽  
Vol 17 (3) ◽  
pp. 59-74
Author(s):  
Qingping He ◽  
Yibing Lv
Keyword(s):  
Particle Swarm Optimization ◽  
Programming Problem ◽  
Multiobjective Programming ◽  
Particle Swarm ◽  
Swarm Optimization ◽  
Pareto Optimal Front ◽  
Lower Level Problem ◽  
Level Problem ◽  
Multiobjective Programming Problem ◽  
Upper Level

Abstract As a metaheuristic, Particle Swarm Optimization (PSO) has been used to solve the Bi-level Multiobjective Programming Problem (BMPP). However, in the existing solving approach based on PSO for the BMPP, the upper level and the lower level problem are solved interactively by PSO. In this paper, we present a different solving approach based on PSO for the BMPP. Firstly, we replace the lower level problem of the BMPP with Kuhn-Tucker optimality conditions and adopt the perturbed Fischer-Burmeister function to smooth the complementary conditions. After that, we adopt PSO approach to solve the smoothed multiobjective programming problem. Numerical results show that our solving approach can obtain the Pareto optimal front of the BMPP efficiently.

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