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

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.

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Inexact Derivative-Free Optimization for Bilevel Learning

Journal of Mathematical Imaging and Vision ◽

10.1007/s10851-021-01020-8 ◽

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).

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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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Proximal Gradient Method for Solving Bilevel Optimization Problems

Mathematical and Computational Applications ◽

10.3390/mca25040066 ◽

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.

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Optimal Road Toll Design from the Perspective of Sustainable Development

Discrete Dynamics in Nature and Society ◽

10.1155/2014/548427 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Lin Cheng ◽

Fei Han

Keyword(s):

Sustainable Development ◽

Design Problem ◽

Bilevel Optimization ◽

Transportation System ◽

Projection Algorithm ◽

The Sustainable Development ◽

Route Choice Behavior ◽

Total Travel Time ◽

Level Problem ◽

Upper Level

This paper investigates the optimal road toll design problem from the perspective of sustainable development at network-wide level. In this paper the sustainable development level of transportation system is quantitatively described with the total vehicular emission, total fuel consumption, and total travel time in the network. In order to simultaneously consider the impacts of all these three indicators on sustainability of transportation system, we integrate them into a sustainable development index (SD-index) by a linear combination, and then we establish the corresponding bilevel optimization model. The upper level problem is the network toll design problem to maximize the SD-index from the viewpoint of traffic managers, and the lower level problem is to depict travelers’ route choice behavior under a certain road toll scheme. Finally, a combined genetic algorithm and gradient projection algorithm (GA-GP) is used to solve the bilevel model, in which the GP algorithm solves the traffic assignment problem with road toll scheme in the lower level. In order to verify the proposed model and algorithm, we take the Nguyen-Dupuis network for the numerical example, and the computing results show that the model and algorithm are effective and efficient.

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Evacuation Network Optimization Model with Lane-Based Reversal and Routing

Mathematical Problems in Engineering ◽

10.1155/2016/1273508 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Xing Zhao ◽

Zhao-yan Feng ◽

Yan Li ◽

Antoine Bernard

Keyword(s):

Emergency Response ◽

Network Optimization ◽

Optimization Model ◽

Simulated Annealing Algorithm ◽

Search Algorithm ◽

Single Arc ◽

Annealing Algorithm ◽

Upper Level ◽

Reversal Design ◽

Lane Based

Sometimes, the evacuation measure may seem to be the best choice as an emergency response. To enable an efficiency evacuation, a network optimization model which integrates lane-based reversal design and routing with intersection crossing conflict elimination for evacuation is constructed. The proposed bilevel model minimizes the total evacuation time to leave the evacuation zone. A tabu search algorithm is applied to find an optimal lane reversal plan in the upper-level. The lower-level utilizes a simulated annealing algorithm to get two types of “a single arc for an intersection approach” and “multiple arcs for an intersection approach” lane-based route plans with intersection crossing conflict elimination. An experiment of a nine-intersection evacuation zone illustrates the validity of the model and the algorithm. A field case with network topology of Jianye District around the Nanjing Olympics Sports Center is studied to show the applicability of this algorithm.

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Hierarchical Scheduling Scheme for AC/DC Hybrid Active Distribution Network Based on Multi-Stakeholders

Energies ◽

10.3390/en11102830 ◽

2018 ◽

Vol 11 (10) ◽

pp. 2830 ◽

Cited By ~ 4

Author(s):

Chang Ye ◽

Shihong Miao ◽

Yaowang Li ◽

Chao Li ◽

Lixing Li

Keyword(s):

Optimization Model ◽

Distribution Network ◽

Hierarchical Optimization ◽

Active Distribution Network ◽

Scheduling Scheme ◽

Multi Stage ◽

Flexible Loads ◽

Level Model ◽

Load Regulation ◽

Upper Level

This paper presents a hierarchical multi-stage scheduling scheme for the AC/DC hybrid active distribution network (ADN). The load regulation center (LRC) is considered in the developed scheduling strategy, as well as the AC and DC sub-network operators. They are taken to be different stakeholders. To coordinate the interests of all stakeholders, a two-level optimization model is established. The flexible loads are dispatched by LRC in the upper-level optimization model, the objective of which is minimizing the loss of the entire distribution network. The lower-level optimization is divided into two sub-optimal models, and they are carried out to minimize the operating costs of the AC/DC sub-network operators respectively. This two-level model avoids the difficulty of solving multi-objective optimization and can clarify the role of various stakeholders in the system scheduling. To solve the model effectively, a discrete wind-driven optimization (DWDO) algorithm is proposed. Then, considering the combination of the proposed DWDO algorithm and the YALMIP toolbox, a hierarchical optimization algorithm (HOA) is developed. The HOA can obtain the overall optimization result of the system through the iterative optimization of the upper and lower levels. Finally, the simulation results verify the effectiveness of the proposed scheduling scheme.

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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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