1,782 research outputs found
Bilevel programming for analysis of low-complexity control of linear systems with constraints
In this paper we use bilevel programming to find the maximum difference between a reference controller and a low-complexity controller in terms of the infinity-norm difference of their control laws. A nominal MPC for linear systems with constraints, and a robust MPC for linear systems with bounded additive noise are considered as reference controllers. For possible low-complexity controllers we discuss partial enumeration (PE), Voronoi/closest point, triangulation, linear controller with saturation, and others. A small difference in the norm between a low-complexity controller and a robust MPC may be used to guarantee closed-loop stability of the low-complexity controller and indicate that the behaviour or performance of the low-complexity controller will be similar to that of the reference one. We further discuss how bilevel programming may be used for closed-loop analysis of model reduction
Time and Location Aware Mobile Data Pricing
Mobile users' correlated mobility and data consumption patterns often lead to
severe cellular network congestion in peak hours and hot spots. This paper
presents an optimal design of time and location aware mobile data pricing,
which incentivizes users to smooth traffic and reduce network congestion. We
derive the optimal pricing scheme through analyzing a two-stage decision
process, where the operator determines the time and location aware prices by
minimizing his total cost in Stage I, and each mobile user schedules his mobile
traffic by maximizing his payoff (i.e., utility minus payment) in Stage II. We
formulate the two-stage decision problem as a bilevel optimization problem, and
propose a derivative-free algorithm to solve the problem for any increasing
concave user utility functions. We further develop low complexity algorithms
for the commonly used logarithmic and linear utility functions. The optimal
pricing scheme ensures a win-win situation for the operator and users.
Simulations show that the operator can reduce the cost by up to 97.52% in the
logarithmic utility case and 98.70% in the linear utility case, and users can
increase their payoff by up to 79.69% and 106.10% for the two types of
utilities, respectively, comparing with a time and location independent pricing
benchmark. Our study suggests that the operator should provide price discounts
at less crowded time slots and locations, and the discounts need to be
significant when the operator's cost of provisioning excessive traffic is high
or users' willingness to delay traffic is low.Comment: This manuscript serves as the online technical report of the article
accepted by IEEE Transactions on Mobile Computin
A new solution algorithm for solving rule-sets based bilevel decision problems
Copyright © 2012 John Wiley & Sons, Ltd. Copyright © 2012 John Wiley & Sons, Ltd. Bilevel decision addresses compromises between two interacting decision entities within a given hierarchical complex system under distributed environments. Bilevel programming typically solves bilevel decision problems. However, formulation of objectives and constraints in mathematical functions is required, which are difficult, and sometimes impossible, in real-world situations because of various uncertainties. Our study develops a rule-set based bilevel decision approach, which models a bilevel decision problem by creating, transforming and reducing related rule sets. This study develops a new rule-sets based solution algorithm to obtain an optimal solution from the bilevel decision problem described by rule sets. A case study and a set of experiments illustrate both functions and the effectiveness of the developed algorithm in solving a bilevel decision problem
Bilevel optimisation with embedded neural networks: Application to scheduling and control integration
Scheduling problems requires to explicitly account for control considerations
in their optimisation. The literature proposes two traditional ways to solve
this integrated problem: hierarchical and monolithic. The monolithic approach
ignores the control level's objective and incorporates it as a constraint into
the upper level at the cost of suboptimality. The hierarchical approach
requires solving a mathematically complex bilevel problem with the scheduling
acting as the leader and control as the follower. The linking variables between
both levels belong to a small subset of scheduling and control decision
variables. For this subset of variables, data-driven surrogate models have been
used to learn follower responses to different leader decisions. In this work,
we propose to use ReLU neural networks for the control level. Consequently, the
bilevel problem is collapsed into a single-level MILP that is still able to
account for the control level's objective. This single-level MILP reformulation
is compared with the monolithic approach and benchmarked against embedding a
nonlinear expression of the neural networks into the optimisation. Moreover, a
neural network is used to predict control level feasibility. The case studies
involve batch reactor and sequential batch process scheduling problems. The
proposed methodology finds optimal solutions while largely outperforming both
approaches in terms of computational time. Additionally, due to well-developed
MILP solvers, adding ReLU neural networks in a MILP form marginally impacts the
computational time. The solution's error due to prediction accuracy is
correlated with the neural network training error. Overall, we expose how - by
using an existing big-M reformulation and being careful about integrating
machine learning and optimisation pipelines - we can more efficiently solve the
bilevel scheduling-control problem with high accuracy.Comment: 18 page
Integer Bilevel Linear Programming Problems: New Results and Applications
Integer Bilevel Linear Programming Problems: New Results and Application
Integer Bilevel Linear Programming Problems: New Results and Applications
Integer Bilevel Linear Programming Problems: New Results and Application
- …