Fuzzy multilevel programming with a hybrid intelligent algorithm

AbstractIn order to model fuzzy decentralized decision-making problem, fuzzy expected value multilevel programming and chance-constrained multilevel programming are introduced. Furthermore, fuzzy simulation, neural network, and genetic algorithm are integrated to produce a hybrid intelligent algorithm for finding the Stackelberg-Nash equilibrium. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm

Stochastic Multilevel Programming with a Hybrid Intelligent Algorithm

A framework of stochastic multilevel programming is proposed for modelling decentralized decision-making problem in stochastic environment. According to different decision criteria, the stochastic decentralized decision-making problem is formulated as expected value multilevel programming, and chanceconstrained multilevel programming. In order to solve the proposed stochastic multilevel programming models for the Stackelberg-Nash equilibriums, genetic algorithms, neural networks and stochastic simulation are integrated to produce a hybrid intelligent algorithm. Finally, two numerical examples are provided to illustrate the effectiveness of the hybrid intelligent algorithm

Fuzzy Bi-level Decision-Making Techniques: A Survey

© 2016 the authors. Bi-level decision-making techniques aim to deal with decentralized management problems that feature interactive decision entities distributed throughout a bi-level hierarchy. A challenge in handling bi-level decision problems is that various uncertainties naturally appear in decision-making process. Significant efforts have been devoted that fuzzy set techniques can be used to effectively deal with uncertain issues in bi-level decision-making, known as fuzzy bi-level decision-making techniques, and researchers have successfully gained experience in this area. It is thus vital that an instructive review of current trends in this area should be conducted, not only of the theoretical research but also the practical developments. This paper systematically reviews up-to-date fuzzy bi-level decisionmaking techniques, including models, approaches, algorithms and systems. It also clusters related technique developments into four main categories: basic fuzzy bi-level decision-making, fuzzy bi-level decision-making with multiple optima, fuzzy random bi-level decision-making, and the applications of bi-level decision-making techniques in different domains. By providing state-of-the-art knowledge, this survey paper will directly support researchers and practitioners in their understanding of developments in theoretical research results and applications in relation to fuzzy bi-level decision-making techniques

A globally convergent neurodynamics optimization model for mathematical programming with equilibrium constraints

summary:This paper introduces a neurodynamics optimization model to compute the solution of mathematical programming with equilibrium constraints (MPEC). A smoothing method based on NPC-function is used to obtain a relaxed optimization problem. The optimal solution of the global optimization problem is estimated using a new neurodynamic system, which, in finite time, is convergent with its equilibrium point. Compared to existing models, the proposed model has a simple structure, with low complexity. The new dynamical system is investigated theoretically, and it is proved that the steady state of the proposed neural network is asymptotic stable and global convergence to the optimal solution of MPEC. Numerical simulations of several examples of MPEC are presented, all of which confirm the agreement between the theoretical and numerical aspects of the problem and show the effectiveness of the proposed model. Moreover, an application to resource allocation problem shows that the new method is a simple, but efficient, and practical algorithm for the solution of real-world MPEC problems

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

Stability Verification of Neural Network Controllers using Mixed-Integer Programming

We propose a framework for the stability verification of Mixed-Integer Linear Programming (MILP) representable control policies. This framework compares a fixed candidate policy, which admits an efficient parameterization and can be evaluated at a low computational cost, against a fixed baseline policy, which is known to be stable but expensive to evaluate. We provide sufficient conditions for the closed-loop stability of the candidate policy in terms of the worst-case approximation error with respect to the baseline policy, and we show that these conditions can be checked by solving a Mixed-Integer Quadratic Program (MIQP). Additionally, we demonstrate that an outer and inner approximation of the stability region of the candidate policy can be computed by solving an MILP. The proposed framework is sufficiently general to accommodate a broad range of candidate policies including ReLU Neural Networks (NNs), optimal solution maps of parametric quadratic programs, and Model Predictive Control (MPC) policies. We also present an open-source toolbox in Python based on the proposed framework, which allows for the easy verification of custom NN architectures and MPC formulations. We showcase the flexibility and reliability of our framework in the context of a DC-DC power converter case study and investigate its computational complexity

Solution of Linear Programming Problems using a Neural Network with Non-Linear Feedback

This paper presents a recurrent neural circuit for solving linear programming problems. The objective is to minimize a linear cost function subject to linear constraints. The proposed circuit employs non-linear feedback, in the form of unipolar comparators, to introduce transcendental terms in the energy function ensuring fast convergence to the solution. The proof of validity of the energy function is also provided. The hardware complexity of the proposed circuit compares favorably with other proposed circuits for the same task. PSPICE simulation results are presented for a chosen optimization problem and are found to agree with the algebraic solution. Hardware test results for a 2–variable problem further serve to strengthen the proposed theory

Modeling Decision Systems via Uncertain Programming

By uncertain programming we mean the optimization theory in generally uncertain (random, fuzzy, rough, fuzzy random, etc.) environments. The main purpose of this paper is to present a brief review on uncertain programming models, and classify them into three broad classes: expected value model, chanceconstrained programming and dependent-chance programming. This presentation is based on the book: B. Liu, Theory and Practice of Uncertain Programming, PhisicaVerlag, Heidelberg, 200