A new algorithm for resolution of the quadratic programming problem with fuzzy relation inequality constraints

The minimization problem of a quadratic objective function with the max-product fuzzy relation inequality constraints is studied in this paper. In this problem, its objective function is not necessarily convex. Hence, its Hessian matrix is not necessarily positive semi-definite. Therefore, we cannot apply the modified simplex method to solve this problem, in a general case. In this paper, we firstly study the structure of its feasible domain. We then use some properties of n × n real symmetric indefinite matrices, Cholesky’s decomposition, and the least square technique, and convert the problem to a separable programming problem. Furthermore, a relation in terms of a closed form is presented to solve it. Finally, an algorithm is proposed to solve the original problem. An application example in the economic area is given to illustrate the problem. Of course, there are other application examples in the area of digital data service and reliability engineering.     

Monomial geometric programming with fuzzy relation inequality constraints with max-product composition

Monomials function has always been considered as a significant and most extensively used function in real living. Resource allocation, structure optimization and technology management can often apply these functions. In optimization problems the objective functions can be considered by monomials. In this paper, we present monomials geometric programming with fuzzy relation inequalities constraint with max-product composition. Simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. Also, an algorithm and a few practical examples are presented to abbreviate and illustrate the steps of the problem resolution.     

Minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations

This paper considers the following optimization problem: minimization of linear objective functions under the constraints expressed by a system of fuzzy relation equations using max-product composition. We first show some properties of minimal solutions of a system of fuzzy relation equations with max-product composition and present some rules for reducing the original problem. Then we derive an algorithm which enables us to find all the optimal solutions for solving the optimization problem and provide some numerical examples to illustrate.     

The boolean quadratic programming problem with generalized upper bound constraints

We consider the boolean quadratic programming problem with generalized upper bound constraints (BQP-GUB) which subsumes the well-known quadratic semi-assignment problem. BQP-GUB has applications in engineering, production planning and biology. We present various complexity results on the problem along with different metaheuristic algorithms. Results of extensive experimental analysis are presented demonstrating the efficacy of our proposed algorithms.     

On solutions of fuzzy random multiobjective quadratic programming with applications in portfolio problem

In this paper, a multiobjective quadratic programming problem fuzzy random coefficients matrix in the objectives and constraints and the decision vector are fuzzy variables is considered. First, we show that the efficient solutions fuzzy quadratic multiobjective programming problems series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. An application fuzzy portfolio optimization problem as a convex quadratic programming approach is discussed and an acceptable solution to such problem is given. At the end, numerical examples are illustrated in the support of the obtained results.     

Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint

The work examines the feasibility of minimizing a linear objective function subject to a max-t fuzzy relation equation constraint, where t is a continuous/Archimedean t-norm. Conventional methods for solving this problem are significantly improved by, first separating the problem into two sub-problems according to the availability of positive coefficients. This decomposition is thus more easily handled than in previous literature. Next, based on use of the maximum solution of the constraint equation, the sub-problem with non-positive coefficients is solved and the size of the sub-problem with positive coefficients reduced as well. This step is unique among conventional methods, owing to its ability to determine as many optimal variables as possible. Additionally, several rules are developed for simplifying the remaining problem. Finally, those undecided optimal variables are obtained using the covering problem rather than the branch-and-bound methods. Three illustrative examples demonstrate that the proposed approach outperforms conventional schemes. Its potential applications are discussed as well.     

二次规划问题的变时滞神经网络模型的全局指数稳定

研究等式约束下二次规划问题最优解神经网络模型的稳定性,提出一种变时滞Lagrange神经网络求解方法.利用线性矩阵不等式(LMI)技术,得到两个变时滞神经网络模型全局指数稳定的条件.分析表明,此稳定判据能够适应慢变时滞和快变时滞两种情况,具有适用范围宽、保守性小且易于验证等特点.数值仿真结果验证了所提方法的有效性.     

Piecewise regression for fuzzy input–output data with automatic change-point detection by quadratic programming

To handle the large variation issues in fuzzy input–output data, the proposed quadratic programming (QP) method uses a piecewise approach to simultaneously generate the possibility and necessity models, as well as the change-points. According to Tanaka and Lee [H. Tanaka, H. Lee, Interval regression analysis by quadratic programming approach, IEEE Transactions on Fuzzy Systems 6 (1998) 473–481], the QP approach gives more diversely spread coefficients than linear programming (LP) does. However, their approach only deals with crisp input and fuzzy output data. Moreover, their method is weak in handling fluctuating data. So far, no method has been developed to cope with the large variation problems in fuzzy input–output data. Hence, we propose a piecewise regression for fuzzy input–output data with a QP approach. There are three advantages in our method. First, the QP technique gives a more diversely spread coefficient than does a linear programming technique. Second, the piecewise approach is used to detect the change-points in the estimated model automatically, and handle the large variation data such as outliers well. Third, the possibility and necessity models with better fitness in data processing are obtained at the same time. Two examples are presented to demonstrate the merits of the proposed method.     

Fuzzy linear optimization in the presence of the fuzzy relation inequality constraints with max-min composition

In this paper, we study the new linear objective function optimization with respect to the fuzzy relational inequalities defined by max-min composition in which fuzzy inequality replaces ordinary inequality in the constraints. Fuzzy inequalities enable us to attain the optimal points that are better solutions than those resulting from the resolution of the similar problems with Ordinary Inequality constraints (denoted by x_OI in this paper). We have presented an algorithm to generate such optimal solutions. Also two examples are given to illustrate the algorithm presented and some real application of this work.     

Deriving minimal solutions for fuzzy relation equations with max-product composition

This work considers fuzzy relation equations with max-product composition. The critical problem in solving such equations is to determine the minimal solutions when an equation is solvable. However, this problem is NP-hard and difficult to solve [A.V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and Systems 153 (2005) 261-273]. This work first examines the attributes of a solvable equation and characteristics of minimal solutions, then reduces the equation to an irreducible form, and converts the problem into a covering problem, for which minimal solutions are correspondingly determined. Furthermore, for theoretical and practical applications, this work presents a novel method for obtaining minimal solutions. The proposed method easily derives a minimal solution, and obtains other minimal solutions from this predecessor using a back-tracking step. The proposed method is compared with an existing algorithm, and some applications are described in detail.     

Improving a Lagrangian decomposition for the unconstrained binary quadratic programming problem

This paper presents a new alternative of Lagrangian decomposition based on column generation technique to solve the unconstrained binary quadratic programming problem. We use a mixed binary linear version of the original quadratic problem with constraints represented by a graph. This graph is partitioned into clusters of vertices forming subproblems whose solutions use the dual variables obtained by a coordinator problem. Computational experiments consider a set of difficult instances and the results are compared against other methods reported recently in the literature.     

Fuzzy two-stage quadratic programming for planning solid waste management under uncertainty

In this study, a fuzzy two-stage quadratic programming (FTSQP) method is developed for planning waste-management systems under uncertainty. It incorporates approaches of fuzzy quadratic programming and two-stage stochastic programming within a general optimization framework, to better reflect uncertainties expressed as probability-density and fuzzy-membership functions. The FTSQP can be used for analyzing various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated. Moreover, using fuzzy quadratic terms rather than linear ones, the proposed method can improve upon the existing fuzzy linear programs through (a) more effectively optimizing the general satisfaction of the objective and constraints, (b) minimizing the variation of satisfaction degrees among the constraints and leading to more robust solutions, and (c) reflecting the trade-off between the system cost and the constraint-violation risk. The developed method is applied to a case study of municipal solid waste management. The results indicate that reasonable solutions have been generated. They will allow in-depth analyses of trade-offs between environmental and economic objectives as well as those between system cost and decision-maker's satisfaction degree.     

An interactive fuzzy programming approach for bi-objective straight and U-shaped assembly line balancing problem

The consideration of this paper is given to address the straight and U-shaped assembly line balancing problem. Although many attempts in the literature have been made to develop deterministic version of the assembly line model, the attention is not considerably given to those in uncertain environment. In this paper, a novel bi-objective fuzzy mixed-integer linear programming model (BOFMILP) is developed so that triangular fuzzy numbers (TFNs) are employed in order to represent uncertainty and vagueness associated with the task processing times in the real production systems. In this proposed model, two conflicting objectives (minimizing the number of stations as well as cycle time) are considered simultaneously with respect to set of constraints. For this purpose, an appropriate strategy in which new two-phase interactive fuzzy programming approach is proposed as a solution method to find an efficient compromise solution. Finally, validity of the proposed model as well as its solution approach are evaluated though numerical examples. In addition, a comparison study is conducted over some test problems in order to assess the performance of the proposed solution approach. The results demonstrate that our proposed interactive fuzzy approach not only can be applied in ALBPs but also is capable to handle any practical MOLP models. Moreover, in light of these results, the proposed model may constitute a framework aiming to assist the decision maker (DM) to deal with uncertainty in assembly line problem.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the “perturbed” convex fuzzy programming is considered. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

求解线性约束的二次规划神经网络学习新算法

提出了一种基于0.618法求解具有线性约束的二次规划问题的神经网络学习新算法。与已有的求解线性约束的二次规划问题的神经网络学习算法相比,该算法的适用范围更广,计算精度更高。其目的是为具有线性约束的二次规划问题的求解提供一种新方法。仿真实验验证了新算法的有效性。     

Global exponential stability of discrete-time recurrent neural network for solving quadratic programming problems subject to linear constraints

In this paper, a discrete-time recurrent neural network with global exponential stability is proposed for solving linear constrained quadratic programming problems. Compared with the existing neural networks for quadratic programming, the proposed neural network in this paper has lower model complexity with only one-layer structure. Moreover, the global exponential stability of the neural network can be guaranteed under some mild conditions. Simulation results with some applications show the performance and characteristic of the proposed neural network.     

A multi-item transportation problem with fuzzy tolerance

This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermann's method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermann's methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.     

Optimization procedures for the bipartite unconstrained 0-1 quadratic programming problem

The bipartite unconstrained 0-1 quadratic programming problem (BQP) is a difficult combinatorial problem defined on a complete graph that consists of selecting a subgraph that maximizes the sum of the weights associated with the chosen vertices and the edges that connect them. The problem has appeared under several different names in the literature, including maximum weight induced subgraph, maximum weight biclique, matrix factorization and maximum cut on bipartite graphs. There are only two unpublished works (technical reports) where heuristic approaches are tested on BQP instances. Our goal is to combine straightforward search elements to balance diversification and intensification in both exact (branch and bound) and heuristic (iterated local search) frameworks. We perform a number of experiments to test individual search components and also to create new benchmarks when comparing against the state of the art, which the proposed procedure outperforms.     

Solving for a quadratic programming with a quadratic constraint based on a neural network frame

In many applications, a class of optimization problems called quadratic programming with a special quadratic constraint (QPQC) often occurs, such as in the fields of maximum entropy spectral estimation, FIR filter design with time–frequency constraint and design of an FIR filter bank with perfect reconstruction property. In order to deal with this kind of optimization problems and be inspired by the computational virtue of analog or dynamic neural networks, a feedback neural network is proposed for solving for this class of QPQC computation problems in real time in this paper. The stability, convergence and computational performance of the proposed neural network have also been analyzed and proved in detail so as to theoretically guarantee the computational effectiveness and capability of the network. From the theoretical analyses it turns out that the solution of a QPQC problem is just the generalized minimum eigenvector of the objective matrix with respect to the constrained matrix. A number of simulation experiments have been given to further support our theoretical analysis and illustrate the computational performance of the proposed network.