A neural network for solving a convex quadratic bilevel programming problem

A neural network is proposed for solving a convex quadratic bilevel programming problem. Based on Lyapunov and LaSalle theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the equilibrium, which corresponds to the optimal solution of a convex quadratic bilevel programming problem. Numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem.     

A new neural network for solving linear programming problems

We propose and analyse a new class of neural network models for solving linear programming (LP) problems in real time. We introduce a novel energy function that transforms linear programming into a system of nonlinear differential equations. This system of differential equations can be solved on-line by a simplified low-cost analog neural network containing only one single artificial neuron with adaptive synaptic weights. The network architecture is suitable for currently available CMOS VLSI implementations. An important feature of the proposed neural network architecture is its flexibility and universality. The correctness and performance of the proposed neural network is illustrated by extensive computer simulation experiments.     

Flying elephants: a general method for solving non-differentiable problems

Flying Elephants (FE) is a generalization and a new interpretation of the Hyperbolic Smoothing approach. The article introduces the fundamental smoothing procedures. It contains a general overview of successful applications of the approach for solving a select set of five important problems, namely: distance geometry, covering, clustering, Fermat–Weber and hub location. For each problem the original non-smooth formulation and the succedaneous completely differentiable one are presented. Computational experiments for all related problems obtained results that exhibited a high level of performance according to all criteria: consistency, robustness and efficiency. For each problem some results to illustrate the performance of FE are also presented.     

一类二层凸规划的分解法

研究了一类二层凸规划和与之相应的凸规划问题的等价性．并讨论了这类凸规划的对偶性和鞍点问题,最后给出了求解这类二层凸规划的一个分解法．     

A novel neural network based on NCP function for solving constrained nonconvex optimization problems

This article presents a novel neural network (NN) based on NCP function for solving nonconvex nonlinear optimization (NCNO) problem subject to nonlinear inequality constraints. We first apply the p‐power convexification of the Lagrangian function in the NCNO problem. The proposed NN is a gradient model which is constructed by an NCP function and an unconstrained minimization problem. The main feature of this NN is that its equilibrium point coincides with the optimal solution of the original problem. Under a proper assumption and utilizing a suitable Lyapunov function, it is shown that the proposed NN is Lyapunov stable and convergent to an exact optimal solution of the original problem. Finally, simulation results on two numerical examples and two practical examples are given to show the effectiveness and applicability of the proposed NN. © 2015 Wiley Periodicals, Inc. Complexity 21: 130–141, 2016     

A novel recurrent nonlinear neural network for solving quadratic programming problems

This paper presents a new neural network for solving quadratic programming problems. The new model has a simple form, furthermore it has a good convergence rate with a less number calculation operation than the old models. It converges very fast to exact solution of the dual problem and by substituting in a formulation, the optimal solution of the original problem is obtained. Neural network model with one of numerical method is solved. Finally, simple numerical examples are provided for more illustration.     

Simplex-inspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose objective function is given as a linear function plus a convex function whose arguments are linear functions of the decision variables and whose feasible region is a polytope. We show that there exists an optimal solution to this class of problems on a face of the constraint polytope of dimension not more than the number of arguments of the convex function. Based on this result, we develop a method to solve this problem that is inspired by the simplex method for linear programming. It is shown that this method terminates in a finite number of iterations in the special case that the convex function has only a single argument. We then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for an arbitrary number of arguments in the convex function. A computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables. The work of T. C. Sharkey was supported by a National Science Foundation Graduate Research Fellowship. The work of H. E. Romeijn was supported by the National Science Foundation under Grant No. DMI-0355533.     

A method for solving a fuzzy linear programming

In this paper a fuzzy linear programming problem is presented. Then using the concept of comparison of fuzzy numbers, by the aid of the Mellin transform, we introduce a method for solving this problem.     

A successive quadratic programming method for a class of constrained nonsmooth optimization problems

In this paper we present an algorithm for solving nonlinear programming problems where the objective function contains a possibly nonsmooth convex term. The algorithm successively solves direction finding subproblems which are quadratic programming problems constructed by exploiting the special feature of the objective function. An exact penalty function is used to determine a step-size, once a search direction thus obtained is judged to yield a sufficient reduction in the penalty function value. The penalty parameter is adjusted to a suitable value automatically. Under appropriate assumptions, the algorithm is shown to produce an approximate optimal solution to the problem with any desirable accuracy in a finite number of iterations.     

求解弱线性双层规划问题的一种全局优化方法

双层规划在经济、交通、生态、工程等领域有着广泛而重要的应用.目前对双层规划的研究主要是基于强双层规划和弱双层规划.然而,针对弱双层规划的求解方法却鲜有研究.研究求解弱线性双层规划问题的一种全局优化方法,首先给出弱线性双层规划问题与其松弛问题在最优解上的关系,然后利用线性规划的对偶理论和罚函数方法,讨论该松弛问题和它的罚问题之间的关系.进一步设计了一种求解弱线性双层规划问题的全局优化方法,该方法的优势在于它仅仅需要求解若干个线性规划问题就可以获得原问题的全局最优解.最后,用一个简单算例说明了所提出的方法是可行的.     

线性半向量二层规划问题的全局优化方法

研究了线性半向量二层规划问题的全局优化方法. 利用下层问题的对偶间隙构造了线性半向量二层规划问题的罚问题, 通过分析原问题的最优解与罚问题可行域顶点之间的关系, 将线性半向量二层规划问题转化为有限个线性规划问题, 从而得到线性半向量二层规划问题的全局最优解. 数值结果表明所设计的全局优化方法对线性半向量二层规划问题是可行的.     

A method for solving fuzzy goal programming problems based on MINMAX approach

Narasimhan incorporated fuzzy set theory within goal programming formulation in 1980. Since then numerous research has been carried out in this field. One of the well-known models for solving fuzzy goal programming problems was proposed by Hannan in 1981. In this paper the conventional MINMAX approach in goal programming is applied to solve fuzzy goal programming problems. It is proved that the proposed model is an extension to Hannan model that deals with unbalanced triangular linear membership functions. In addition, it is shown that the new model is equivalent to a model proposed in 1991 by Yang et al. Moreover, a weighted model of the new approach is introduced and is compared with Kim and Whang’s model presented in 1998. A numerical example is given to demonstrate the validity and strengths of the new models.     

Value on a class of non-differentiable market games

We prove the existence of a (unique) Aumann-Shapley value on the space on non-atomic gamesQ ⁿ generated byn-handed glove games. (These are the minima ofn non-atomic mutually singular probability measures.) It is also shown that this value can be extended to a value on the smallest space containingQ ⁿ andpNA.     

Global optimization for a class of fractional programming problems

This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the “problem-defining” matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.     

An algorithm for solving a structured class of linear programming problems

We present a procedure for solving a class of specially structured linear programs. Our approach consists of solving a sequence of continuous knapsack problems each of which requires linear time to solve. The computational effort required by the procedure is proportional to the number of non-zero entries in the constraint matrix. The model arises in portfolio selection, advertising, inventory and production planning and interval linear programming.     

Satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem

This paper proposes a satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem. The aim of this presented approach is to make the more important objective achieving the higher desirable satisfying degree. For different fuzzy relations and fuzzy importance, the reformulated optimization models based on goal programming is proposed. Not only the satisfying results of all the objectives can be acquired, but also the fuzzy importance requirement can be simultaneously actualized. The balance between optimization and relative importance is realized. We demonstrate the efficiency, flexibility and sensitivity of the proposed method by numerical examples.     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

Shooting method for solving equilibrium programming problems

A new iterative method is proposed for solving equilibrium programming problems. The sequence of points it generates is proved to converge weakly to the solution set of the equilibrium problem under study. If the initial point has at least one projection onto the solution set of the equilibrium problem, the sequence generated by the method is shown to converge strongly to the set of these projections. The partial gradient of the initial data is assumed to be invertible and strictly monotone, which differs from the classical skew-symmetry condition.     

A DC programming approach for solving multicast network design problems via the Nesterov smoothing technique

This paper continues our recent effort in applying continuous optimization techniques to study optimal multicast communication networks modeled as bilevel hierarchical clustering problems. Given a finite number of nodes, we consider two different models of multicast networks by identifying a certain number of nodes as cluster centers, and at the same time, locating a particular node that serves as a total center so as to minimize the total transportation cost throughout the network. The fact that the cluster centers and the total center have to be among the given nodes makes these problems discrete optimization problems. Our approach is to reformulate the discrete problems as continuous ones and to apply Nesterov’s smoothing approximation techniques on the Minkowski gauges that are used as distance measures. This approach enables us to propose two implementable DCA-based algorithms for solving the problems. Numerical results and practical applications are provided to illustrate our approach.