Fuzzy Linear Programming with Interactive Uncertain Parameters

In this paper, we treat fuzzy linear programming problems with uncertain parameters whose ranges are specified as fuzzy polytopes. The problem is formulated based on fractile optimization model using a necessity measure. It is shown that the problem can be reduced to a semi-infinite linear programming problem and that a solution algorithm based on a relaxation procedure can be applied. A simple numerical example is given to illustrate the solution procedure. This revised version was published online in June 2006 with corrections to the Cover Date.     

Logarithmic Barrier Decomposition Methods for Semi-infinite Programming

A computational study of some logarithmic barrier decomposition algorithms for semi-infinite programming is presented in this paper. The conceptual algorithm is a straightforward adaptation of the logarithmic barrier cutting plane algorithm which was presented recently by den Hartog et al. ( Annals of Operations Research , 58 , 69–98, 1995), to solve semi-infinite programming problems. Usually decomposition (cutting plane methods) use cutting planes to improve the localization of the given problem. In this paper we propose an extension which uses linear cuts to solve large scale, difficult real world problems. This algorithm uses both static and (doubly) dynamic enumeration of the parameter space and allows for multiple cuts to be simultaneously added for larger/difficult problems. The algorithm is implemented both on sequential and parallel computers. Implementation issues and parallelization strategies are discussed and encouraging computational results are presented.     

Rational suboptimal continuous-time controller design

We obtain rational suboptimal continuous-time solutions to some optimal control problems specified via time domain performance criteria. These include ℒ₁ and ℒ_∞ norms and peak overshoot. The approach uses linear semi-infinite programming to compute weights for a given finite set of rational basis functions and makes the best possible use of the basis. The closed-loop transfer functions obtained satisfy appropriate interpolation constraints for internal stability and the formulation allows additional linear constraints to be incorporated     

Semi-infinite relaxation of joint chance constraints in chance-constrained programming Part 1. Zero-order stochastic decision rules†

A new class of semi-infinite deterministic (‘determinizations’) dominants and relaxations of joint chance-constraints in chance-constrained programming is developed and specialized to zero-order stochastic decision rule situations. Tight constraint relaxations are obtained where only the partial information of means and variances is known. The tight non-linear semi-infinite relaxations are related to an accessible finite subsystem. When the chance constraints involve linear inequalities, for a large class, the non-linear tight system is proved equivalent to a linear program. Its solutions for the Prekopa-Szantai reservoir construction examples agree well with theirs     

On the use of ε-most-active constraints in an exact penalty function method for nonlinear optimization

This note presents an algorithm for nonlinear programming problems, which utilizes the ε-most-active constraint strategy in an exact penalty function method with trust region. The algorithm is particularly suitable for problems containing a large number of constraints. The global convergence of the proposed algorithm is proved. The results of limited computational experiments for discretized semi-infinite programming problems are also reported to demonstrate the effectiveness of the present approach.     

Solving quality control problems with an algorithm for minimax programs with coupled constraints

We propose a systematic algorithm to tackle a set of acceptance sampling problems introduced by Seidel [1] and their generalization when no prior knowledge is assumed. The problems are modeled as minimax problems with coupled or decoupled constraints. We use ideas from recent work on bi-level programming, reformulating the problem as a semi-infinite program with disjunctive constraints and employing a two phase discretization method to solve it. We use the KKT conditions of the inner problem of minimax to tighten the relaxation of the semi-infinite problem obtained by discretization. In addition, to avoid convergence trouble, a strategy based on a feasibility test relative to the objective value of the outer program is used.     

Dynamic optimization of large scale systems: Case study

We present the analysis of the steady state backoff problem with state and dynamic constraints of a non-linear chemical process described by almost 3000 differential algebraic equations. The dynamic optimization is carried out using a new approach based on an SQP algorithm for semi-infinite non-linear programming problems. The system equations are integrated with an implicit Runge-Kutta method and 'reduced' gradients are evaluated by adjoint equations. The high performance of the algorithm is analysed and compared to fully non-linear programming proposals in which discretized system equations are treated as general non-linear equality constraints.     

A fast algorithm for the optimal design of high accuracy windows in signal processing

This paper presents a new optimal window design method with a general window design specification for the passband and stopband. The design problem is formulated as a semi-infinite linear programming problem. With suitable discretizations, an exchange algorithm is employed. The convergence of the proposed algorithm is established. In the formulation, since the stopband is minimized, the method can be employed for the design of very highly optimized windows.     

一种线性约束规划问题的神经网络模型及其应用

提出了一种基于梯度投影矩阵下的求解线性约束下规划问题的神经网络。针对解的稳定性问题,导出了该网络相关参数之间的关系。由文中定义可知,该网络不但适合于求解线性约束下线性或非二次规划问题,而且也用于求解线性或非线性方程组问题,比其它规划问题的神经网络方法更具有一般性。     

Optimal rejection of persistent disturbances, robust stability, andmixed sensitivity minimization

The problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case. The minimum value of the objective function can be obtained by solving a semi-infinite linear programming problem, and an iterative procedure for obtaining approximate solutions is introduced. Application of the l¹ -optimal problem to robustness is discussed. A mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations     

微粒群优化算法在相关新产品组合投入的应用

引入新产品最佳投入期和相关收益的概念，在产品生命周期量化描述前提下，提出一个非线性半无限规划的相关新产品组合投入模型，并用微粒群优化算法(PS0)求解，仿真实例表明，微粒群优化算法在求解组合优化问题时简便且易于实现，具有很强的实用性。     

基于进化能力的多策略粒子群优化算法

针对粒子群优化算法易早熟收敛、求解精度低等缺点，提出基于进化能力的多策略粒子群优化算法（multistrategy particle swarm optimization algorithm based on evolution ability）。将粒子按照适应值变化方向分为进步粒子和停退粒子。对于进步粒子按照原始进化策略更新，保留原算法的优点。对于停退粒子进一步根据粒子活性分为暂时停退粒子和长久停退粒子，针对暂时停退的粒子，减小对个体历史速度的依赖甚至向相反方向学习，针对长久停退粒子，根据粒子的适应值优劣采用不同的进化策略，提高全局寻优能力。同时，设计一种带随机波动的惯性权重，使粒子在算法后期仍然具有跳出当前区域的能力，利于全局搜索。通过与其他算法在10个测试函数不同维度上的优化结果对比表明，该算法无论对低维还是高维问题求解的收敛速度和求解精度均有优势。将EAMSPSO算法应用于半无限规划问题的求解，实验结果表明，该算法可以用于半无限规划问题的求解，且具有优势。     

基于线性约束规划问题的神经网络模型

提出了一种基于梯度投影矩阵下的求解线性约束下规划问题的神经网络模型，并导出了线性约束下规划问题的稳定解法，该网络模型既适合于求解线性约束下线性或非二次规划问题，又适合于求解线性或非线性方程组，与其它规划问题的神经网络相比，更具有一般性。     

The generalized inverse in parametric programming

In this paper we are concerned with parametric programming problems. The main results are two: the first one is an explicit representation of the general optimal solution of particular parametric programming problems; the second one is an unified approach to general parametric programming problems. The method suggested here is based on the utilization of the generalized inverse of a matrix: it allows a fast and simple solution of parametric problems and has several advantages over the classical parametric methods. We refer to interval linear programming problems but a fruitful direction of research may be its application to linear programming problems in the standard from. Research supported in part by the Gruppo Nazionale per l'Analisi Funzionale e le sne applicazioni del C. N. R.     

A stochastic dual dynamic programming method for two-stage distributionally robust optimization problems

This paper studies a class of two-stage distributionally robust optimization (TDRO) problems which comes from many practical application fields. In order to set up some implementable solution method, we first transfer the TDRO problem to its equivalent robust counterpart (RC) by the duality theorem of optimization. The RC reformulation of TDRO is a semi-infinite stochastic programming. Then we construct a conditional value-at-risk-based sample average approximation model for the RC problem. Furthermore, we analyse the error bound of the approximation model and obtain the convergent results with respect to optimal value and optimal solution set. Finally, a so-called stochastic dual dynamic programming approach is proposed to solve the approximate model. Numerical results validate the solution approach of this paper.     

免疫遗传算法及在新产品投入计划中的应用

提出一种新的求解复杂约束优化问题的免疫遗传算法. 算法首先产生大量抗原来训练抗体, 从而建立起一个具有自体和非自体识别能力的免疫系统. 将该免疫系统嵌入到遗传算法中, 即可在遗传过程中不经解码就能识别非法或不可行的染色体. 算法有效地改进了遗传算法求解复杂约束优化问题的性能. 算法用于求解用半无限规划模型描述的新产品投入计划问题, 获得了满意的结果.     

准时化生产计划的半无限规划模型与模拟退火方法

半无限规划模型是描述ＪＩＴ环境下，提前／拖期生产计划问题的一种数学模型。因为目标函数是非线性的、有无限多约束和约束的非凸性，所以求解比较困难。用模型退火结合启发式方法和最速下降法求解半无限规划模型，与遗传算法进行比较，计算结果表明用模拟退火方法求解更有效。     

Computation of probabilistic linear programming problems involving normal and log-normal random variables with a joint constraint

A method for solving single- and multi-objective probabilistic linear programming problems with a joint constraint is presented. It is assumed that the parameters in the probabilistic linear programming problems are random variables, and the probabilistic problem is converted to an equivalent deterministic mathematical programming problem. In this paper the parameters are generally considered as normal and log-normal random variables. A non-linear programming method is used to solve the single-objective deterministic problem, and a fuzzy programming method is used to solve the multi-objective deterministic problem. Finally, a numerical example is presented to illustrate the methodology.     

A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems

There are several methods, in the literature, for solving fuzzy variable linear programming problems (fuzzy linear programming in which the right-hand-side vectors and decision variables are represented by trapezoidal fuzzy numbers). In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings a new method based on the bounded dual simplex method is proposed to determine the fuzzy optimal solution of that kind of fuzzy variable linear programming problems in which some or all variables are restricted to lie within lower and upper bounds. To illustrate the proposed method, an application example is solved and the obtained results are given. The advantages of the proposed method over existing methods are discussed. Also, one application of this algorithm in solving bounded transportation problems with fuzzy supplies and demands is dealt with. The proposed method is easy to understand and to apply for determining the fuzzy optimal solution of bounded fuzzy variable linear programming problems occurring in real-life situations.     

Majorization: a computational complexity reduction technique incontrol system design

A methodology is presented for transcribing worst-case control-system design specifications into semi-infinite inequalities of low computational complexity, which are tractable by semi-infinite programming algorithms. These algorithms can be used to find a design which satisfies specifications or to find a design which satisfies specifications and minimizes a cost. A computational example is given