串联式多级火箭成本问题初论

本文以火箭最大速度值的一般变化规律为基础, 改进了以前考虑火箭发射的成本问题的常用数学模型：最省的最省推进剂方案, 详细研究了各种情况下串联式多级火箭的成本问题,并以算例验证了所得的新成本计算模型的有效性.     

最省砝码设置问题的数学模型

通过对砝码的最省设置问题的研究,建立了解决这一类问题的数学模型,并给出了一个有效、简便、操作程序化的求解算法.     

2011年中考函数应用题分类解析

函数是初中数学的主干知识,历届中考都重视对函数应用的考查,近年来更是如此.综观2011年全国各地数学中考试卷,大多数省市都要求考生用函数知识对日常生活中普遍存在的成本最低、利润最高、产量最大、效益最好、用料最省等实际问题进行信息的加工与分析,建立相应的目标函数,确定变量的限制条     

一个最值问题的推广

最值问题是微积分应用的典型问题之一.某些经典教材中,应用微分工具解决货物运输过程中运费最省问题的方法,将被拓展成一般的解法原理,同时给出这种原理的实际应用.     

易拉罐形状和尺寸的最优设计

从用料最省的角度研究了易拉罐的形状和尺寸的优化设计问题,首先通过多次测量取平均值的方法得到了题目所需的数据．然后就问题二和问题三分别建立了优化模型,并借助数学软件进行了求解,得到了最优设计的尺寸．最后设计出了椭球形状的易拉罐作为自己的最优设计．     

一类同时含有预防接种和治疗的传染病最优控制模型

建立和分析一个同时含有预防接种和治疗的传染病最优控制模型.首先计算基本再生数,并对无病平衡点和地方病平衡点进行稳定性分析.为了同时降低被感染者人数以及治疗成本,一方面使用最优控制理论和Pontryagin原理分析最优控制策略;另一方面从经济角度出发,综合考虑预防接种和治疗的花费,计算成本效益,使得疾病控制过程中的总成本最省.最后通过数值模拟和敏感性分析,验证理论结果以及寻求对传染病流行起决定性作用的参数.     

探析如何利用导数解决实际问题中的“优化”问题

在实际生活中的优化问题一般为利润最大、用料最省、效率最高等问题.这类优化问题可归结为求函数的最值问题,导数正是求最值的有力工具,利用导数处理优化问题的基本思路是将题目中的实际问题转化为数学问题进行求解.笔者通过对几道试题的评析谈谈如何灵活运用导数处理     

输油管线的最优设计方案

以输油管线铺设费用最省为目标,通过建立非线性规划数学模型,得到了不同情形下管线铺设的最优设计方案.综合运用多元函数极值和三角函数相关知识得出了模型的精确解.     

建立数学模型解应用问题

建立数学模型解应用问题孙罗超（广东省吴川市第二中学５２４５００）如何帮助学生掌握解应用问题的方法呢？本文试着利用建立数学模型来解答．１建立“函数模型”解应用问题在实际生活中，有关用料最省、造价最低、利润最大、容积（面积）最大等问题，往往可以通过分析、...     

最省和最奢的2—竞赛矩阵

本文考虑以允许平局的单循环比赛为模型的2－竞赛图（二重完全图的定向图0和它的邻接矩阵（2－竞赛矩阵）。得到了得分向量与2－圈数,3－圈数之间的关系;给出了构造最省和最奢的2－竞赛矩阵的方法;部分地回答了文献[4]中的一个问题。     

Distributionally robust chance constrained problem under interval distribution information

This paper is concerned with distributionally robust chance constrained problem under interval distribution information. Using worst-case CVaR approximation, we present a tractable convex programming approximation for distributionally robust individual chance constrained problem under interval sets of mean and covariance information. We prove the worst-case CVaR approximation problem is an exact form of the distributionally robust individual chance constrained problem. Then, our result is applied to worst-case Value-at-Risk optimization problem. Moreover, we discuss the problem under several ambiguous distribution information and investigate tractable approximations for distributionally robust joint chance constrained problem. Finally, we provide an illustrative example to show our results.     

Robust optimization approximation for joint chance constrained optimization problem

Chance constraint is widely used for modeling solution reliability in optimization problems with uncertainty. Due to the difficulties in checking the feasibility of the probabilistic constraint and the non-convexity of the feasible region, chance constrained problems are generally solved through approximations. Joint chance constrained problem enforces that several constraints are satisfied simultaneously and it is more complicated than individual chance constrained problem. This work investigates the tractable robust optimization approximation framework for solving the joint chance constrained problem. Various robust counterpart optimization formulations are derived based on different types of uncertainty set. To improve the quality of robust optimization approximation, a two-layer algorithm is proposed. The inner layer optimizes over the size of the uncertainty set, and the outer layer optimizes over the parameter t which is used for the indicator function upper bounding. Numerical studies demonstrate that the proposed method can lead to solutions close to the true solution of a joint chance constrained problem.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

一类分布鲁棒线性决策随机优化研究

随机优化广泛应用于经济、管理、工程和国防等领域,分布鲁棒优化作为解决分布信息模糊下的随机优化问题近年来成为学术界的研究热点.本文基于φ-散度不确定集和线性决策方式研究一类分布鲁棒随机优化的建模与计算,构建了易于计算实现的分布鲁棒随机优化的上界和下界问题.数值算例验证了模型分析的有效性.     

A new distributionally robust p-hub median problem with uncertain carbon emissions and its tractable approximation method

The p-hub median problem is to determine the optimal location for p hubs and assign the remaining nodes to hubs so as to minimize the total transportation costs. Under the carbon cap-and-trade policy, we study this problem by addressing the uncertain carbon emissions from the transportation, where the probability distributions of the uncertain carbon emissions are only partially available. A novel distributionally robust optimization model with the ambiguous chance constraint is developed for the uncapacitated single allocation p-hub median problem. The proposed distributionally robust optimization problem is a semi-infinite chance-constrained optimization model, which is computationally intractable for general ambiguity sets. To solve this hard optimization model, we discuss the safe approximation to the ambiguous chance constraint in the following two types of ambiguity sets. The first ambiguity set includes the probability distributions with the bounded perturbations with zero means. In this case, we can turn the ambiguous chance constraint into its computable form based on tractable approximation method. The second ambiguity set is the family of Gaussian perturbations with partial knowledge of expectations and variances. Under this situation, we obtain the deterministic equivalent form of the ambiguous chance constraint. Finally, we validate the proposed optimization model via a case study from Southeast Asia and CAB data set. The numerical experiments indicate that the optimal solutions depend heavily on the distribution information of carbon emissions. In addition, the comparison with the classical robust optimization method shows that the proposed distributionally robust optimization method can avoid over-conservative solutions by incorporating partial probability distribution information. Compared with the stochastic optimization method, the proposed method pays a small price to depict the uncertainty of probability distribution. Compared with the deterministic model, the proposed method generates the new robust optimal solution under uncertain carbon emissions.     

Moment inequalities for sums of random matrices and their applications in optimization

In this paper, we consider various moment inequalities for sums of random matrices—which are well-studied in the functional analysis and probability theory literature—and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization. First, we show that the validity of a recent conjecture of Nemirovski is actually a direct consequence of the so-called non-commutative Khintchine’s inequality in functional analysis. Using this result, we show that an SDP-based algorithm of Nemirovski, which is developed for solving a class of quadratic optimization problems with orthogonality constraints, has a logarithmic approximation guarantee. This improves upon the polynomial approximation guarantee established earlier by Nemirovski. Furthermore, we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities. Secondly, we consider a recent result of Delage and Ye on the so-called data-driven distributionally robust stochastic programming problem. One of the assumptions in the Delage–Ye result is that the underlying probability distribution has bounded support. However, using a suitable moment inequality, we show that the result in fact holds for a much larger class of probability distributions. Given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems, we expect that the moment inequalities discussed in this paper will find further applications in optimization.     

Distributionally robust discrete optimization with Entropic Value-at-Risk

We study the discrete optimization problem under the distributionally robust framework. We optimize the Entropic Value-at-Risk, which is a coherent risk measure and is also known as Bernstein approximation for the chance constraint. We propose an efficient approximation algorithm to resolve the problem via solving a sequence of nominal problems. The computational results show that the number of nominal problems required to be solved is small under various distributional information sets.     

A model of distributionally robust two-stage stochastic convex programming with linear recourse

We consider distributionally robust two-stage stochastic convex programming problems, in which the recourse problem is linear. Other than analyzing these new models case by case for different ambiguity sets, we adopt a unified form of ambiguity sets proposed by Wiesemann, Kuhn and Sim, and extend their analysis from a single stochastic constraint to the two-stage stochastic programming setting. It is shown that under a standard set of regularity conditions, this class of problems can be converted to a conic optimization problem. Numerical results are presented to show the efficiency of the distributionally robust approach.     

A robust optimization approach to wine grape harvesting scheduling

Optimization models are increasingly being used in agricultural planning. However, the inherent uncertainties present in agriculture make it difficult. In recent years, robust optimization has emerged as a methodology that allows dealing with uncertainty in optimization models, even when probabilistic knowledge of the phenomenon is incomplete. In this paper, we consider a wine grape harvesting scheduling optimization problem subject to several uncertainties, such as the actual productivity that can be achieved when harvesting. We study how effective robust optimization is solving this problem in practice. We develop alternative robust models and show results for some test problems obtained from actual wine industry problems.     

Data-Driven Robust Chance Constrained Problems: A Mixture Model Approach

This paper discusses the mixture distribution-based data-driven robust chance constrained problem. We construct a data-driven mixture distribution-based uncertainty set from the perspective of simultaneously estimating higher-order moments. Then, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex programming problem, we propose new and tight convex approximations based on the piecewise linear approximation method. We establish the theoretical foundation for these approximations. Finally, numerical results show that the proposed approximations are practical and efficient.