整数规划新进展

整数规划是对全部或部分决策变量为整数的最优化问题的模型、算法及应用等的研究, 是运筹学和管理科学中应用最广泛的优化模型之一. 首先简要回顾整数规划的历史和发展进程, 概述线性和非线性整数规划的一些经典方法. 然后着重讨论整数规划若干新进展, 包括0-1二次规划的半定规划~(SDP)~松弛和随机化方法, 带半连续变量和稀疏约束的优化问题的整数规划模型和方法, 以及0-1二次规划的协正锥规划表示和协正锥的层级半定规划~(SDP)~逼近. 最后, 对整数规划未来研究方向进行展望并对一些公开问题进行讨论.     

概率约束规划的稳定性分析

本文对概率约束规划问题的稳定性进行了探讨,得出了当随机向量序列{ξ^(k)(ω）}分布收敛于ξ（ω）时,相应于ξ^(k)(ω)的概率约束规划问题的最优值收敛于原问题的最优值,这个结果为设计逼近算法和改进逼近解提供了一个理论基础。     

一类概率约束规划逼近最优解集的上半收敛性

对一类概率约束规划逼近最优解集的上半收敛性进行了研究.利用概率测度弱收敛的特征,给出了概率约束规划可行集的收敛性条件,得到了概率约束规划逼近最优解集的上半收敛性.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

概率约束规划逼近最优解集的稳定性和最优值的连续性

在原始规划可行集上引入了正则的概念,并在此正则条件下,研究了更一般的概率约束规划问题的稳定性.在一定的条件下,得到了概率约束规划逼近最优解集的稳定性和最优值的连续性,从而对近似求解这类问题提供了某种理论依据.     

概率约束规划逼近最优解集的上半收敛性

本文讨论了概率约束规划目标函数的连续收敛性,并利用概率测度弱收敛的特征给出了概率约束规划可行集的收敛性条件,得到了概率约束规划逼近最优解集的上半收敛性.     

线性无关假设如何影响非线性约束最优化算法

首先综述非线性约束最优化最近的一些进展. 首次定义了约束最优化算法的全局收敛性. 注意到最优性条件的精确性和算法近似性之间的差异, 并回顾等式约束最优化的原始的Newton 型算法框架, 即可理解为什么约束梯度的线性无关假设应该而且可以被弱化. 这些讨论被扩展到不等式约束最优化问题. 然后在没有线性无关假设条件下, 证明了一个使用精确罚函数和二阶校正技术的算法可具有超线性收敛性. 这些认知有助于接下来开发求解包括非线性半定规划和锥规划等约束最优化问题的更加有效的新算法.     

凹整数规划的分枝定界解法

凹整数规划是一类重要的非线性整数规划问题，也是在经济和管理中有着广泛应用的最优化问题．本文主要研究用分枝定界方法求解凹整数规划问题，这一方法的基本思想是对目标函数进行线性下逼近，然后用乘子搜索法求解连续松弛问题．数值结果表明，用这种分枝定界方法求解凹整数规划是有效的．     

有限维逼近无限维总极值的水平值估计方法

变分计算、最优控制、微分对策等常常要求考虑无限维空间中的总极值问题,但实际计算中只能得出有限维空间中的解.本文用有限维逼近无限维的方法来讨论函数空间中的总体最优化问题.用水平值估计和变侧度方法来求得有限维逼近总体最优化问题.对于有约束问题,用不连续精确罚函数法将其转化为无约束问题求解.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

线性与非线性规划算法与理论

线性规划与非线性规划是数学规划中经典而重要的研究方向. 主要介绍该研究方向的背景知识,并介绍线性规划、无约束优化和约束优化的最新算法与理论以及一些前沿与热点问题. 交替方向乘子法是一类求解带结构的约束优化问题的方法,近年来倍受重视. 全局优化是一个对于应用优化领域非常重要的研究方向. 因此也试图介绍这两个方面的一些最新研究进展和问题.     

关于二层规划的逼近问题

下层问题以上层决策变量作为参数,而上层是以下层问题的最优值作为响应 的一类最优化问题——二层规划问题。我们给出了由一系列此类二层规划去逼近原二层规划的逼近法,得到了这种逼近的一些有趣的结果.     

A multi-parametric programming approach for constrained dynamic programming problems

In this work, we present a new algorithm for solving complex multi-stage optimization problems involving hard constraints and uncertainties, based on dynamic and multi-parametric programming techniques. Each echelon of the dynamic programming procedure, typically employed in the context of multi-stage optimization models, is interpreted as a multi-parametric optimization problem, with the present states and future decision variables being the parameters, while the present decisions the corresponding optimization variables. This reformulation significantly reduces the dimension of the original problem, essentially to a set of lower dimensional multi-parametric programs, which are sequentially solved. Furthermore, the use of sensitivity analysis circumvents non-convexities that naturally arise in constrained dynamic programming problems. The potential application of the proposed novel framework to robust constrained optimal control is highlighted.     

Minimization of the Ratio of Functions Defined as Sums of the Absolute Values

This paper addresses a new class of linearly constrained fractional programming problems where the objective function is defined as the ratio of two functions which are the sums of the absolute values of affine functions. This problem has an important application in financial optimization. This problem is a convex-convex type of fractional program which cannot be solved by standard algorithms. We propose a branch-and-bound algorithm and an integer programming algorithm. We demonstrate that a fairly large scale problem can be solved within a practical amount of time. The research of the first author was supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Culture and Sports of the Government of Japan, B(2) 15310122 and 15656025.     

关于加权广义逆的上确界

设P为一类半正定实对称矩阵的集合，满足本文推导此上确界的若干等价性表达式，并具体讨论几种特殊情形．该问题在数学规划和约束最优化问题中具有重要意义     

MONOTONIZATION IN GLOBAL OPTIMIZATION

A general monotonization method is proposed for converting a constrained programming problem with non-monotone objective function and monotone constraint functions into a monotone programming problem. An equivalent monotone programming problem with only inequality constraints is obtained via this monotonization method. Then the existing convexification and concavefication methods can be used to convert the monotone programming problem into an equivalent better-structured optimization problem.     

On the global optimality of generalized trust region subproblems

Quadratically constrained quadratic programming is an important class of optimization problems. We consider the case with one quadratic constraint. Since both the objective function and its constraint can be neither convex nor concave, it is also known as the ‘generalized trust region subproblem.’ The theory and algorithms for this problem have been well studied under the Slater condition. In this article, we analyse the duality property between the primal problem and its Lagrangian dual problem, and discuss the attainability of the optimal primal solution without the Slater condition. The relations between the Lagrangian dual and semidefinite programming dual is also given.     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.     

An algebraic geometry algorithm for scheduling in presence of setups and correlated demands

We study here a problem of schedulingn job types onm parallel machines, when setups are required and the demands for the products are correlated random variables. We model this problem as a chance constrained integer program.Methods of solution currently available—in integer programming and stochastic programming—are not sufficient to solve this model exactly. We develop and introduce here a new approach, based on a geometric interpretation of some recent results in Gröbner basis theory, to provide a solution method applicable to a general class of chance constrained integer programming problems.Out algorithm is conceptually simple and easy to implement. Starting from a (possibly) infeasible solution, we move from one lattice point to another in a monotone manner regularly querying a membership oracle for feasibility until the optimal solution is found. We illustrate this methodology by solving a problem based on a real system.Corresponding author.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.