A dynamic programming branch and bound algorithm for pure integer programming

Several algorithms for linear integer programming have been proposed in which one or more of the conditions are relaxed to produce a solvable problem form. Reimposing these relaxed conditions can lead to a branch and bound process. In this paper an alternative relaxation is proposed in which the integer conditions are maintained but the feasibility conditions are relaxed in a special way. Reimposing feasibility by sequentially setting variables creates the branch and bound process. In special cases, the algorithm reverts to the normal form of dynamic programming.The algorithm is applicable to both linear and non-linear pure integer problems. It has been programmed to solve pure $\text{0}\text{1}$ problems and results of computational experience with some linear problems previously used in the literature are given.     

An efficient branch and bound algorithm to solve the quadratic integer programming problem

The quadratic integer programming problem is considered. It will be shown in which order the variablesx ₁, ...,x _n should be ramified in order to reduce the number of knots being studied to a minimum. There areO(n ³) operations necessary to determine a favourable ramification. Numerical tests confirm the efficiency of the given algorithm.     

高效求解整数线性规划问题的分支算法

为了提高求解一般整数线性规划问题的效率,提出了一种基于目标函数超平面移动的分支算法。对于给定的目标函数整数值,首先利用线性规划松弛问题的最优单纯形表确定变量的上、下界,然后将变量的上、下界条件加入约束条件中对相应的目标函数超平面进行切割,最后应用分支定界算法中的分支方法来搜寻目标函数超平面上的可行解。通过对一些经典的数值例子的求解计算并与经典的分支定界算法进行比较,结果表明,该算法减少了分支数和单纯形迭代数,具有较大的实用价值。     

整数线性规划的切割与分支算法

基于整数线性规划问题的分支定界方法,以子问题或根问题的目标最优值作为参数,构造了一种新的切割不等式,能够方便地切割子问题或根问题的非整数最优解.在分支之前进行这种切割,产生了一种新的求解整数线性规划问题的切割与分支算法.将该算法应用于求解一些经典的数值例子,实验结果表明,与经典的分支定界方法相比,该算法大大减少了分支的数量,提高了计算效率.随着问题规模的增大,该算法的计算优越性体现得更加明显.     

混合整数非线性规划问题的改进差分进化算法*

提出一种改进差分进化算法求解混合整数非线性规划问题。该算法利用同态映射方法,解决差分进化算法无法直接处理整数决策变量问题;提出改进的自适应交替变异算子,提高算法的搜索性能;提出一种自适应保留不可行解的方法处理约束条件,并对差分进化算法的选择算子进行改进,提出一种直接处理约束条件的新选择算子。六个常用的混合整数非线性规划问题的实验结果表明了该方法的有效性和适用性。     

非凸二次规划全局极小问题的新型分枝定界算法

针对求解多面集上二次函数的全局近似最优解问题,利用逐步缩小对偶间隙的处理办法,提出了一个新型分枝定界算法。新算法的主要改进之处是利用了Lagrange 对偶性获取下界。最后,用构造和随机产生的问题实例,对提出的新算法和传统的分枝定界算法做了初步的数值比较实验。计算实验表明算法对求解中大规模非凸二次规划问题的有效性。     

Genetic algorithm for non-linear mixed integer programming problems and its applications

In this paper we propose a method for solving non-linear mixed integer programming (NMIP) problems using genetic algorithm (GAs) to get an optimal or near optimal solution. The penalty function method was used to evaluate those infeasible chromosomes generated from genetic reproduction. Also, we apply the method for solving several optimization problems of system reliability which belong to non-linear integer programming (NIP) or (NMIP) problems, using the proposed method. Numerical experiments and comparisons with previous works are illustrated to demonstrate the efficiency of the proposed method.     

A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming

In this paper we describe a branch and bound algorithm for solving the unconstrained quadratic 0–1 programming problem. The salient features of it are the use of quadratic programming heuristics in the transformation of subproblems and exploiting some classes of facets of the polytope related to the quadratic problem in deriving upper bounds on the objective function. We develop facet selection procedures that form a basis of the bound computation algorithm. We present computational experience on four series of randomly generated problems and 14 real instances of a quadratic problem arising in design automation. We remark that the same ideas can also be applied to some other combinatorial optimization problems.     

Computational aspects of a branch and bound algorithm for quadratic zero-one programming

In this paper we describe computational experience in solving unconstrained quadratic zero-one problems using a branch and bound algorithm. The algorithm incorporates dynamic preprocessing techniques for forcing variables and heuristics to obtain good starting points. Computational results and comparisons with previous studies on several hundred test problems with dimensions up to 200 demonstrate the efficiency of our algorithm.     

一种求解整数规划与混合整数规划非线性罚函数方法

证明了任何一个变量有界的整数规划问题（IP）和混合整数规划问题（MIP）都可以转化为一个等价的非整数（或连续化）规划问题（NIP），并给出一个用非线性精确罚函数法来求解该等价NIP的方法，从而达到求解IP或MIP的目的，数值实验表明了算法的可行性。该方法可广泛用于各应用领域里IP和MIP的求解，特别是为非线性IP和MIP问题提供了一条通用 的求解途径，对解决许多实际优化问题具有重要意义。     

Many-valued logic and mixed integer programming

We generalize prepositional semantic tableaux for classical and many-valued logics toconstraint tableaux. We show that this technique is a generalization of the standard translation from CNF formulas into integer programming. The main advantages are (i) a relatively efficient satisfiability checking procedure for classical, finitely-valued and, for the first time, for a wide range of infinitely-valued propositional logics; (ii) easy NP-containment proofs for many-valued logics. The standard translation of two-valued CNF formulas into integer programs and Tseitin's structure preserving clause form translation are obtained as a special case of our approach.Part of the research reported here was carried out while the author was supported by a grant within the DFG Schwerpunktprogramm Deduktion. Preliminary and partial versions of this paper were published as [15, 16].     

一种求非线性整数规划最优解的仿生算法

从大自然植物生长中得到启发,提出了一种求解非线性整数规划全局最小解的仿生算法。该算法将植物生长过程及生长模式应用到非线性整数规划问题的求解,能够快速得到最优解。通过对各种不同类型非线性整数规划问题的具体求解,表明了该方法十分有效。     

整数规划的花授粉算法

整数规划是NP困难(Non-deterministic Polynomial-time hard,NP-hard)的经典问题之一。整数规划的花授粉算法(Integer Flower Pollination Algorithm,IFPA)是采用截断取整的方法,将最近开发的花授粉算法(Flower Pollination Algorithm,FPA)扩展到求解整数规划问题。通过对测试函数集进行仿真实验,结果表明IFPA拥有很好的性能和很强的全局寻优能力,可以作为一种实用方法用于求解无约束整数规划和有约束整数规划问题。     

整数线性规划的改进分支定界算法

分支定界(B&B)算法是求解整数线性规划(ILP)问题的一种最常用的方法,如何划分问题(分支)和按何种策略选择子问题进行扩展是影响算法效率的两个重要因素.提出了一种改进的分支定界算法,采用伪费用分支策略划分问题,采用深度优先搜索(DFS)策略选择子问题进行扩展,并在Matlab中编程实现.数值实验表明,改进的算法能够有效提高求解效率,当问题规模较大时,改进效果尤其明显.     

整数规划的量子行为蝙蝠算法

蝙蝠算法是一种新型群体智能算法,传统的蝙蝠算法在解决整数规划问题时容易陷入局部最优并出现早熟收敛现象,为了解决这些弊端,提出了一种基于势阱的具有量子行为的蝙蝠算法。论述了算法的优化原理和实现方式,并通过仿真实验,与粒子群算法和量子行为粒子群算法进行性能对比。实验结果表明,量子行为蝙蝠算法不仅能够有效地解决整数规划问题,而且比其他算法具有更好的性能。     

Detecting infeasibility and generating cuts for mixed integer programming using constraint programming

We study a hybrid MIP/CP solution approach in which CP is used for detecting infeasibilities and generating cuts within a branch-and-cut algorithm for MIP. Our framework applies to MIP problems augmented by monotone constraints that can be handled by CP. We illustrate our approach on a generic multiple machine scheduling problem, and present a number of computational experiments.     

A branch and bound algorithm for numerical Max-CSP

The Constraint Satisfaction Problem (CSP) framework allows users to define problems in a declarative way, quite independently from the solving process. However, when the problem is over-constrained, the answer “no solution” is generally unsatisfactory. A Max-CSP \(\mathcal{P}_m = \langle V, \textbf{D}, C \rangle\) is a triple defining a constraint problem whose solutions maximize the number of satisfied constraints. In this paper, we focus on numerical CSPs, which are defined on real variables represented as floating point intervals and which constraints are numerical relations defined in intension. Solving such a problem (i.e., exactly characterizing its solution set) is generally undecidable and thus consists in providing approximations. We propose a Branch and Bound algorithm that provides under and over approximations of a solution set that maximize the maximum number \({m_{\mathcal P}}\) of satisfied constraints. The technique is applied on three numeric applications and provides promising results.     

An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions

We present an exact algorithm for the bilevel mixed integer linear programming (BMILP) problem under three simplifying assumptions. Although BMILP has been studied for decades and widely applied to various real world problems, there are only a few BMILP algorithms. Compared to these existing ones, our new algorithm relies on fewer and weaker assumptions, explicitly considers finite optimal, infeasible, and unbounded cases, and is proved to terminate finitely and correctly. We report results of our computational experiments on a small library of BMILP test instances, which we created and made publicly available online.     

改进的分支定界算法

如何进行最优的特征选择是模式识别的研究重点之一。目前比较常用的最优特征选择方法是BAB和BAB＋算法,然而此算法搜索时间比较长。在此基础上详细地阐述改进的分支定界的原理以及算法,该算法的基本思想是通过剪切那些肯定不会产生最优解的分支,同时引入了部分路径和父路径的概念,以达到决策树能够快速搜索到最优解的目的。实验结果证明了该算法的有效性及优越性。