Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method

In this paper, we propose a convergent Lagrangian and objective level cut method for computing exact solution to two classes of nonlinear integer programming problems: separable nonlinear integer programming and polynomial zero-one programming. The method exposes an optimal solution to the convex hull of a revised perturbation function by successively reshaping or re-confining the perturbation function. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain. Computational results are reported for a variety of nonlinear integer programming problems and demonstrate that the proposed method is promising in solving medium-size nonlinear integer programming problems.     

多约束非线性整数规划的一种改进的算法

多约束非线性整数规划是一类非常重要的问题,非线性背包问题是它的一类特殊而重要的问题.定义在有限整数集上极大化一个可分离非线性函数的多约束最优化问题.这类问题常常用于资源分配、工业生产及计算机网络的最优化模型中,运用一种新的割平面法来求解对偶问题以得到上界,不仅减少了对偶间隙,而且保证了算法的收敛性.利用区域割丢掉某些整数箱子,并把剩下的区域划分为一些整数箱子的并集,以便使拉格朗日松弛问题能有效求解,且使算法在有限步内收敛到最优解.算法把改进的割平面法用于求解对偶问题并与区域分割有效结合解决了多约束非线性背包问题的求解.数值结果表明了改进的割平面方法对对偶搜索更加有效.     

The multi-product newsboy problem with supplier quantity discounts and a budget constraint

This paper considers the multi-product newsboy problem with both supplier quantity discounts and a budget constraint, while each feature has been addressed separately in the literature. Different from most previous nonlinear optimization models on the topic, the problem is formulated as a mixed integer nonlinear programming model due to price discounts. A Lagrangian relaxation approach is presented to solve the problem. Computational results on both small and large-scale test instances indicate that the proposed algorithm is extremely effective for the problem. An extension to multiple constraints and preliminary computational results are also reported.     

Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.     

New results on a class of exact augmented Lagrangians

In this paper, a new continuously differentiable exact augmented Lagrangian is introduced for the solution of nonlinear programming problems with compact feasible set. The distinguishing features of this augmented Lagrangian are that it is radially unbounded with respect to the multiplier and that it goes to infinity on the boundary of a compact set containing the feasible region. This allows one to establish a complete equivalence between the unconstrained minimization of the augmented Lagrangian on the product space of problem variables and multipliers and the solution of the constrained problem.The author wishes to thank Dr. L. Grippo for having suggested the topic of this paper and for helpful discussions.     

不可分离凸背包问题的拉格朗日分解和区域分割方法

本文对线性约束不可分离凸背包问题给出了一种精确算法．该算法是拉格朗日分解和区域分割结合起来的一种分枝定界算法．利用拉格朗日分解方法可以得到每个子问题的一个可行解，一个不可行解，一个下界和一个上界．区域分割可以把一个整数箱子分割成几个互不相交的整数子箱子的并集，每个整数子箱子对应一个子问题．通过区域分割可以逐步减小对偶间隙并最终经过有限步迭代找到原问题的最优解．数值结果表明该算法对不可分离凸背包问题是有效的．     

Approximate Optimal Solutions and Nonlinear Lagrangian Functions*

There is an increasing interest in the study of optimality conditions of approximate solutions for nonlinear optimization problems. In this paper, relationships between approximate optimal values and approximate roots of a nonlinear function are explored via a nonlinear Lagrangian function. Almost approximate optimal solutions are investigated by means of nonlinear Lagrangian functions.     

Numerical Comparison of Augmented Lagrangian Algorithms for Nonconvex Problems

Augmented Lagrangian algorithms are very popular tools for solving nonlinear programming problems. At each outer iteration of these methods a simpler optimization problem is solved, for which efficient algorithms can be used, especially when the problems are large. The most famous Augmented Lagrangian algorithm for minimization with inequality constraints is known as Powell-Hestenes-Rockafellar (PHR) method. The main drawback of PHR is that the objective function of the subproblems is not twice continuously differentiable. This is the main motivation for the introduction of many alternative Augmented Lagrangian methods. Most of them have interesting interpretations as proximal point methods for solving the dual problem, when the original nonlinear programming problem is convex. In this paper a numerical comparison between many of these methods is performed using all the suitable problems of the CUTE collection.This author was supported by ProNEx MCT/CNPq/FAPERJ 171.164/2003, FAPESP (Grants 2001/04597-4 and 2002/00094-0 and 2003/09169-6) and CNPq (Grant 302266/2002-0).This author was partially supported by CNPq-Brasil and CDCHT-Venezuela.This author was supported by ProNEx MCT/CNPq/FAPERJ 171.164/2003, FAPESP (Grant 2001/04597-4) and CNPq.     

A nonlinear Lagrangian dual for integer programming

Nonlinear Lagrangian theory offers a success guarantee for the dual search via construction of a nonlinear support of the perturbation function at the optimal point. In this paper, a new nonlinear dual formulation of an exponential form is proposed for bounded integer programming. This new formulation possesses an asymptotic strong duality property and guarantees a success in identifying a primal optimum solution. No actual dual search is needed in the solution process when the parameter of the nonlinear Lagrangian formulation is set to be large enough.     

An Exact Augmented Lagrangian Function for Nonlinear Programming with Two-Sided Constraints

This paper is aimed toward the definition of a new exact augmented Lagrangian function for two-sided inequality constrained problems. The distinguishing feature of this augmented Lagrangian function is that it employs only one multiplier for each two-sided constraint. We prove that stationary points, local minimizers and global minimizers of the exact augmented Lagrangian function correspond exactly to KKT pairs, local solutions and global solutions of the constrained problem.     

带有不等式约束的非线性规划问题的一个精确增广Lagrange函数

对求解带有不等式约束的非线性非凸规划问题的一个精确增广Lagrange函数进行了研究．在适当的假设下,给出了原约束问题的局部极小点与增广Lagrange函数,在原问题变量空间上的无约束局部极小点之间的对应关系．进一步地,在对全局解的一定假设下,还提供了原约束问题的全局最优解与增广Lagrange函数,在原问题变量空间的一个紧子集上的全局最优解之间的一些对应关系．因此,从理论上讲,采用该文给出的增广Lagrange函数作为辅助函数的乘子法,可以求得不等式约束非线性规划问题的最优解和对应的Lagrange乘子．     

A new augmented Lagrangian function for inequality constraints in nonlinear programming problems

In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.     

About Lagrangian Methods in Integer Optimization

It is well-known that the Lagrangian dual of an Integer Linear Program (ILP) provides the same bound as a continuous relaxation involving the convex hull of all the optimal solutions of the Lagrangian relaxation. It is less often realized that this equivalence is effective, in that basically all known algorithms for solving the Lagrangian dual either naturally compute an (approximate) optimal solution of the “convexified relaxation”, or can be modified to do so. After recalling these results we elaborate on the importance of the availability of primal information produced by the Lagrangian dual within both exact and approximate approaches to the original (ILP), using three optimization problems with different structure to illustrate some of the main points.     

On a pair of nonlinear mixed integer programming problems

We consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense. Under weaker (pseudo-convex/pseudo-concave) assumptions, we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem. As a particular case, this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems. Further, this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming.     

求解约束规划的一个非线性Lagrange函数

本文提出了一个求解不等式约束优化问题的非线性Lagrange函数,并构造了基于该函数的对偶算法.证明了当参数σ小于某一阈值σ_0时,由算法生成的原始-对偶点列是局部收敛的,并给出了原始-对偶解的误差估计.此外,建立了基于该函数的对偶理论.最后给出了算法的数值结果.     

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．     

Comment on “A nonlinear Lagrangian dual for integer programming”

We present a counterexample and correction to the contention by Xu and Li that the nonlinear Lagrangian dual problem they propose [Oper. Res. Lett. 30 (2002) 401] asymptotically has no duality gap.     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

对等式约束非线性规划问题的Hestenes-Powell增广拉格朗日函数的进一步研究

本文对用无约束极小化方法求解等式约束非线性规划问题的Hestenes-Powell 增广拉格朗日函数作了进一步研究．在适当的条件下,我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系,并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系．因此,从理论的观点来看,原约束问题的解和对应的拉格朗日乘子值不仅可以用众所周知的乘子法求得,而且可以通过对Hestenes-Powell 增广拉格朗日函数在原问题变量和乘子变量的积空间上执行一个单一的无约束极小化来获得．     

非线性半定规划若干算法介绍

介绍近几年国际上求解非线性半定规划的若干有效新算法, 包括增广Lagrangian函数法、序列半定规划法、序列线性方程组法以及交替方向乘子法. 最后, 对非线性半定规划的算法研究前景进行了探讨.