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1.
We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. Research supported by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E, and the National Natural Science Foundation of China under Grants 79970107 and 10571116.  相似文献   

2.
We consider a convexification method for a class of nonsmooth monotone functions. Specifically, we prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. The results derived in this paper lay a theoretical base to extend the reach of convexification methods in monotone optimization to nonsmooth situations. Communicated by X. Q. Yang This research was partially supported by the National Natural Science Foundation of China under Grants 70671064 and 60473097 and by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.  相似文献   

3.
For a class of global optimization (maximization) problems, with a separable non-concave objective function and a linear constraint a computationally efficient heuristic has been developed.The concave relaxation of a global optimization problem is introduced. An algorithm for solving this problem to optimality is presented. The optimal solution of the relaxation problem is shown to provide an upper bound for the optimal value of the objective function of the original global optimization problem. An easily checked sufficient optimality condition is formulated under which the optimal solution of concave relaxation problem is optimal for the corresponding non-concave problem. An heuristic algorithm for solving the considered global optimization problem is developed.The considered global optimization problem models a wide class of optimal distribution of a unidimensional resource over subsystems to provide maximum total output in a multicomponent systems.In the presented computational experiments the developed heuristic algorithm generated solutions, which either met optimality conditions or had objective function values with a negligible deviation from optimality (less than 1/10 of a percent over entire range of problems tested).  相似文献   

4.
Homogeneous programming is an important class of optimization problems. The purpose of this note is to give a truly equivalent characterization of KKT points of homogeneous programming problems, correcting a result given by Lasserre and Hiriart-Urruty in Ref. 1.Communicated by P. TsengThis work was partially supported by the National Natural Science Foundation of China, Grants 10201032 and 70221001, and by the Research Grants Council, Hong Kong, Grant CUHK4180/03E.The authors thank two anonymous referees for valuable remarks and insights that have helped improving the paper.  相似文献   

5.
We study a single-machine stochastic scheduling problem with n jobs, in which each job has a random processing time and a general stochastic cost function which may include a random due date and weight. The processing times are exponentially distributed, whereas the stochastic cost functions and the due dates may follow any distributions. The objective is to minimize the expected sum of the cost functions. We prove that a sequence in an order based on the product of the rate of processing time with the expected cost function is optimal, and under certain conditions, a sequence with the weighted shortest expected processing time first (WSEPT) structure is optimal. We show that this generalizes previous known results to more general situations. Examples of applications to practical problems are also discussed.This work was partially supported by the Research Grants Council of Hong Kong under Earmarked Grants No. CUHK4418/99E and No. PolyU 5081/00E.  相似文献   

6.
In this paper we propose an extension of the so-called Iri-Imai method to solve constrained convex programming problems. The original Iri-Imai method is designed for linear programs and assumes that the optimal objective value of the optimization problem is known in advance. Zhang (Ref. 9) extends the method for constrained convex optimization but the optimum value is still assumed to be known in advance. In our new extension this last requirement on the optimal value is relaxed; instead only a lower bound of the optimal value is needed. Our approach uses a multiplicative barrier function for the problem with a univariate parameter that represents an estimated optimum value of the original optimization problem. An optimal solution to the original problem can be traced down by minimizing the multiplicative barrier function. Due to the convexity of this barrier function the optimal objective value as well as the optimal solution of the original problem are sought iteratively by applying Newtons method to the multiplicative barrier function. A new formulation of the multiplicative barrier function is further developed to acquire computational tractability and efficiency. Numerical results are presented to show the efficiency of the new method.His research supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.Communicated by Z. Q. Luo  相似文献   

7.
讨论了整体目标函数关于各子系统不具有可加形式的大规模稳态系统的优化问题,将混沌优化算法应用于其最优值的求解,利用混沌运动的遍历性来得到优化问题的全局最优值.仿真结果表明,该算法简单易行,求解精度和可靠性较高,是解决不可分稳态大系统优化问题的一种有效方法.  相似文献   

8.
Integer programming problems with a concave cost function are often encountered in optimization models involving economics of scale. In this paper, we propose an efficient exact algorithm for solving concave knapsack problems. The algorithm consists of an iterative process between finding lower and upper bounds by linearly underestimating the objective function and performing domain cut and partition by exploring the special structure of the problem. The lower bound is improved iteratively via cutting and partitioning the domain. This iteration process converges to the optimality in a finite number of steps. Promising computational results are reported for large-scale concave knapsack problems with up to 1200 integer variables. Comparison results with other existing methods in the literature are also presented. *Research supported by the National Natural Science Foundation of China under Grants 79970107 and 10271073,and the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.  相似文献   

9.
Most existing methods of global optimization for generalized geometric programming (GGP) actually compute an approximate optimal solution of a linear or convex relaxation of the original problem. However, these approaches may sometimes provide an infeasible solution, or far from the true optimum. To overcome these limitations, a robust solution algorithm is proposed for global optimization of (GGP) problem. This algorithm guarantees adequately to obtain a robust optimal solution, which is feasible and close to the actual optimal solution, and is also stable under small perturbations of the constraints.  相似文献   

10.
In this paper, we study a gradient-based continuous method for large-scale optimization problems. By converting the optimization problem into an ODE, we are able to show that the solution trajectory of this ODE tends to the set of stationary points of the original optimization problem. We test our continuous method on large-scale problems available in the literature. The simulation results are very attractive.This research was supported in part by Grants FRG/99-00/II-23 and FRG/00-0l/II-63 of Hong Kong Baptist University and the Research Grant Council of Hong Kong.  相似文献   

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