A comparison between a primal and a dual cutting plane algorithm for posynomial geometric programming problems

In this paper, primal and dual cutting plane algorithms for the solution of posynomial geometric programming problems are presented. It is shown that these cuts are deepest, in the sense that they cut off as much of the infeasible set as possible. Problems of nondifferentiability in the dual cutting plane are circumvented by the use of a subgradient. Although the resulting dual problem seems easier to solve, the computational experience seems to show that the primal cutting plane outperforms the dual.     

Subgradient optimization applied to a discrete nonlinear problem in engineering design

The particular application considered here is the design of relief-header systems, involving compressible fluid flow through tree-networks. The flowrates are specified, and the design problem involves the choice of discrete pipe sizes to minimize total cost while satisfying pressure-drop constraints, which are highly nonlinear. The problem is solved in two stages. Firstly the problem of obtaining the optimal set of continuous pipe sizes is addressed; it turns out that a dual formulation provides and extremely rapid solution. Next, a subgradient optimization procedure is used on the dual in order to solve for the discrete pipe sizes. Networks of up to 78 paths and 205 sections, each involving 50 discrete pipe sizes, have been solved.     

Lagrangian heuristics for the two-echelon,single-source,capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.     

用次微分表达的对偶问题

本利用次微分建立了多目标规划的一个新的对偶问题，并给出其弱、强和逆对偶性，得到了一个新的次梯度的定义，并用其建立了一个新的对偶问题。     

An algorithm for set covering problem

In this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent, subgradient optimisation and linear programming. Computational results are presented for problems involving up to 400 rows and 4000 columns.     

Approximate Subgradient Methods for Nonlinearly Constrained Network Flow Problems

The minimization of nonlinearly constrained network flow problems can be performed by using approximate subgradient methods. The idea is to solve this kind of problem by means of primal-dual methods, given that the minimization of nonlinear network flow problems can be done efficiently exploiting the network structure. In this work, it is proposed to solve the dual problem by using ε-subgradient methods, as the dual function is estimated by minimizing approximately a Lagrangian function, which includes the side constraints (nonnetwork constraints) and is subject only to the network constraints. Some well-known subgradient methods are modified in order to be used as ε-subgradient methods and the convergence properties of these new methods are analyzed. Numerical results appear very promising and effective for this kind of problems This research was partially supported by Grant MCYT DPI 2002-03330.     

Subgradient Methods for Saddle-Point Problems

We study subgradient methods for computing the saddle points of a convex-concave function. Our motivation comes from networking applications where dual and primal-dual subgradient methods have attracted much attention in the design of decentralized network protocols. We first present a subgradient algorithm for generating approximate saddle points and provide per-iteration convergence rate estimates on the constructed solutions. We then focus on Lagrangian duality, where we consider a convex primal optimization problem and its Lagrangian dual problem, and generate approximate primal-dual optimal solutions as approximate saddle points of the Lagrangian function. We present a variation of our subgradient method under the Slater constraint qualification and provide stronger estimates on the convergence rate of the generated primal sequences. In particular, we provide bounds on the amount of feasibility violation and on the primal objective function values at the approximate solutions. Our algorithm is particularly well-suited for problems where the subgradient of the dual function cannot be evaluated easily (equivalently, the minimum of the Lagrangian function at a dual solution cannot be computed efficiently), thus impeding the use of dual subgradient methods.     

On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

Dual formulations and subgradient optimization strategies for linear programming relaxations of mixed-integer programs

This paper examines algorithmic strategies relating to the formulation of Lagrangian duals, and their solution via subgradient optimization, in the context of solving linear programming relaxations of mixed-integer programs. As in the current trend in integer programming, several researchers have found it beneficial to generate and add additional constraints to the model, prior to its solution, in order to tighten its linear programming relaxation. It becomes necessary, therefore, to be able to efficiently derive strong surrogate constraints and bounds based on such reformulations. This paper addresses the design and testing of the most viable technique available for exploiting such tight reformulations, namely, using Lagrangian relaxation in coordination with subgradient optimization. A computational study is conducted to test the efficiency of various Lagrangian dual formulations, and to investigate in the context of using subgradient optimization the effects of step size choices, problem scaling, conducting pattern moves, and projecting dual solutions onto subsets of violated dual constraints. Based on this study, certain recommendations are made regarding the manner in which one should implement such an approach.     

A new model and hybrid approach for large scale inventory routing problems

This paper studies an inventory routing problem (IRP) with split delivery and vehicle fleet size constraint. Due to the complexity of the IRP, it is very difficult to develop an exact algorithm that can solve large scale problems in a reasonable computation time. As an alternative, an approximate approach that can quickly and near-optimally solve the problem is developed based on an approximate model of the problem and Lagrangian relaxation. In the approach, the model is solved by using a Lagrangian relaxation method in which the relaxed problem is decomposed into an inventory problem and a routing problem that are solved by a linear programming algorithm and a minimum cost flow algorithm, respectively, and the dual problem is solved by using the surrogate subgradient method. The solution of the model obtained by the Lagrangian relaxation method is used to construct a near-optimal solution of the IRP by solving a series of assignment problems. Numerical experiments show that the proposed hybrid approach can find a high quality near-optimal solution for the IRP with up to 200 customers in a reasonable computation time.     

Dual extragradient algorithms extended to equilibrium problems

In this paper we propose two iterative schemes for solving equilibrium problems which are called dual extragradient algorithms. In contrast with the primal extragradient methods in Quoc et al. (Optimization 57(6):749–776, 2008) which require to solve two general strongly convex programs at each iteration, the dual extragradient algorithms proposed in this paper only need to solve, at each iteration, one general strongly convex program, one projection problem and one subgradient calculation. Moreover, we provide the worst case complexity bounds of these algorithms, which have not been done in the primal extragradient methods yet. An application to Nash-Cournot equilibrium models of electricity markets is presented and implemented to examine the performance of the proposed algorithms.     

基于异步次梯度法的LR算法及其在多阶段HFSP的应用

为降低求解复杂度和缩短计算时间,针对多阶段混合流水车间总加权完成时间问题,提出了一种结合异步次梯度法的改进拉格朗日松弛算法。建立综合考虑有限等待时间和工件释放时间的整数规划数学模型,将异步次梯度法嵌入到拉格朗日松弛算法中,从而通过近似求解拉格朗日松弛问题得到一个合理的异步次梯度方向,沿此方向进行搜索,逐渐降低到最优点的距离。通过仿真实验,验证了所提算法的有效性。对比所提算法与传统的基于次梯度法的拉格朗日松弛算法,结果表明,就综合解的质量和计算效率而言,所提算法能在较短的计算时间内获得更好的近优解,尤其是对大规模问题。     

A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process

One of the largest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The SCC scheduling is a complex hybrid flowshop (HFS) scheduling problem with the following features: job grouping and precedence constraints, no idle time within the same group of jobs and setup time constraints on the casters. This paper first models the scheduling problem as a mixed-integer programming (MIP) problem with the objective of minimizing the total weighted earliness/tardiness penalties and job waiting. Next, a Lagrangian relaxation (LR) approach relaxing the machine capacity constraints is presented to solve the MIP problem, which decomposes the relaxed problem into two tractable subproblems by separating the continuous variables from the integer ones. Additionally, two methods, i.e., the boundedness detection method and time horizon method, are explored to handle the unboundedness of the decomposed subproblems in iterations. Furthermore, an improved subgradient level algorithm with global convergence is developed to solve the Lagrangian dual (LD) problem. The computational results and comparisons demonstrate that the proposed LR approach outperforms the conventional LR approaches in terms of solution quality, with a significantly shorter running time being observed.     

An integrated cutting stock and sequencing problem

In this paper an integrated problem formulated as an integer linear programming problem is presented to find an optimal solution to the cutting stock problem under particular pattern sequencing constraints. The solution uses a Lagrangian approach. The dual problem is solved using a modified subgradient method. A heuristic for the integrated problem is also presented. The computational results obtained from a set of unidimensional instances that use these procedures are reported.     

Augmented Lagrangian Duality and Nondifferentiable Optimization Methods in Nonconvex Programming

In this paper we present augmented Lagrangians for nonconvex minimization problems with equality constraints. We construct a dual problem with respect to the presented here Lagrangian, give the saddle point optimality conditions and obtain strong duality results. We use these results and modify the subgradient and cutting plane methods for solving the dual problem constructed. Algorithms proposed in this paper have some advantages. We do not use any convexity and differentiability conditions, and show that the dual problem is always concave regardless of properties the primal problem satisfies. The subgradient of the dual function along which its value increases is calculated without solving any additional problem. In contrast with the penalty or multiplier methods, for improving the value of the dual function, one need not to take the penalty like parameter to infinity in the new methods. In both methods the value of the dual function strongly increases at each iteration. In the contrast, by using the primal-dual gap, the proposed algorithms possess a natural stopping criteria. The convergence theorem for the subgradient method is also presented.     

Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

An outer approximate subdifferential method for piecewise affine optimization

Piecewise affine functions arise from Lagrangian duals of integer programming problems, and optimizing them provides good bounds for use in a branch and bound method. Methods such as the subgradient method and bundle methods assume only one subgradient is available at each point, but in many situations there is more information available. We present a new method for optimizing such functions, which is related to steepest descent, but uses an outer approximation to the subdifferential to avoid some of the numerical problems with the steepest descent approach. We provide convergence results for a class of outer approximations, and then develop a practical algorithm using such an approximation for the compact dual to the linear programming relaxation of the uncapacitated facility location problem. We make a numerical comparison of our outer approximation method with the projection method of Conn and Cornuéjols, and the bundle method of Schramm and Zowe. Received September 10, 1998 / Revised version received August 1999?Published online December 15, 1999     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

An equilibrium model of resource scarcity

A new equilibrium model of resource scarcity is discussed, whose solution is the intersection point of two maps: the gradient (subgradient) of the sensitivity function of a convex programming problem and the gradient (subgradient) of an a priori given resource cost function. In fact, we compute the equilibrium consumption volumes of resources with equal internal and external marginal costs. The proposed method of computing the equilibrium solution of the model reduces to solving a saddle-point problem. Convergence of the proposed method is proved.