A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

Hierarchical generating method for large-scale multiobjective systems

This paper investigates large-scale multiobjective systems in the context of a general hierarchical generating method which considers the problem of how to find the set of all noninferior solutions by decomposition and coordination. A new, unified framework of the hierarchical generating method is developed by integrating the envelope analysis approach and the duality theory that is used in multiobjective programming. In this scheme, the vector-valued Lagrangian and the duality theorem provide the basis of a decomposition of the overall multiobjective system into several multiobjective subsystems, and the envelope analysis gives an efficient approach to deal with the coordination at a high level. The following decomposition-coordination schemes for different problems are developed: (i) a spatial decomposition and envelope coordination algorithm for large-scale multiobjective static systems; (ii) a temporal decomposition and envelope coordination algorithm for multiobjective dynamic systems; and (iii) a three-level structure algorithm for large-scale multiobjective dynamic systems.This work was supported by NSF Grant No. CEE-82-11606.     

A Dynamic Programming Algorithm for the κ-Haplotyping Problem

The Minimum Fragments Removal （MFR） problem is one of the haplotyping problems： given a set of fragments, remove the minimum number of fragments so that the resulting fragments can be partitioned into k classes of non-conflicting subsets. In this paper, we formulate the κ-MFR problem as an integer linear programming problem, and develop a dynamic programming approach to solve the κ-MFR problem for both the gapless and gap eases.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

多目标半定规划的加权中心路径法

对于线性型多目标半定规划问题，引进加权中心路径的概念，并利用单目标半定规划的中心路径法，提出了求解多目标半定规划问题的加权中心路径法，先得型对一个叔向量的有效解，然后在此基础上，提出了通过一次迭代得到对应一定范围内其他任意权向量的有效解的一步修正方法．     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part II. Duality

This paper represents the second part of a study concerning the so-called G-multiobjective programming. A new approach to duality in differentiable vector optimization problems is presented. The techniques used are based on the results established in the paper: On G-invex multiobjective programming. Part I. Optimality by T.Antczak. In this work, we use a generalization of convexity, namely G-invexity, to prove new duality results for nonlinear differentiable multiobjective programming problems. For such vector optimization problems, a number of new vector duality problems is introduced. The so-called G-Mond–Weir, G-Wolfe and G-mixed dual vector problems to the primal one are defined. Furthermore, various so-called G-duality theorems are proved between the considered differentiable multiobjective programming problem and its nonconvex vector G-dual problems. Some previous duality results for differentiable multiobjective programming problems turn out to be special cases of the results described in the paper.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

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.     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

On fuzzy random multiobjective quadratic programming

In this paper, a multiobjective quadratic programming problem having fuzzy random coefficients matrix in the objective and constraints and the decision vector are fuzzy pseudorandom variables is considered. First, we show that the efficient solutions of fuzzy quadratic multiobjective programming problems are resolved into series-optimal-solutions of relative scalar fuzzy quadratic programming. Some theorems are proved to find an optimal solution of the relative scalar quadratic multiobjective programming with fuzzy coefficients, having decision vectors as fuzzy variables. At the end, numerical examples are illustrated in the support of the obtained results.     

On noninferior performance index vectors

The noninferior vector index problem of optimal control theory is investigated in an effort to establish some basic properties of the noninferior index surface in the generalN-dimensional index problem. The vector performance index problem is first converted to a family of scalar index problems by forming an auxiliary scalar index as a function of the vector index and a vector of weighting parameters. The functional relationship between noninferior vectors and the weighting vectors of the auxiliary index problem is investigated for the particular case in which the auxiliary index is a weighted sum of the vector index elements. Special attention is devoted to the noninferior index problem for whichN = 2.     

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．     

Nodal aggregation of resource constraints in a shortest path problem

The shortest path problem with resource constraints consists of finding the minimum cost path between two specified points while respecting constraints on resource consumption. Its solving by a dynamic programming algorithm requires a computation time increasing with the number of resources. With the aim of producing rapidly a good heuristic solution we propose to reduce the state space by aggregating resources. Our approach consists of projecting the resources on a vector of smaller dimension and then to dynamically adjust the projection matrix to get a better approximation of the optimal solution. We propose an adjustment based on Lagrangian and surrogate relaxations in a column generation framework, in which the sub-problems are shortest path problems with resource constraints. We adjust the multipliers only one time at each column generation iteration. This permit to obtain good solutions of the scheduling problem in few time.     

A Lagrangian search method for the <Emphasis Type="Italic">P</Emphasis>-median problem

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.     

Success Guarantee of Dual Search in Integer Programming: p-th Power Lagrangian Method

Although the Lagrangian method is a powerful dual search approach in integer programming, it often fails to identify an optimal solution of the primal problem. The p-th power Lagrangian method developed in this paper offers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations. One other prominent feature of the p-th power Lagrangian method is that the dual search only involves a one-dimensional search within [0,1]. Some potential applications of the method as well as the issue of its implementation are discussed.     

Extended convergence results for the method of multipliers for nonstrictly binding inequality constraints

In this paper, we extend the classical convergence and rate of convergence results for the method of multipliers for equality constrained problems to general inequality constrained problems, without assuming the strict complementarity hypothesis at the local optimal solution. Instead, we consider an alternative second-order sufficient condition for a strict local minimum, which coincides with the standard one in the case of strict complementary slackness. As a consequence, new stopping rules are derived in order to guarantee a local linear rate of convergence for the method, even if the current Lagrangian is only asymptotically minimized in this more general setting. These extended results allow us to broaden the scope of applicability of the method of multipliers, in order to cover all those problems admitting loosely binding constraints at some optimal solution. This fact is not meaningless, since in practice this kind of problem seems to be more the rule rather than the exception.In proving the different results, we follow the classical primaldual approach to the method of multipliers, considering the approximate minimizers for the original augmented Lagrangian as the exact solutions for some adequate approximate augmented Lagrangian. In particular, we prove a general uniform continuity property concerning both their primal and their dual optimal solution set maps, a property that could be useful beyond the scope of this paper. This approach leads to very simple proofs of the preliminary results and to a straight-forward proof of the main results.The author gratefully acknowledges the referees for their helpful comments and remarks. This research was supported by FONDECYT (Fondo Nacional de Desarrollo Científico y Technológico de Chile).     

On <Emphasis Type="Italic">G</Emphasis>-invex multiobjective programming. Part I. Optimality

In this paper, a generalization of convexity, namely G-invexity, is considered in the case of nonlinear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. The modified Karush-Kuhn-Tucker necessary optimality conditions for a certain class of multiobjective programming problems are established. To prove this result, the Kuhn-Tucker constraint qualification and the definition of the Bouligand tangent cone for a set are used. The assumptions on (weak) Pareto optimal solutions are relaxed by means of vector-valued G-invex functions.     

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms.The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined.The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients.The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.     

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.