An algorithm for systems welfare interactive goal programming modelling

Every human system is faced with the problem of choosing between alternative options, and methods of interactive programming have been suggested as the best way to lead decision makers reach decisions that are consistent with their preferences. However, even though a large number of interactive algorithms have been proposed for multiobjective decision making (MODM), there is yet no truly interactive goal programming (GP) algorithm, despite the preference of GP over other MODM methodologies. The current paper presents an algorithm for interactive GP modelling called SWIGP (systems welfare interactive GP) which ensures that the overall welfare of the system under consideration is adequately taken into account in the interactive process. To achieve this, this paper distinguishes between technical, allocative and economic efficiencies and combines an economic efficiency index with interactive GP process. Besides being of wide applicability, the algorithm exerts little cognitive burden on the decision maker (DM). Indeed, even if the DM is assumed to operate under conditions of complete ignorance, SWIGP provides the direction for searching the “best” compromise solution. Moreover, the algorithm converges very fast because of the economic efficiency index that complements the interactive process in aiding the DM arrive at a most preferred solution.     

Goal programming in the context of the assignment problem and a computationally effective solution method

This paper develops the goal programming technique to solve the multiple objective assignment problem. The required model is formulated and an appropriate solution method is presented. The proposed method, which is a decomposition method, exploits the total unimodularity feature of the assignment problem and effectively reduces the computational efforts. Some issues related to the efficiency of a GP solution are stated and some specialized techniques for detecting and restoring efficiency are proposed.     

Goal programming for decision making: An overview of the current state-of-the-art

There have been significant advances in the theory of goal programming (GP) in recent years, particularly in the area of intelligent modelling and solution analysis. The intention of this paper is to provide an overview of these developments, to detail and assess the current state-of-the-art in the subject, and to highlight areas which seem promising for future research. Modelling techniques such as detection and restoration of pareto efficiency, normalisation, redundancy checking, and non-standard utility function modelling are overviewed. The connection between GP and other multi-objective-programming techniques as well as a utility interpretation of GP are examined. The rationality of ranking Multi-Criteria Decision Making techniques, and of placing GP in such a ranking, is discussed.     

Solving a class of geometric programming problems by an efficient dynamic model

In this paper, a neural network model is constructed on the basis of the duality theory, optimization theory, convex analysis theory, Lyapunov stability theory and LaSalle invariance principle to solve geometric programming (GP) problems. The main idea is to convert the GP problem into an equivalent convex optimization problem. A neural network model is then constructed for solving the obtained convex programming problem. By employing Lyapunov function approach, it is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the original problem. The simulation results also show that the proposed neural network is feasible and efficient.     

On the computational utility of posynomial geometric programming solution methods

Ten codes or code variants were used to solve the five equivalent posynomial GP problem formulations. Four of these codes were general NLP codes; six were specialized GP codes. A total of forty-two test problems was solved with up to twenty randomly generated starting points per problem. The convex primal formulation is shown to be intrinsically easiest to solve. The general purpose GRG code called OPT appears to be the most efficient code for GP problem solution. The reputed superiority of the specialized GP codes GGP and GPKTC appears to be largely due to the fact that these codes solve the convex primal formulation. The dual approaches are only likely to be competitive for small degree of difficulty, tightly constrained problems.     

Optimal Replacement Planning by Geometric Programming

Geometric programming (GP) is suggested as an analytical toolfor solving replacement problems with infinite time horizon.The GP solution method is described and explained through theformulation and solution of a typical replacement problem. Asimple example is worked out to demonstrate the pint that GPhas potential as an appropriate mathematical tool for the analysisof certain types of replacement problems.     

Goal programming using multiple objective hybrid metaheuristic algorithm

In this paper, a Goal Programming (GP) model is converted into a multi-objective optimization problem (MOO) of minimizing deviations from fixed goals. To solve the resulting MOO problem, a hybrid metaheuristic with two steps is proposed to find the Pareto set's solutions. First, a Record-to-Record Travel with an adaptive memory is used to find first non-dominated Pareto frontier solutions preemptively. Second, a Variable Neighbour Search technique with three transformation types is used to intensify every non dominated solution found in the first Pareto frontier to produce the final Pareto frontier solutions. The efficiency of the proposed approach is demonstrated by solving two nonlinear GP test problems and three engineering design problems. In all problems, multiple solutions to the GP problem are found in one single simulation run. The results prove that the proposed algorithm is robust, fast and simply structured, and manages to find high-quality solutions in short computational times by efficiently alternating search diversification and intensification using very few user-defined parameters.     

Existence of efficient solutions for vector maximization problems

The vector maximization problem arises when more than one objective function is to be maximized over a given feasibility region. The concept of efficiency has played a useful role in analyzing this problem. In order to exclude efficient solutions of a certain anomalous type, the concept of proper efficiency has also been utilized. In this paper, an examination of the existence of efficient and properly efficient solutions for the vector maximization problem is undertaken. Given a feasible solution for the vector maximization problem, a related single-objective mathematical programming problem is investigated. Any optimal solution to this program, if one exists, yields an efficient solution for the vector maximization problem. In many cases, the unboundedness of this problem shows that no properly efficient solutions exist. Conditions are pointed out under which the latter conclusion implies that the set of efficient solutions is null. As a byproduct of our results, conditions are derived which guarantee that the outcome of any improperly efficient point is the limit of the outcomes of some sequence of properly efficient points. Examples are provided to illustrate these results.The author would like to thank Professor T. L. Morin for his helpful comments. Thanks also go to an anonymous reviewer for his useful comments concerning an earlier version of this paper.The author would like to acknowledge a useful discussion with Professor G. Bitran which helped in motivating Example 4.1.     

The 0-1 bidimensional knapsack problem: Toward an efficient high-level primitive tool

Efficient codes exist for exactly solving the 0-1 knapsack problem, which is a common primitive structure in relaxation and decomposition techniques for the solution of general models. We suggest moving to a higher primitive level by using the bidimensional knapsack, which can be used to enhance linear programming or Lagrangean type classical relaxations.With the ultimate aim of providing an exact and efficient solution to the bidimensional knapsack problem, we describe here a heuristic approach based on surrogate duality. In particular, we consider the usefulness of a specific preprocessing phase before a possible enumerative phase.Extensive numerical experiments, based on test problems from the literature as well as randomly generated instances, show that our code compares favorably with the GP procedure developed by Gavish and Pirkul for the multidimensional case.     

Shrinking gradient descent algorithms for total variation regularized image denoising

Total variation regularization introduced by Rudin, Osher, and Fatemi (ROF) is widely used in image denoising problems for its capability to preserve repetitive textures and details of images. Many efforts have been devoted to obtain efficient gradient descent schemes for dual minimization of ROF model, such as Chambolle’s algorithm or gradient projection (GP) algorithm. In this paper, we propose a general gradient descent algorithm with a shrinking factor. Both Chambolle’s and GP algorithm can be regarded as the special cases of the proposed methods with special parameters. Global convergence analysis of the new algorithms with various step lengths and shrinking factors are present. Numerical results demonstrate their competitiveness in computational efficiency and reconstruction quality with some existing classic algorithms on a set of gray scale images.     

Enhanced Efficiency in Multi-objective Optimization

For a given multi-objective optimization problem, we introduce and study the notion of α-proper efficiency. We give two characterizations of such proper efficiency: one is in terms of exact penalization and the other is in terms of stability of associated parametric problems. Applying the aforementioned characterizations and recent results on global error bounds for inequality systems, we obtain verifiable conditions for α-proper efficiency. For a large class of polynomial multi-objective optimization problems, we show that any efficient solution is α-properly efficient under some mild conditions. For a convex quadratically constrained multi-objective optimization problem with convex quadratic objective functions, we show that any efficient solution is α-properly efficient with a known estimate on α whenever its constraint set is bounded. Finally, we illustrate our achieved results with examples, and give an example to show that such an enhanced efficiency property may not hold for multi-objective optimization problems involving C ^∞-functions as objective functions.     

Solving Posynomial Geometric Programming Problems via Generalized Linear Programming

This paper revisits an efficient procedure for solving posynomial geometric programming (GP) problems, which was initially developed by Avriel et al. The procedure, which used the concept of condensation, was embedded within an algorithm for the more general (signomial) GP problem. It is shown here that a computationally equivalent dual-based algorithm may be independently derived based on some more recent work where the GP primal-dual pair was reformulated as a set of inexact linear programs. The constraint structure of the reformulation provides insight into why the algorithm is successful in avoiding all of the computational problems traditionally associated with dual-based algorithms. Test results indicate that the algorithm can be used to successfully solve large-scale geometric programming problems on a desktop computer.     

A linear programming approach to test efficiency in multi-objective linear fractional programming problems

In a multi-objective linear fractional programming problem (MOLFPP), it is often useful to check the efficiency of a given feasible solution, and if the solution is efficient, it is useful to check strong or weak efficiency. In this paper, by applying a geometrical interpretation, a linear programming approach is achieved to test weak efficiency. Also, in order to test strong efficiency for a given weakly efficient point, a linear programming approach is constructed.     

Isolated efficiency in nonsmooth semi-infinite multi-objective programming

ABSTRACT

In this paper, isolated efficient solutions of a given nonsmooth Multi-Objective Semi-Infinite Programming problem (MOSIP) are studied. Two new Data Qualifications (DQs) are introduced and it is shown that these DQs are, to a large extent, weaker than already known Constraint Qualifications (CQs). The relationships between isolated efficiency and some relevant notions existing in the literature, including robustness, are established. Various necessary and sufficient conditions for characterizing isolated efficient solutions of a general problem are derived. It is done invoking the tangent cones, the normal cones, the generalized directional derivatives, and some gap functions. Using these characterizations, the (strongly) perturbed Karush-Kuhn-Tucker (KKT) optimality conditions for MOSIP are analyzed. Furthermore, it is shown that each isolated efficient solution is a Geoffrion properly efficient solution under appropriate assumptions. Moreover, Kuhn-Tucker (KT) and Klinger properly efficient solutions for a nonsmooth MOSIP are defined and it is proved that each isolated efficient solution is a KT properly efficient solution in general, and a Klinger properly efficient solution under a DQ. Finally, in the last section, the largest isolated efficiency constant for a given isolated efficient solution is determined.     

Modified geometric programming problem and its applications

In this paper, we propose unconstrained and constrained posynomial Geometric Programming (GP) problem with negative or positive integral degree of difficulty. Conventional GP approach has been modified to solve some special typer of GP problems. In specific case, when the degree of difficulty is negative, the normality and the orthogonality conditions of the dual program give a system of linear equations. No general solution vector exists for this system of linear equations. But an approximate solution can be determined by the least square and also max-min method. Here, modified form of geometric programming method has been demonstrated and for that purpose necessary theorems have been derived. Finally, these are illustrated by numerical examples and applications.     

Global efficiency in multiobjective programming *

In this paper, we are concerned with global efficiency in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove that a local weakly efficient solution, a local efficient solution and a local properly efficient solution are respectively a global weakly efficient solution, a global efficient solution and a global properly efficient solution of a multiobjective programming problem if cone- subconvexlikeness or cone-pre-invexity is assumed     

An additive achievement scalarizing function for multiobjective programming problems

The success of the reference point scheme within interactive techniques for multiobjective programming problems is unquestionable. However, so far, the different achievement scalarizing functions are, more or less, extensions of the Tchebychev distance. The reason for this is the ability of this function to determine efficient solutions and to support every efficient solution of the problem. For the same reasons, no additive scheme has yet been used in reference point-based interactive methods. In this paper, an additive achievement scalarizing function is proposed. Theoretical results prove that this function supports every efficient solution, and conditions are given under which the efficiency of each solution is guaranteed. Some examples and computational tests show the different behaviours of the Tchebychev and additive approaches, and an additive reference point interactive algorithm is proposed.     

基于平均树值的无圈图博弈有效解

本文对无圈图博弈进行了研究,考虑了大联盟收益不小于各分支收益之和的情况。通过引入剩余公平分配性质,也就是任意两个分支联盟的平均支付变化相等,给出了一个基于平均树值的无圈图博弈有效解。同时,结合有效性和分支公平性对该有效解进行了刻画。特别地,若无圈图博弈满足超可加性时,证明了该有效解一定是核中的元素,说明此时的解是稳定的。最后,通过一案例分析了该有效解的特点,即越大的分支分得的剩余越多,并且关键参与者,也就是具有较大度的参与者可获得相对多的支付。     

一类多目标优化问题的有效性

本文研究了一类带不等式约束的多目标优化问题,给出了该类问题的有效解的一些充分必要条件,在适当条件下利用线性标量化方法证明了其有效解和真有效解的等价性。本文的主要结论是对最近一些文献中相应结果的改进与推广。     

Determining maximal efficient faces in multiobjective linear programming problem

Finding an efficient or weakly efficient solution in a multiobjective linear programming (MOLP) problem is not a difficult task. The difficulty lies in finding all these solutions and representing their structures. Since there are many convenient approaches that obtain all of the (weakly) efficient extreme points and (weakly) efficient extreme rays in an MOLP, this paper develops an algorithm which effectively finds all of the (weakly) efficient maximal faces in an MOLP using all of the (weakly) efficient extreme points and extreme rays. The proposed algorithm avoids the degeneration problem, which is the major problem of the most of previous algorithms and gives an explicit structure for maximal efficient (weak efficient) faces. Consequently, it gives a convenient representation of efficient (weak efficient) set using maximal efficient (weak efficient) faces. The proposed algorithm is based on two facts. Firstly, the efficiency and weak efficiency property of a face is determined using a relative interior point of it. Secondly, the relative interior point is achieved using some affine independent points. Indeed, the affine independent property enable us to obtain an efficient relative interior point rapidly.