Solving Multi-Objective Linear Programming Problem by Statistical Averaging Method with the Help of Fuzzy Programming Method

A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.

A COMBINED PARAMETRIC QUADRATIC PROGRAMMING AND PRECISE INTEGRATION METHOD BASED DYNAMIC ANALYSIS OF ELASTIC-PLASTIC HARDENING/SOFTENING PROBLEMS

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

Dynamic programming methodology for multi-criteria group decision-making under ordinal preferences

A method of minimizing rankings inconsistency is proposed for a decision-making problem with rankings of alternatives given by multiple decision makers according to multiple criteria. For each criteria, at first, the total inconsistency between the rankings of all alternatives for the group and the ones for every decision maker is defined after the decision maker weights in respect to the criteria are considered. Similarly, the total inconsistency between their final rankings for the group and the ones under every criteria is determined after the criteria weights are taken into account. Then two nonlinear integer programming models minimizing respectively the two total inconsistencies above are developed and then transformed to two dynamic programming models to obtain separately the rankings of all alternatives for the group with respect to each criteria and their final rankings. A supplier selection case illustrated the proposed method, and some discussions on the results verified its effectiveness. This work develops a new measurement of ordinal preferences’ inconsistency in multi-criteria group decision-making （MCGDM） and extends the cook-seiford social selection function to MCGDM considering weights of criteria and decision makers and can obtain unique ranking result.

Global convergent algorithm for the bilevel linear fractional-linear programming based on modified convex simplex method

A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equaling to zero, the bilevel linear fractional-linear programming is transformed into a traditional sin- gle level programming problem, which can be transformed into a series of linear fractional programming problem. Thus, the modi- fied convex simplex method is used to solve the infinite linear fractional programming to obtain the global convergent solution of the original bilevel linear fractional-linear programming. Finally, an example demonstrates the feasibility of the proposed algorithm.

A Combined Homotopy Infeasible Interior-Point Method for Convex Nonlinear Programming

In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex nonlinear programming, without strict convexity for the logarithmic barrier function, we get different solutions of the convex programming in different cases by CHIIP method.

THE CONVERGENCE OF APPROACH PENALTY FUNCTION METHOD FOR APPROXIMATE BILEVEL PROGRAMMING PROBLEM

In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem. the evaluation of the objective function is very difficult, so that only their approximate values can be obtained. This algorithm is obtained by combining penalty function method and approximation in bilevel programming. The presented algorithm is completely different from existing methods. That convergence for this algorithm is proved.

MIXED ENERGY METHOD FOR SOLUTION OF QUADRATIC PROGRAMMING PROBLEMS AND ELASTIC-PLASTIC ANALYSIS OF TRUSS STRUCTURES

A new algorithm for the solution of quadratic programming problemsis put forward in terms of the mixed energy theory and is furtherused for the incremental solution of elastic-plastic trussstructures. The method proposed is different from the traditionalone, for which the unknown variables are selected just in one classsuch as displacements or stresses. The present method selects thevariables in the mixed form with both displacement and stress. As themethod is established in the hybrid space, the information found inthe previous incremental step can be used for the solution of thepresent step, making the algorithm highly effi- cient in thenumerical solution process of quadratic programming problems. Theresults obtained in the exm- ples of the elastic-plastic solution ofthe truss structures verify what has been predicted in thetheoretical anal- ysis.

Solving a Class of Nonlinear Programming Problems via a Homotopy Continuation Method

In this paper we present a homotopy continuation method for finding the Karush-Kuhn-Tucker point of a class of nonlinear non-convex programming problems. Two numerical examples are given to show that this method is effective. It should be pointed out that we extend the results of Lin et al. （see Appl. Math. Comput., 80（1996）, 209-224） to a broader class of non-convex programming problems.

Modified Exact Jacobian Semidefinite Programming Relaxation for Celis-Dennis-Tapia Problem

A modified exact Jacobian semidefinite programming(SDP)relaxation method is proposed in this paper to solve the Celis-Dennis-Tapia(CDT)problem using the Jacobian matrix of objective and constraining polynomials.In the modified relaxation problem,the number of introduced constraints and the lowest relaxation order decreases significantly.At the same time,the finite convergence property is guaranteed.In addition,the proposed method can be applied to the quadratically constrained problem with two quadratic constraints.Moreover,the efficiency of the proposed method is verified by numerical experiments.

A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.

An Augmented Lagrangian based Semismooth Newton Method for a Class of Bilinear Programming Problems

This paper proposes a semismooth Newton method for a class of bilinear programming problems(BLPs)based on the augmented Lagrangian,in which the BLPs are reformulated as a system of nonlinear equations with original variables and Lagrange multipliers.Without strict complementarity,the convergence of the method is studied by means of theories of semismooth analysis under the linear independence constraint qualification and strong second order sufficient condition.At last,numerical results are reported to show the performance of the proposed method.

NEWTON METHOD FOR SOLVING A CLASS OF SMOOTH CONVEX PROGRAMMING

An algorithm for solving a class of smooth convex programming is given. Using smooth exact multiplier penalty function, a smooth convex programming is minimized to a minimizing strongly convex function on the compact set was reduced. Then the strongly convex function with a Newton method on the given compact set was minimized.

Novel Method to Handle Inequality Constraints for Nonlinear Programming

By redefining the multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, say, u2_i, i=1, 2, ..., m, nonnegative constraints imposed on inequality constraints in Karush-Kuhn-Tucker necessary conditions are removed. For constructing the Lagrange neural network and Lagrange multiplier method, it is no longer necessary to convert inequality constraints into equality constraints by slack variables in order to reuse those results dedicated to equality constraints, and they can be similarly proved with minor modification. Utilizing this technique, a new type of Lagrange neural network and a new type of Lagrange multiplier method are devised, which both handle inequality constraints directly. Also, their stability and convergence are analyzed rigorously.

An Interactive Fuzzy Satisficing Method for Multiobjective Stochastic Integer Programming with Simple Recourse

This paper considers multiobjective integer programming problems involving random variables in constraints. Using the concept of simple recourse, the formulated multiobjective stochastic simple recourse problems are transformed into deterministic ones. For solving transformed deterministic problems efficiently, we also introduce genetic algorithms with double strings for nonlinear integer programming problems. Taking into account vagueness of judgments of the decision maker, an interactive fuzzy satisficing method is presented. In the proposed interactive method, after determineing the fuzzy goals of the decision maker, a satisficing solution for the decision maker is derived efficiently by updating the reference membership levels of the decision maker. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Approximation-Exact Penalty Function Method for Solving a Class of Stochastic Programming

We present an approximation-exact penalty function method for solving the single stage stochastic programming problem with continuous random variable. The original problem is transformed into a determinate nonlinear programming problem with a discrete random variable sequence, which is obtained by some discrete method. We construct an exact penalty function and obtain an unconstrained optimization. It avoids the difficulty in solution by the rapid growing of the number of constraints for discrete precision. Under lenient conditions, we prove the equivalence of the minimum solution of penalty function and the solution of the determinate programming, and prove that the solution sequences of the discrete problem converge to a solution to the original problem.

A Numerical Method for Rigid-plastic FEM Analysis Basing on Mathematical Programming

The rigid-plastic analysis of mental forming simulation is formulated as a discrete nonlinear mathematical programming problem with equality and inequality constraints by means of the finite element technique. An iteration algorithm is used to solve this formulation, which distinguishes the integration points of the rigid zones and the plastic zones and solves a series of the quadratic programming to overcome the difficulties caused by the nonsmoothness and the nonlinearity of the objective function. This method has been used to carry out the rigid-plastic FEM analysis. An example is given to demonstrate the effectiveness of this method.

A Weaker Constraint Qualification of Globally Convergent Homotopy Method for a Multiobjective Programming Problem

In this paper, we prove that the combined homotopy interior point method for a multiobjective programming problem introduced in Ref. [1] remains valid under a weaker constrained qualification—the Mangasarian-Fromovitz constrained qualification, instead of linear independence constraint qualification. The algorithm generated by this method associated to the Karush-Kuhn-Tucker points of the multiobjective programming problem is proved to be globally convergent.

A New Numerical Method for Solving the Stokes Problem Using Quadratic Programming

In this paper we present a new method for solving the Stokes problem which is a constrained optimization method. The new method is simpler and requires less computation than the existing methods. In this method we transform the Stokes problem into a quadratic programming problem and by solving it, the velocity and the pressure are obtained.

Programming Logic Modeling and Cross-Program Defect Detection Method for Object-Oriented Code

Code defects can lead to software vulnerability and even produce vulnerability risks.Existing research shows that the code detection technology with text analysis can judge whether object-oriented code files are defective to some extent.However,these detection techniques are mainly based on text features and have weak detection capabilities across programs.Compared with the uncertainty of the code and text caused by the developer’s personalization,the programming language has a stricter logical specification,which reflects the rules and requirements of the language itself and the developer’s potential way of thinking.This article replaces text analysis with programming logic modeling,breaks through the limitation of code text analysis solely relying on the probability of sentence/word occurrence in the code,and proposes an object-oriented language programming logic construction method based on method constraint relationships,selecting features through hypothesis testing ideas,and construct support vector machine classifier to detect class files with defects and reduce the impact of personalized programming on detection methods.In the experiment,some representative Android applications were selected to test and compare the proposed methods.In terms of the accuracy of code defect detection,through cross validation,the proposed method and the existing leading methods all reach an average of more than 90%.In the aspect of cross program detection,the method proposed in this paper is superior to the other two leading methods in accuracy,recall and F1 value.