Hybrid particle swarm optimization with chaotic search for solving integer and mixed integer programming problems

A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.The performances of CLSPSO are compared with those of other five hybrid algorithms combining PSO with chaotic search methods.Experimental results indicate that in terms of robustness and final convergence speed,CLSPSO is better than other five algorithms in solving many of these problems.Furthermore,CLSPSO exhibits good performance in solving two high-dimensional problems,and it finds better solutions than the known ones.A performance index(PI) is introduced to fairly compare the above six algorithms,and the obtained values of(PI) in three cases demonstrate that CLSPSO is superior to all the other five algorithms under the same conditions.

Improved genetic algorithm for nonlinear programming problems

An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.

Solution for integer linear bilevel programming problems using orthogonal genetic algorithm

An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorithm is developed for solving the binary linear implicit programming problem based on the orthogonal design. The orthogonal design with the factor analysis, an experimental design method is applied to the genetic algorithm to make the algorithm more robust, statistical y sound and quickly convergent. A crossover operator formed by the orthogonal array and the factor analysis is presented. First, this crossover operator can generate a smal but representative sample of points as offspring. After al of the better genes of these offspring are selected, a best combination among these offspring is then generated. The simulation results show the effectiveness of the proposed algorithm.

Orthogonal genetic algorithm for solving quadratic bilevel programming problems

A quadratic bilevel programming problem is transformed into a single level complementarity slackness problem by applying Karush-Kuhn-Tucker（KKT） conditions.To cope with the complementarity constraints,a binary encoding scheme is adopted for KKT multipliers,and then the complementarity slackness problem is simplified to successive quadratic programming problems,which can be solved by many algorithms available.Based on 0-1 binary encoding,an orthogonal genetic algorithm,in which the orthogonal experimental design with both two-level orthogonal array and factor analysis is used as crossover operator,is proposed.Numerical experiments on 10 benchmark examples show that the orthogonal genetic algorithm can find global optimal solutions of quadratic bilevel programming problems with high accuracy in a small number of iterations.

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.

Discrete differential evolution algorithm for integer linear bilevel programming problems

A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.

Saddle Point Criteria in Nonsmooth Semi-Infinite Minimax Fractional Programming Problems

This paper considers a nonsmooth semi-infinite minimax fractional programming problem（SIMFP） involving locally Lipschitz invex functions. The authors establish necessary optimality conditions for SIMFP. The authors establish the relationship between an optimal solution of SIMFP and saddle point of scalar Lagrange function for SIMFP. Further, the authors study saddle point criteria of a vector Lagrange function defined for SIMFP.

Higher-order Symmetric Duality in Multiobjective Programming Problems

In this paper, a pair of Mond-Weir type higher-order symmetric dual programs over arbitrary cones is formulated. The appropriate duality theorems, such as weak duality theorem, strong duality theorem and converse duality theorem, are established under higher-order （strongly） cone pseudoinvexity assumptions.

Optimization Methods for Mixed Integer Weakly Concave Programming Problems

In this paper,we consider a class of mixed integer weakly concave programming problems(MIWCPP)consisting of minimizing a difference of a quadratic function and a convex function.A new necessary global optimality conditions for MIWCPP is presented in this paper.A new local optimization method for MIWCPP is designed based on the necessary global optimality conditions,which is different from the traditional local optimization method.A global optimization method is proposed by combining some auxiliary functions and the new local optimization method.Furthermore,numerical examples are also presented to show that the proposed global optimization method for MIWCPP is efficient.

Optimality Conditions of Approximate Solutions for Nonsmooth Semi-infinite Programming Problems

In this paper,we study optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems.Three new classes of functions,namelyε-pseudoconvex functions of type I and type II andε-quasiconvex functions are introduced,respectively.By utilizing these new concepts,sufficient optimality conditions of approximate solutions for the nonsmooth semi-infinite programming problem are established.Some examples are also presented.The results obtained in this paper improve the corresponding results of Son et al.(J Optim Theory Appl 141:389–409,2009).

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.

A Trust Region Algorithm for Solving Bilevel Programming Problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.

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 PRIMAL-DUAL INTERIOR POINT METHOD FOR PARAMETRIC SEMIDEFINITE PROGRAMMING PROBLEMS

On the basis of primal-dual approach, we present in this paper an interior point method that gives parametric E-approximate solutions to parametric semi-definite programming problems. The method is finite, and the number of its iterations is quasi-polynomially bounded.

Intelligent Tutoring System of Linear Programming

There is a growing technological development in intelligent teaching systems. This field has become interesting to many researchers. In this paper, we present an intelligent tutoring system for teaching mathematics that helps students understand the basics of linear programming using Linear Program Solver and Service for Solving Linear Programming Problems, through which students will be able to solve economic problems. It comes down to determining the minimum or maximum value of a linear function, which is called the objective function, according to pre-set limiting conditions expressed by linear equations and inequalities. The goal function and the limiting conditions represent a mathematical model of the observed problem. Working as a professor of mathematics in high school, I felt the need for one such work and dealing with the study of linear programming as an integral part of mathematics. There are a number of papers in this regard, but exclusively related to traditional ways of working, as stated in the introductory part of the paper. The center of work as well as the final part deals with the study of linear programming using programs that deal with this topic.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

ON ALTERNATIVE OPTIMAL SOLUTIONS TO QUASIMONOTONIC PROGRAMMING WITH LINEAR CONSTRAINTS

In this paper, the nonlinear programming problem with quasimonotonic （ both quasiconvex and quasiconcave ）objective function and linear constraints is considered. With the decomposition theorem of polyhedral sets, the structure of optimal solution set for the programming problem is depicted. Based on a simplified version of the convex simplex method, the uniqueness condition of optimal solution and the computational procedures to determine all optimal solutions are given, if the uniqueness condition is not satisfied. An illustrative example is also presented.

The Optimal Conditions of the Linear Fractional Programming Problem with Constraint

In this article,the authors discuss the optimal conditions of the linear fractionalprogramming problem and prove that a locally optional solution is a globally optional solution and the locally optimal solution can be attained at a basic feasible solution withconstraint condition.

An Optimal DASH Diet Model for People with Hypertension Using Linear Programming Approach

Selecting diets by quantitative techniques is becoming increasingly common. Linear programming is the most popular technique for the selection of least cost mixes of food to meet specific nutritional requirements for a particular group of persons for either general health or disease-related reason. Hypertension is a silent killer and its prevalence rate especially in the developing countries, which has been mostly associated to demographic, environmental and genetic factors, is becoming alarming. The DASH diet has been clinically proven to prevent and control hypertension. In this paper, a model that provides a Daily Optimal (minimum cost) DASH Diet plan for people with hypertension is formulated. The objective is to obtain daily minimum cost diet plans that satisfy the DASH Diets’ nutrients Tolerable Upper and Lower Intake for different daily Calorie Levels. The formulated DASH diet model was further illustrated using real data set with food samples gotten from the DASH eating plan chart. A DASH diet model for a hypertensive person with a 2000-daily-caloric need was formulated and its optimal diet plan for a day obtained with a total cost of 944.41 Naira. Optimal diet plans for other recommended daily calorie levels were also obtained.

Semidefinite programming approach for TDOA/GROA based source localization

Time-differences-of-arrival （TDOA） and gain-ratios-of- arrival （GROA） measurements are used to determine the passive source location. Based on the measurement models, the con- strained weighted least squares （CWLS） estimator is presented. Due to the nonconvex nature of the CWLS problem, it is difficult to obtain its globally optimal solution. However, according to the semidefinite relaxation, the CWLS problem can be relaxed as a convex semidefinite programming problem （SDP）, which can be solved by using modern convex optimization algorithms. Moreover, this relaxation can be proved to be tight, i.e., the SDP solves the relaxed CWLS problem, and this hence guarantees the good per- formance of the proposed method. Furthermore, this method is extended to solve the localization problem with sensor position errors. Simulation results corroborate the theoretical results and the good performance of the proposed method.