An improved PSO algorithm for solving nonlinear programing problems with constrained conditions

Engineering optimization problems can be always classified into two main categories including the linear programming(LP)and nonlinear programming(NLP)problems.Each programming problem further involves the unconstrained conditions and constrained conditions for design variables of the optimized system.This paper will focus on the issue about the design problem of NLP with the constrained conditions.The employed method for such NLP problems is a variant of particle swarm optimization(PSO),named improved particle swarm optimization(IPSO).The developed IPSO is to modify the velocity updating formula of the algorithm to enhance the search ability for given optimization problems.In this work,many different kinds of physical engineering optimization problems are examined and solved via the proposed IPSO algorithm.Simulation results compared with various optimization methods reported in the literature will show the effectiveness and feasibility for solving NLP problems with the constrained 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.

A DUAL-RELAX PENALTY FUNCTION APPROACH FOR SOLVING NONLINEAR BILEVEL PROGRAMMING WITH LINEAR LOWER LEVEL PROBLEM

The penalty function method, presented many years ago, is an important nu- merical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty func- tion approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.

Further study on a class of augmented Lagrangians of Di Pillo and Grippo in nonlinear programming

In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo （DGALs） was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discussed between the uneonstrained minimizers of DGALs on the product space of problem variables and multipliers, and the solutions of the eonstrained problem and the corresponding values of the Lagrange multipliers. The resulting properties indicate more precisely that this class of DGALs is exact multiplier penalty functions. Therefore, a solution of the equslity-constralned problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of a DGAL on the product space of problem variables and multipliers.

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.

A TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods. The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods. The advantages of the above two methods can be combined to form a more powerful method for constrained optimization. The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound. At the same time, the new algorithm still possesses robust global properties. The global convergence of the new algorithm under standard conditions is established.

An efficient algorithm for multi-dimensional nonlinear knapsack problems

Multi-dimensional nonlinear knapsack problem is a bounded nonlinear integer programming problem that maximizes a separable nondecreasing function subject to multiple separable nondecreasing constraints. This problem is often encountered in resource allocation, industrial planning and computer network. In this paper, a new convergent Lagrangian dual method was proposed for solving this problem. Cutting plane method was used to solve the dual problem and to compute the Lagrangian bounds of the primal problem. In order to eliminate the duality gap and thus to guarantee the convergence of the algorithm, domain cut technique was employed to remove certain integer boxes and partition the revised domain to a union of integer boxes. Extensive computational results show that the proposed method is efficient for solving large-scale multi-dimensional nonlinear knapsack problems. Our numerical results also indicate that the cutting plane method significantly outperforms the subgradient method as a dual search procedure.

Penalized interior point approach for constrained nonlinear programming

A penalized interior point approach for constrained nonlinear programming is examined in this work. To overcome the difficulty of initialization for the interior point method, a problem equivalent to the primal problem via incorporating an auxiliary variable is constructed. A combined approach of logarithm barrier and quadratic penalty function is proposed to solve the problem. Based on Newton＇s method, the global convergence of interior point and line search algorithm is proven. Only a finite number of iterations is required to reach an approximate optimal solution. Numerical tests are given to show the effectiveness of the method.

Global Convergence of a New Restarting Conjugate Gradient Method for Nonlinear Optimizations

Conjugate gradient optimization algorithms depend on the search directions with different choices for the parameters in the search directions. In this note, by combining the nice numerical performance of PR and HS methods with the global convergence property of the class of conjugate gradient methods presented by HU and STOREY(1991), a class of new restarting conjugate gradient methods is presented. Global convergences of the new method with two kinds of common line searches, are proved. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continously dif ferentiable function with Curry-Altman's step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with other kind of step size rule are established. Numerical experiments show that the new method is efficient by comparing with FR conjugate gradient method.

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.

EXACT AUGMENTED LAGRANGIAN FUNCTION FOR NONLINEAR PROGRAMMING PROBLEMS WITH INEQUALITY CONSTRAINTS

An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.

Optimal Budget Spending for Software Testing under the Condition of Nonlinear Constraint

Software testing is a very important phase of the software development process. It is a very difficult job for a software manager to allocate optimally the financial budget to a software project during testing. In this paper the problem of optimal allocation of the software testing cost is studied. There exist several models focused on the development of software costs measuring the number of software errors remaining in the software during testing. The purpose of this paper is to use these models to formulate the optimization problems of resource allocation: Minimization of the total number of software errors remaining in the system. On the assumption that a software project consists of some independent modules, the presented approach extends previous work by defining new goal functions and extending the primary assumption and precondition.

A UNIVERSAL APPROACH FOR CONTINUOUS OR DISCRETE NONLINEAR PROGRAMMINGS WITH MULTIPLE VARIABLES AND CONSTRAINTS

A universal numerical approach for nonlinear mathematic programming problems is presented with an application of ratios of first-order differentials/differences of objective functions to constraint functions with respect to design variables. This approach can be efficiently used to solve continuous and, in particular, discrete programmings with arbitrary design variables and constraints. As a search method, this approach requires only computations of the functions and their partial derivatives or differences with respect to design variables, rather than any solution of mathematic equations. The present approach has been applied on many numerical examples as well as on some classical operational problems such as one-dimensional and two-dimensional knap-sack problems, one-dimensional and two-dimensional resource-distribution problems, problems of working reliability of composite systems and loading problems of machine, and more efficient and reliable solutions are obtained than traditional methods. The present approach can be used without limitation of modeling scales of the problem. Optimum solutions can be guaranteed as long as the objective function, constraint functions and their First-order derivatives/differences exist in the feasible domain or feasible set. There are no failures of convergence and instability when this approach is adopted.

Nonlinear Programming Algorithm and Its Convergence Rate Analysis

In this paper,we improve the algorithm proposed by T.F.Colemen and A.R.Conn in paper [1]. It is shown that the improved algorithm is possessed of global convergence and under some conditions it can obtain locally supperlinear convergence which is not possessed by the original algorithm.

A Primal-dual Interior Point Method for Nonlinear Programming

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.

SIMPLE NONLINEAR OPTIMIZATION-BASED SELECTION OF INSULATION MATERIAL AND WINDOW TYPE IN TURKEY:EFFECT OF HEATING AND COOLING BASE TEMPERATURES

The energy-savings of four hypothetical households in different climatic regions of Turkey were calculated via a nonlinear mixed integer optimization model.The ideal insulation material,its optimum thickness,and the ideal window type were determined.The standard degree days method was used with five different base temperatures for heating and five different base temperatures for cooling.The climatic conditions of the region,the properties of the insulation options,the unit price of fuel and electricity and the base temperature are used as model inputs,whereas the combination of selected insulation material with its optimum thickness and window type are given as model outputs.Stone Wool was found to be the ideal wall insulation material in all scenarios.The optimum window type was found to depend on the heating or cooling requirements of the house,as well as the lifetime of insulation.The region where the energy saving actions are deemed most feasible has been identified as Erzurum(Region 4),followed by Antalya(Region 1).Finally,the effect of changing the base temperature on energy savings was investigated and the results showed that an approximate average increase of$15/℃ in annual savings is possible.Our model can be used by any prospective home-owner who would like to maximize their energy savings.

Nonlinear Programming Based Preamble Design for OFDM Systems

A new preamble structure and design method for orthogonal frequency division multiplexing（OFDM）systems is described,which results a two-symbol long training preamble.The preamble contains four parts,the first part is the same as the third,and the four parts are calculated by using nonlinear programming（NLP）model such that the moving correlation of the preamble results a steep rectangular-like pulse of certain width,whose step-down indicates the timing offset.Simulation results in AWGN channel are given to evaluate the perf o rmance of the proposed preamble design.

Necessary and Sufficient Conditions in Smooth Programming with Generalized Convexity

Hanson and Mond have grven sets of necessary and sufficient conditions for optimality in constrained optimization by introducing classes of generalized functions, called type Ⅰ functions. Recently, Bector definded univex functions, a new class of functions that unifies several concepts of generalized convexity. In this paper, additional conditions are attached to the Kuhn Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications.

Analysis of Mine Ventilation Network Using Genetic Algorithm

Aim To determine the global optimal solution for a mine ventilation network under given network topology and airway characteristics. Methods\ The genetic algorithm was used to find the global optimal solution of the network. Results\ A modified genetic algorithm is presented with its characteristics and principle. Instead of working on the conventional bit by bit operation, both the crossover and mutation operators are handled in real values by the proposed algorithms. To prevent the system from turning into a premature problem, the elitists from two groups of possible solutions are selected to reproduce the new populations. Conclusion\ The simulation results show that the method outperforms the conventional nonlinear programming approach whether from the viewpoint of the number of iterations required to find the optimum solutions or from the final solutions obtained.

Comprehensive decision-making method for optimal location of logistics hubs

A mathematical optimal model is developed to address the problem of logistics hubs location.The method integrates nonlinear programming models and fuzzy comprehensive evaluation,and handles both quantitative and qualitative factors that influence the location of logistics hubs.A mathematical programming model is used to find the optimal candidate sites of the logistics hub based on profit-maximizing criteria.Then a fuzzy comprehensive evaluation method is adopted to determine the final optimal location of those candidate sites by taking account of criteria including stability of policy environment,convenience of transportation(e.g.,easy access to highways/expressways,etc.),rationality of economy,competition and sustainability.Finally,the methodology is demonstrated by a case study of the Transfar Logistics Base in Zhejiang Province.The results verify the method application for logistics hub locations and show how to locate a logistics hub in practice from the case of Transfar Logistics Base.