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In this paper, an algorithm of barrier objective penalty function for inequality constrained optimization is studied and a conception–the stability of barrier objective penalty function is presented. It is proved that an approximate optimal solution may be obtained by solving a barrier objective penalty function for inequality constrained optimization problem when the barrier objective penalty function is stable. Under some conditions, the stability of barrier objective penalty function is proved for convex programming. Specially, the logarithmic barrier function of convex programming is stable. Based on the barrier objective penalty function, an algorithm is developed for finding an approximate optimal solution to an inequality constrained optimization problem and its convergence is also proved under some conditions. Finally, numerical experiments show that the barrier objective penalty function algorithm has better convergence than the classical barrier function algorithm. [ABSTRACT FROM PUBLISHER]

Penalty function is an important tool in solving many constrained optimization problems in areas such as industrial design and management. In this paper, we study exactness and algorithm of an objective penalty function for inequality constrained optimization. In terms of exactness, this objective penalty function is at least as good as traditional exact penalty functions. Especially, in the case of a global solution, the exactness of the proposed objective penalty function shows a significant advantage. The sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved. Based on the objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the objective penalty function is proved for a local solution. An algorithm is presented in the paper in finding a local solution, with its convergence proved under some conditions. Finally, numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm. [ABSTRACT FROM AUTHOR]

Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming, with its convergence proved under some conditions. [ABSTRACT FROM AUTHOR]

In this article, a smoothing objective penalty function for inequality constrained optimization problems is presented. The article proves that this type of the smoothing objective penalty functions has good properties in helping to solve inequality constrained optimization problems. Moreover, based on the penalty function, an algorithm is presented to solve the inequality constrained optimization problems, with its convergence under some conditions proved. Two numerical experiments show that a satisfactory approximate optimal solution can be obtained by the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: The min-bisection problem is an NP-hard combinatorial optimization problem. In this paper an equivalent linearly constrained continuous optimization problem is formulated and an algorithm is proposed for approximating its solution. The algorithm is derived from the introduction of a logarithmic-cosine barrier function, where the barrier parameter behaves as temperature in an annealing procedure and decreases from a sufficiently large positive number to zero. The algorithm searches for a better solution in a feasible descent direction, which has a desired property that lower and upper bounds are always satisfied automatically if the step length is a number between zero and one. We prove that the algorithm converges to at least a local minimum point of the problem if a local minimum point of the barrier problem is generated for a sequence of descending values of the barrier parameter with a limit of zero. Numerical results show that the algorithm is much more efficient than two of the best existing heuristic methods for the min-bisection problem, Kernighan–Lin method with multiple starting points (MSKL) and multilevel graph partitioning scheme (MLGP). [Copyright &y& Elsevier]

Abstract: In this paper, the dynamic multi-objective optimization problem (DMOP) is first approximated by a series of static multi-objective optimization problems (SMOPs) by dividing the time period into several equal subperiods. In each subperiod, the dynamic multi-objective optimization problem is seen as a static multi-objective optimization problem by taking the time parameter fixed. Then, to decrease the amount of computation and efficiently solve the static problems, each static multi-objective optimization problem is transformed into a two-objective optimization problem based on two re-defined objectives. Finally, a new crossover operator and mutation operator adapting to the environment changing are designed. Based on these techniques, a new evolutionary algorithm is proposed. The simulation results indicate that the proposed algorithm can effectively track the varying Pareto fronts with time. [Copyright &y& Elsevier]

A Lagrange multiplier and Hopfield-type barrier function method is proposed for approximating a solution of the traveling salesman problem. The method is derived from applications of Lagrange multipliers and a Hopfield-type barrier function and attempts to produce a solution of high quality by generating a minimum point of a barrier problem for a sequence of descending values of the barrier parameter. For any given value of the barrier parameter, the method searches for a minimum point of the barrier problem in a feasible descent direction, which has a desired property that lower and upper bounds on variables are always satisfied automatically if the step length is a number between zero and one. At each iteration, the feasible descent direction is found by updating Lagrange multipliers with a globally convergent iterative procedure. For any given value of the barrier parameter, the method converges to a stationary point of the barrier problem without any condition on the objective function. Theoretical and numerical results show that the method seems more effective and efficient than the softassign algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new fuzzy clustering algorithm for categorical data. In the algorithm, the objective function of the fuzzy k-modes algorithm is modified by adding the between-cluster information so that we can simultaneously minimize the within-cluster dispersion and enhance the between-cluster separation. For obtaining the local optimal solutions of the modified objective function, the corresponding update formulas of the membership matrix and the cluster prototypes are strictly derived. The convergence of the proposed algorithm under the optimization framework is proved. On several real data sets from UCI, the performance of the proposed algorithm is studied. The experimental results illustrate that the algorithm is effective and suitable for categorical data sets. [Copyright &y& Elsevier]

Abstract: Due to data sparseness and attribute redundancy in high-dimensional data, clusters of objects often exist in subspaces rather than in the entire space. To effectively address this issue, this paper presents a new optimization algorithm for clustering high-dimensional categorical data, which is an extension of the k-modes clustering algorithm. In the proposed algorithm, a novel weighting technique for categorical data is developed to calculate two weights for each attribute (or dimension) in each cluster and use the weight values to identify the subsets of important attributes that categorize different clusters. The convergence of the algorithm under an optimization framework is proved. The performance and scalability of the algorithm is evaluated experimentally on both synthetic and real data sets. The experimental studies show that the proposed algorithm is effective in clustering categorical data sets and also scalable to large data sets owning to its linear time complexity with respect to the number of data objects, attributes or clusters. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new algorithm using the nonmonotone second-order Wolfe’s line search. By using the negative curvature information from the Hessian, we prove that the generated sequence converges to the stationary points that satisfy the second-order optimality conditions. We also report numerical results which show the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel objective penalty function for inequality constrained optimization problems. The objective penalty function differs from any existing penalty function and also has two desired features: exactness and smoothness if the constraints and objective function are differentiable. An exact penalty result is proved for the objective penalty function. In addition to these results, based on the objective penalty function, we develop an algorithm for solving the original problem and show its convergence under some mild conditions. [Copyright &y& Elsevier]