General Exact Penalty Functions in Integer Programming

In this paper, the general exact penalty functions in integer programming were studied. The conditions which ensure the exact penalty property for the general penalty function with one penalty parameter were given and a general penalty function with two parameters was proposed.

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 New Kind of Simple Smooth Exact Penalty Function of Constrained Nonlinear Programming

The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offers a new kind of simple smooth approximative exact penalty function of general constrained nonlinear programmings and analyzes its properties.

A SIMPLE SMOOTH EXACT PENALTY FUNCTION FOR SMOOTH OPTIMIZATION PROBLEM

For smooth optimization problem with equMity constraints, new continuously differentiable penalty function is derived. It is proved exact in the sense that local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function under some nondegeneracy assumption. It is simple in the sense that the penalty function only includes the objective function and constrained functions, and it doesn＇t include their gradients. This is achieved by augmenting the dimension of the program by a variable that controls the weight of the penalty terms.

Exact Penalty Function and Asymptotic Strong Nonlinear Duality in Integer Programming

In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].

Exactness of penalization for exact minimax penalty function method in nonconvex programming

The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η （with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η）. Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.

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.

An Exact Logarithmic-exponential Multiplier Penalty Function

In this paper, we give a solving approach based on a logarithmic-exponential multiplier penalty function for the constrained minimization problem. It is proved exact in the sense that the global optimizers of a nonlinear problem are precisely the global optimizers of the logarithmic-exponential multiplier penalty problem.

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.

An Efficient Approach to a Class of Non-smooth Optimization Problems

This paper presents an entropy-based smoothing technique for solving a class of non-smooth optimization problems that are in some way related to the maximum function.Basic ideas concerning this approach are that we replace the non-smooth maximum function by a smooth one,called aggregate function,which is derived by employing the maximum entropy principle and its useful properties are proved.Wilh this smoothing technique,both unconstrained and constrained mimma.x problems are transformed into unconstrained optimization problems of smooth functions such that this class of non-smooth optimization problems can be solved by some existing unconstrained optimization softwares for smooth functions The present approach can be very easily implemented on computers with very fast and-.Inhie convergence.

AN ADAPTIVE TRUST REGION METHOD FOR EQUALITY CONSTRAINED OPTIMIZATION

In this paper, a trust region method for equality constrained optimizationbased on nondifferentiable exact penalty is proposed. In this algorithm, the trail step ischaracterized by computation of its normal component being separated from computation of itstangential component, i.e., only the tangential component of the trail step is constrained by trustradius while the normal component and trail step itself have no constraints. The other maincharacteristic of the algorithm is the decision of trust region radius. Here, the decision of trustregion radius uses the information of the gradient of objective function and reduced Hessian.However, Maratos effect will occur when we use the nondifferentiable exact penalty function as themerit function. In order to obtain the superlinear convergence of the algorithm, we use the twiceorder correction technique. Because of the speciality of the adaptive trust region method, we usetwice order correction when p = 0 (the definition is as in Section 2) and this is different from thetraditional trust region methods for equality constrained optimization. So the computation of thealgorithm in this paper is reduced. What is more, we can prove that the algorithm is globally andsuperlinearly convergent.

An SQP Algorithm with Cautious Updating Criteria for Nonlinear Degenerate Problems

An efficient SQP algorithm for solving nonlinear degenerate problems is proposed in the paper. At each iteration of the algorithm, a quadratic programming subproblem, which is always feasible by introducing a slack variable, is solved to obtain a search direction. The steplength along this direction is computed by employing the 1∞ exact penalty function through Armijo-type line search scheme. The algorithm is proved to be convergent globally under mild conditions.

A SQP Method for Inequality Constrained Optimization

In this paper, a new SQP method for inequality constrained optimization is proposed and the global convergence is obtained under very mild conditions.