A Class of Algorithms for Solving LP Problems by Prioritizing the Constraints

Linear programming is a method for solving linear optimization problems with constraints, widely met in real-world applications. In the vast majority of these applications, the number of constraints is significantly larger than the number of variables. Since the crucial subject of these problems is to detect the constraints that will be verified as equality in an optimal solution, there are methods for investigating such constraints to accelerate the whole process. In this paper, a technique named proximity technique is addressed, which under a proposed theoretical framework gives an ascending order to the constraints in such a way that those with low ranking are characterized of high priority to be binding. Under this framework, two new Linear programming optimization algorithms are introduced, based on a proposed Utility matrix and a utility vector accordingly. For testing the addressed algorithms firstly a generator of 10,000 random linear programming problems of dimension n with m constraints, where , is introduced in order to simulate as many as possible real-world problems, and secondly, real-life linear programming examples from the NETLIB repository are tested. A discussion of the numerical results is given. Furthermore, already known methods for solving linear programming problems are suggested to be fitted under the proposed framework.

Stationary flow fields prediction of variable physical domain based on proper orthogonal decomposition and kriging surrogate model

In this paper a new flow field prediction method which is independent of the governing equations, is developed to predict stationary flow fields of variable physical domain. Predicted flow fields come from linear superposition of selected basis modes generated by proper orthogonal decomposition（POD）. Instead of traditional projection methods, kriging surrogate model is used to calculate the superposition coefficients through building approximate function relationships between profile geometry parameters of physical domain and these coefficients. In this context,the problem which troubles the traditional POD-projection method due to viscosity and compressibility has been avoided in the whole process. Moreover, there are no constraints for the inner product form, so two forms of simple ones are applied to improving computational efficiency and cope with variable physical domain problem. An iterative algorithm is developed to determine how many basis modes ranking front should be used in the prediction. Testing results prove the feasibility of this new method for subsonic flow field, but also prove that it is not proper for transonic flow field because of the poor predicted shock waves.

Optimality Conditions for Rank-Constrained Matrix Optimization

In this paper,we comprehensively study optimality conditions for rank-constrained matrix optimization(RCMO).By calculating the Clarke tangent and normal cones to a rank-constrained set,along with the given Fréchet,Mordukhovich normal cones,we investigate four kinds of stationary points of the RCMO and analyze the relations between each stationary point and local/global minimizer of the RCMO.Furthermore,the second-order optimality condition of the RCMO is achieved with the help of the Clarke tangent cone.