Properties Of Indefinite Matrix Constraints For Linear Programming In Optimal Solution

Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints.This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the available resources.One approach to solve optimization is to use formulation problem in linear form and subjects to linear constraints,the problem can be deliberated as linear programming problem.The linear constraints can be in a form of a matrix.There are limited researches that discuss the effect of the properties of matrix constraint to the solution.In fact,the matrix constraint has significant influence to the existent of the optimal solution to the optimization problem.This research focused on the investigation of characteristics of non-symmetric indefinite square matrices of linear programming problems which represent the constraints of linear programming problems.The non-symmetric indefinite square matrices are generated randomly by the MATLAB simulation software and its indefinite properties are verified through the principal minor test,quadratic form test and eigenvalues test.The solutions of the primal and dual linear programming problem are simulated and discussed.Optimization software,LINGO,is used to validate the solutions to assure the reliability of the simulated solutions in the MATLAB software.Based on the simulation results,some of the non-symmetric indefinite random matrices found duality gap and those matrices could not provide optimal solution to the problem.Whereas,some indefinite matrices with certain characteristics could achieve optimal solution and no duality gap presented.An indefinite random matrix with all positive off-diagonal entries and the determinant of leading principal minors with positive sign at odd orders and negative sign at even orders surely deliver the optimal solution to the linear programming problems.This research may contribute to the advancement of linear programming solution particularly when the constraints form an indefinite matrix

Making Indefinite Kernel Learning Practical

In this paper we embed evolutionary computation into statistical learning theory. First, we outline the connection between large margin optimization and statistical learning and see why this paradigm is successful for many pattern recognition problems. We then embed evolutionary computation into the most prominent representative of this class of learning methods, namely into Support Vector Machines (SVM). In contrast to former applications of evolutionary algorithms to SVM we do not only optimize the method or kernel parameters. We rather use evolution strategies in order to directly solve the posed constrained optimization problem. Transforming the problem into the Wolfe dual reduces the total runtime and allows the usage of kernel functions just as for traditional SVM. We will show that evolutionary SVM are at least as accurate as their quadratic programming counterparts on eight real-world benchmark data sets in terms of generalization performance. They always outperform traditional approaches in terms of the original optimization problem. Additionally, the proposed algorithm is more generic than existing traditional solutions since it will also work for non-positive semidefinite or indefinite kernel functions. The evolutionary SVM variants frequently outperform their quadratic programming competitors in cases where such an indefinite Kernel function is used. --

Computational complexity of μ calculation

The structured singular value μ measures the robustness of uncertain systems. Numerous researchers over the last decade have worked on developing efficient methods for computing μ. This paper considers the complexity of calculating μ with general mixed real/complex uncertainty in the framework of combinatorial complexity theory. In particular, it is proved that the μ recognition problem with either pure real or mixed real/complex uncertainty is NP-hard. This strongly suggests that it is futile to pursue exact methods for calculating μ of general systems with pure real or mixed uncertainty for other than small problems

On implicit-factorization constraint preconditioners

Recently Dollar and Wathen [14] proposed a class of incomplete factorizations for saddle-point problems, based upon earlier work by Schilders [40]. In this paper, we generalize this class of preconditioners, and examine the spectral implications of our approach. Numerical tests indicate the efficacy of our preconditioners

Estimation of Markov Chain via Rank-Constrained Likelihood

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank-constrained non-smooth optimization problem. Convergence results are established. Experiments show that the proposed estimator achieves better empirical performance than other popular approaches.Comment: Accepted at ICML 201