Programming of Finite Element Methods in MATLAB

We discuss how to implement the linear finite element method for solving the Poisson equation. We begin with the data structure to represent the triangulation and boundary conditions, introduce the sparse matrix, and then discuss the assembling process. We pay special attention to an efficient programming style using sparse matrices in MATLAB

Algorithms and literate programs for weighted low-rank approximation with missing data

Linear models identification from data with missing values is posed as a weighted low-rank approximation problem with weights related to the missing values equal to zero. Alternating projections and variable projections methods for solving the resulting problem are outlined and implemented in a literate programming style, using Matlab/Octave's scripting language. The methods are evaluated on synthetic data and real data from the MovieLens data sets

A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

A new mathematical model for tiling finite regions of the plane with polyominoes

We present a new mathematical model for tiling finite subsets of $\mathbb{Z}^2$ using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear programming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB

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

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximization for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimization problems with the help of convex semidefinite programming (optimization over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach

Решение математических задач на языке Visual C# с использованием пакета MATLAB

В работе рассмотрены особенности и этапы интеграции программного обеспечения, написанного на языке программирования Visual C#, с математическим пакетом MATLAB на примере решения задачи линейного программирования. Приведены листинги программ в MATLAB и Visual C#. Сделаны выводы о преимуществах и недостатках изложенного решения.The article deals with the features and integration stages application in Visual C# and the mathematical package MATLAB. Solution overview organized on the example solving a linear programming problem. Listings of programs in MATLAB and Visual C# given. Conclusions about the advantages and disadvantages of the above solutions done

The estimation of electric power losses in electrical networks by fuzzy regression model using genetic algorithm

This paper presents the comparative study for fuzzy regression model using linear programming, fuzzy regression model using genetic algorithms and standard regression model. The fuzzy and standard models were developed for estimation of electric power losses in electrical networks. Simulation was carried out with a tool developed in MATLAB