VOL. 2- LINEAR AND NON-LINEAR PROGRAMMING
EMANUEL EDUARDO PIRES VAZ
Preface.
The experience of the last years has shown that the prodigious power of the humanity since who makes use of electronic machines, far from suppressing the numerical calculation that burnt the generations of astronomers, geodesists, for the opposite it increases its necessity. This way of calculation is indispensable for all the engineers. The great schools give however an important teaching, following practical works and programming, obliging their students to an enough experience of the numerical calculation. This text, independently of the name that it has, (numerical calculation, applied mathematics…), can extremely be significant; since it demands, at the same time beyond a serious knowledge of numerous parts of the program, an understanding of the relations of “formulas” of the mathematical beings. First under an abstract form, symbolic, and whose sense of true is not completely shown excepting “for putting in practice” a problem with numerical data linked to the inquiry of a numerical result. In our schools, professors oblige the students to make in a fruitful way, many varied exercises during the sessions of practical works. And the students of Prof Emanuel Vaz are honoured for finding in his books the theoretical questions that are displayed to them during the year, perhaps in a dispersed sequence. It is necessary to understand the favours of Prof Emanuel Vaz who has taught with rigour for example, the necessary notations to assign approaching values of a real number, to distinguish errors, corrections and uncertainties. This publication is fighting for the disappearance of inexact expressions many times for façade. For example when we say that the measure of a given length is known with two exacts significant digits instead of this usual but much incorrect way of saying, we should say: “with a relative error that is inferior to 0.01”. The students will also recognize every detail on the examples of many questions that due the lack of time, are impossible to examine in a deep way with other context. But what will we need to say in such a way? Prof Emanuel Vaz is not an unknown. He has already published sixteen books. This text has an aim to prove that the knowledge of the practical work for the engineers, technicians and also for the mathematical students that must choose to run the wave of the future. Firstly it is intended to be useful applying the mathematical speech to areas of practical meaning. Computer science here is used to the resolution of the problems stated using the simplex algorithm. I forever being one of his friends and his disciple, I hope that the reader finds these discussed prospects in the text as interesting as I found them while I made the revision for this book.
Porto, 17th June 2010
The Best Luck.
JOÃO PEDRO BARBOSA VAZ
This book is ready both in English and in Portuguese versions and, attention is being given to French and German versions which are the other languages which these lessons will be presented in. For the edition in English language it is foreseen that the economic profits with this book will revert for an entity of the NYAS, the human rights. The intention of the author is accompanied by his mother, too. The author declines his rights on English edition after publishing it, in favour of this aim.
Porto, 16th June 2013
Emanuel Eduardo Pires Vaz
"LINEAR INTERPOLATION.
1. LINEAR INTERPOLATION FORMULA.
1.1. Let continuous function f be defined by a numerical table (to study the errors we will suppose that there are derivatives f´ and f´´). It is evaluated the numerical value of the function for a value x, ranging between two consecutive entries a and b of the table. The simplest process consists in replacing the function f by the polynomial P, of degree one, such as:"
"5.5. Graphical technique to solve Models of Linear Programming with two variable of Decision.
• This technique consists on representing in a Cartesian plan the set of possible solutions of the problem
• These solutions are really the set of points (x1, x2) which obey the restrictions imposed for the system in study
• The performance of the model is evaluated through the graphical representation of the objective-function, evaluating each possible solution in accordance with its position in the graph
• It is known that the graphical representation of a linear equation is a straight line
• From this, the graphical representation of a linear inequality is one of the sub-plans delimited for the straight line of the equivalent linear equation."
INDEX
1.1. Direct linear interpolation.
Direct linear interpolation formula.
1.2. Error of the method in linear interpolation.
Definition
Evaluation of the error of method.
Sign of error from method.
1.3. Uncertainty error for direct interpolation
1.4. Systematic error in direct linear interpolation.
Definition
Evaluation of the Systematic error.
1.5. Miscalculation.
1.5.1. Definitions.
1.5.1.1. Objective function.
1.5.1.2. Conditions.
1.5.1.3. Conditions of the problem.
1.5.1.4. Non-negative conditions.
1.5.1.5 Possible region.
1.5.1.6. Turning point.
1.5.1.7. Without –boundary region.
1.5.1.8. Basic theorem of the linear programming.
1.5.1.9. Basic theorem of the arithmetic.
1.5.1.10. Basic theorem of algebra.
1.5.1.11. Basic principle of the counting.
1.5.1.12. Basic theorem of the linear programming.
1.5.1.13. Boundary region.
1.6. What the optimization is.
- As the linear programming appeared.
2.1. What linear programming is?
2.2. Limitation of the linear programming.
Constant coefficient
Divisibility
Addictively
2.3. Mathematical formularization.
3. The standard form of the Simplex method.
Non-negativity Restrictions
Objective function
- The method to Solve of Excel.
4.1. Presentation of the example.
4.2. Formularization of the problem of linear programming.
4.3. Graphical solution.
4.4. Analysis of sensibility.
4.4.1. Variation of the coefficients of the objective function.
4.4.2. Analysis of the second members of equations.
4.4.3. Price shade.
5. Attainment of the solution with the resource of excel.
5.1. Resolution of a problem with four variable of decision with the method “To Solve of Excel”.
Some problems of analysis of activities
5.2. Objective function.
5.2.1. Restrictions.
5.2.2. Compact form.
5.2.3. Compact form to the restrictions.
5.2.4. Parameters
5.2.5. Variables.
5.3. Problem of the Diet.
5.4. Problem of Transport.
Graphical Technique to solve models of linear programming with two variables of decision
5.5. Examples of reduction of a model of linear programming for the standard form.
5.5.1. Standard Form.
5.5.2. Occurring the inequalities.
5.5.3. Graphical solution.
Graph for sensibility analyze.
- Least squares.
6.-1. Problem statement
6.2. Solving the least squares problem.
6.3. Least squares and regression analysis.
6.4. Estimating parameter.
6.5. Summarizing the data.
6.5.1. Estimating beta (the slope).
6.5.2. Estimating alpha (The intercept).
6.6. Limitations
6.7. Weighted least squares.
6.7.1. Linear Algebraic Derivation.
6.7.2. Weighted case.
7. Nonlinear programming.
7.1. Introduction.
7.2. The generic problem.
7.3. A Multi-Variate Calculus Review.
7.4. Taylor’s Theorem.
Mean value theorem.
7.5. Implicit function theorem.
7.5.1. Unconstrained problems.
7.5.2. Feasible direction.
7.5.3. First order necessary conditions.
7.5.4. Second order sufficient conditions (unconstrained case).
7.5. Convex programming.
7.5.5.1. Theorem 1.
7.5.5.2. Corollary.
7.6. The Lagrange Multiplier theorem.
7.6.1. Lemma 1.
7.6.2. Lemma 2 (theorem of alternatives).
7.6.3. The Lagrange Multiplier Theorem.
7.7.1. Theorem 2.
7.7.2. Lemma 3.
7.7.4. Lemma4.
7.9. Karush-Kuhn-Tucker theorem.
7.9.1. Sufficient conditions.
8. Problems.
Non linear programming
Prob 4. Gasoline blending.
Prob. 5. Site location.
Prob.6. Production quantities
Prob.7. Flow in Pipes
Prob.8. Robot motion planning
Prob.9. Design parameters for coil spring
9. Issues with non-linear programs.
- Approaches to non-linear programs.
9.2. Polynomial interpolation and approximation.
- Lagrange interpolation formula.
9.3.1. Theorem 1.
9.3.2. Theorem 2.
9.4. Linear rational interpolation in barycentric form
.
9.4.1. Lemma 1.
9.4.2. Theorem 2.
Appendix.
Formula of the two levels.
Resolution of a system of equations with a calculating machine.
Bibliography.
The author of this book would like to know the opinion about its design: Use the following contacts: Emanuel Eduardo Pires Vaz, address: Rua Augusto Lessa 268 --- 4200-097 Porto, Portugal. eepv@yahoo.com