Optimization of Design. Lecturer:Dung-An Wang Lecture 12
|
|
- Kristopher Hart
- 7 years ago
- Views:
Transcription
1 Optimization of Design Lecturer:Dung-An Wang Lecture 12
2 Lecture outline Reading: Ch12 of text Today s lecture 2
3 Constrained nonlinear programming problem Find x=(x1,..., xn), a design variable vector of dimension n, to minimize f=f(x) subject to 3
4 The method is based on linearization of the problem about the current estimate of the optimum design. Therefore, linearization of the problem is quite important and is discussed in detail. Once the problem has been linearized, it is natural to ask if it can be solved using linear programming methods. The answer is yes, and we first describe a method that is a simple extension of the Simplex method of linear programming. Then we describe the constrained steepest-descent method. 4
5 Definitions of the status of a constraint at a design point Active constraint Inactive constraint Violated constraint ε-active inequality constraint 5
6 Constraint Normalization Usually one value for ε (say 0.01) is used in checking the status of all of the constraints to check for the ε- active constraint condition. Since different constraints involve different orders of magnitude, it is not proper to use the same ε for all of the constraints unless they are normalized 6
7 In some cases, it may be better to use the constraints in their original form, especially the equality constraints. Thus, in numerical calculations some experimentation with normalization of constraints may be needed for some constraint forms 7
8 The Descent Function A function used to monitor progress toward the minimum is called the descent, or merit, function The descent function also has the property that its minimum value is the same as that of the original cost function 8
9 Convergence of an Algorithm An algorithm is said to be convergent if it reaches a local minimum point starting from an arbitrary point. The feasible set is closed if all of the boundary points are included in the set; that is, there are no strict inequalities in the problem formulation 9
10 12.2 LINEARIZATION OF THE CONSTRAINED PROBLEM At each iteration, most numerical methods for constrained optimization compute design change by solving a subproblem that is obtained by writing linear Taylor s expansions for the cost and constraint functions. Writing Taylor s expansion of the cost and constraint functions about the point x (k), we obtain the linearized subproblem 10
11 NOTATION FOR THE LINEARIZED SUBPROBLEM 11
12 DEFINITION OF THE LINEARIZED SUBPROBLEM Drop f k in the linearized cost function, the approximate subproblem, Note that f k is a constant, which does not affect solution of the linearized subproblem 12
13 EXAMPLE 12.2 DEFINITION OF A LINEARIZED SUBPROBLEM Linearize the cost and constraint functions about the point x (0) =(1,1) and write the approximate problem 13
14 Function gradients 14
15 Linearized subproblem Using the Taylor s expansion, the linearized cost function at the point (1,1) linearizing the constraint functions 15
16 16
17 Linearization in Terms of Original Variables the linearized subproblem is in terms of the design changes d 1 and d 2. We may also write the subproblem in terms of the original variables x 1 and x 2. To do this we substitute d=(x-x(0)) in all of the foregoing expressions 17
18 Since the linearized cost function is parallel to the linearized first constraint g1, the optimum solution for the linearized subproblem is any point on the line DE 18
19 EXAMPLE 12.3 Linearize the rectangular beam design problem formulated that is in Section 3.8 at the point (50,200) mm 19
20 Evaluation of problem functions At the given point 20
21 Evaluation of gradients In the following calculations, we will ignore constraints g4 and g5, assuming that they will remain satisfied; that is, the design will remain in the first quadrant gradients of the functions 21
22 Linearized subproblem 22
23 The linearized constraint functions the linearized cost function is parallel to constraint. The optimum solution lies at the point H, which is at the intersection of constraints For this point, the original constraints g1 and g2 are still violated. Apparently, for nonlinear constraints, iterations are needed to correct constraint violations and reach the feasible set 23
24 12.3 THE SEQUENTIAL LINEAR PROGRAMMING ALGORITHM Linearized functions in the variables d i can be solved by linear programming methods. Such procedures where linear programming is used to compute design change are referred to as sequential linear programming, or SLP for short The sequential linear programming algorithm is a simple and straightforward approach to solving constrained optimization problems 24
25 Move Limits in SLP To solve the LP by the standard Simplex method, the right-side parameters ei and bj must be nonnegative the problem may not have a bounded solution, or the changes in design may become too large, thus invalidating the linear approximations. Therefore, limits must be imposed on changes in design. Such constraints are usually called move limits, 25
26 Selection of Proper Move Limits Selecting proper move limits is of critical importance because it can mean success or failure of the SLP algorithm. the user should try different move limits if one specification leads to failure or improper design. Usually Δil (k) and Δiu (k) are selected as some fraction of the current design variable values (this may vary from 1 to 100 percent). If the resulting LP problem turns out to be infeasible, the move limits will need to be relaxed (i.e., larger changes in the design must be allowed 26
27 the current values of the design variables can increase or decrease. To allow for such a change, we must treat the LP variables di as free in sign. This can be done as 27
28 stopping criteria 28
29 Sequential linear programming algorithm 29
30 EXAMPLE 12.4 Consider the problem given in Example Define the linearized subproblem at the point (3, 3) and discuss its solution after imposing the proper move limits 30
31 functions and their gradients are calculated at the given point (3, 3) The given point is in the infeasible region, as the first constraint is violated 31
32 Linearized subproblem 32
33 The subproblem has only two variables, so it can be solved using the graphical solution We may choose d1=-1 and d2=-1 as the solution that satisfies all of the linearized constraints (note that the linearized change in cost is 6). If 100 percent move limits are selected then the solution to the LP subproblem must lie in the region ADEF. If the move limits are set as 20 percent of the current value of design variables, the solution must satisfy 33
34 EXAMPLE 12.5 Consider the problem given in Example Perform one iteration of the SLP algorithm, choose move limits such that a 15 percent design change is permissible 34
35 Solution The given point represents a feasible solution 35
36 linearized subproblem The linearized subproblem with 15 percent move limits on design changes d1 and d2 at the point x(0) is obtained in Example
37 solve the problem using the graphical solution Move limits of 15 percent define the solution region as DEFG. The optimum solution for the problem is at point F where d1=0.15 and d2=
38 solve the problem using the Simplex method in the linearized subproblem, the design changes d1 and d2 are free in sign. If we wish to solve the problem by the Simplex method, we must define new variables, A, B, C, and D such that 38
39 39
40 The solution to the foregoing LP problem with the Simplex method is obtained as: A=0.15,B=0, C=0.15, and D=0. Therefore, is larger than the permissible tolerance (0.001), we need to go through more iterations to satisfy the stopping criterion 40
41 The SLP Algorithm: Some Observations The method should not be used as a black box approach for engineering design problems. The selection of move limits is one of trial and error and can be best achieved in an interactive mode. The method may not converge to the precise minimum since no descent function is defined, and line search is not performed along the search direction to compute a step size. The method can cycle between two points if the optimum solution is not a vertex of the feasible set. The method is quite simple conceptually as well as numerically. Although it may not be possible to reach the precise local minimum point with it, it may be used to obtain improved designs in practice. 41
42 12.4 SEQUENTIAL QUADRATIC PROGRAMMING the SLP has some limitations, the major one being its lack of robustness. To overcome SLP s drawbacks, several other derivativebased methods have been developed to solve smooth nonlinear programming problems. gradient projection method the feasible directions method the generalized reduced gradient method Sequential quadratic programming (SQP) methods are relatively new and have become quite popular as a result of their generality, robustness, and efficiency. 42
43 SQP methods Step 1. A search direction in the design space is calculated by utilizing the values and the gradients of the problem functions; a quadratic programming subproblem is defined and solved. Step 2. A step size along the search direction is calculated to minimize a descent function; a step size calculation subproblem is defined and solved 43
44 12.5 SEARCH DIRECTION CALCULATION 1. The QP subproblem is strictly convex and therefore its minimum (if one exists) is global and unique. 2. The cost function represents a equation of a hypersphere with its center at -c (circle in two dimensions, sphere in three dimensions). 44
45 EXAMPLE 12.6 DEFINITION OF A QP SUBPROBLEM Linearize the cost and constraint functions about a point (1, 1) and define the QP subproblem 45
46 Solution constraints for the problem are already written in the normalized form graph 46
47 Evaluation of functions cost and the constraint functions are evaluated at the point (1, 1) 47
48 Gradient evaluation gradients of the cost and constraint functions 48
49 Linearized subproblem Substituting Eqs. (f) and (i) into Eq. (12.9), linearized cost function, linearized forms of the constraint functions can be written and the linearized subproblem the constant 5 has been dropped from the linearized cost function because it does not affect the solution to the subproblem. Also, the constants 3 and in the linearized constraints have been transferred to the right side 49
50 QP subproblem 50
51 Compart LP with QP subproblems Solution to LP subproblem Solution to QP subproblem 51
52 Solving the QP Subproblem EXAMPLE 12.7 SOLUTION TO THE QP SUBPROBLEM 52
53 Solution linearized cost function is modified to a quadratic function 53
54 Graphical solution 54
55 Analytical solution A numerical method must generally be used to solve the subproblem. However, since the present problem is quite simple, it can be solved by writing the KKT necessary conditions of Theorem
56 12.6 THE STEP SIZE CALCULATION SUBPROBLEM The Descent Function Recall that in unconstrained optimization methods the cost function is used as the descent function to monitor the progress of the algorithms toward the optimum point. Although the cost function can be used as a descent function with some constrained optimization methods, it cannot be used for general SQP-type methods. For most methods, the descent function is constructed by adding a penalty for constraint violations to the current value of the cost function. to determine the step size, we minimize the descent function. 56
57 EXAMPLE 12.8 CALCULATION OF DESCENT FUNCTION Taking the penalty parameter R as 10,000, calculate the value of the descent function at the point 57
58 Solution The cost and constraint functions at the given point x(0)=(40, 0.5) maximum constraint violation is determined 58
59 the descent function is calculated 59
60 Step Size Calculation: Line Search Once the search direction d(k)is determined at the current point x(k) the descent function becomes Thus the step size calculation subproblem becomes finding α to Minimize 60
61 EXAMPLE 12.9 CALCULATION OF THE DESCENT FUNCTION, follow Example 12.8 Descent function value at the initial point, α=0 The necessary condition of Eq. (12.33) is satisfied if we select the penalty parameter R as the descent function value at the starting point 61
62 Descent function value at the first trial point the maximum constraint violation 62
63 the descent function at the trial step size of α0=0.1 is given (note that the value of the penalty parameter R is not changed during step size calculation): need to continue the process of initial bracketing of the optimum step size. 63
64 Descent function value at the second trial point In the golden section search procedure, the next trial step size has an increment of (1.618xthe previous increment) next trial design point At the point (46.70, 0.618) 64
65 the minimum for the descent function has not been surpassed yet. Therefore we need to continue the initial bracketing process. The next trial step size with an increment of (1.618xthe previous increment) Value of the penalty parameter R is calculated at the beginning of the line search along the search direction and then kept fixed during all subsequent calculations for step size determination. 65
66 12.7 THE CONSTRAINED STEEPEST-DESCENT METHOD constrained steepest-descent (CSD) method has been proved to be convergent to a local minimum point starting from any point. This is considered a model algorithm that illustrates how most optimization algorithms work. Since the search direction is a modification of the steepest-descent direction to satisfy constraints, it is called the constrained steepest-descent direction. It is actually a direction obtained by projecting the steepestdescent direction on to the constraint hyperplane 66
67 The CSD Algorithm 67
Nonlinear Programming Methods.S2 Quadratic Programming
Nonlinear Programming Methods.S2 Quadratic Programming Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard A linearly constrained optimization problem with a quadratic objective
More informationSupport Vector Machine (SVM)
Support Vector Machine (SVM) CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationLinear Programming Notes V Problem Transformations
Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material
More informationNumerisches Rechnen. (für Informatiker) M. Grepl J. Berger & J.T. Frings. Institut für Geometrie und Praktische Mathematik RWTH Aachen
(für Informatiker) M. Grepl J. Berger & J.T. Frings Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2010/11 Problem Statement Unconstrained Optimality Conditions Constrained
More informationLecture 3. Linear Programming. 3B1B Optimization Michaelmas 2015 A. Zisserman. Extreme solutions. Simplex method. Interior point method
Lecture 3 3B1B Optimization Michaelmas 2015 A. Zisserman Linear Programming Extreme solutions Simplex method Interior point method Integer programming and relaxation The Optimization Tree Linear Programming
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationNonlinear Optimization: Algorithms 3: Interior-point methods
Nonlinear Optimization: Algorithms 3: Interior-point methods INSEAD, Spring 2006 Jean-Philippe Vert Ecole des Mines de Paris Jean-Philippe.Vert@mines.org Nonlinear optimization c 2006 Jean-Philippe Vert,
More informationLecture 2: August 29. Linear Programming (part I)
10-725: Convex Optimization Fall 2013 Lecture 2: August 29 Lecturer: Barnabás Póczos Scribes: Samrachana Adhikari, Mattia Ciollaro, Fabrizio Lecci Note: LaTeX template courtesy of UC Berkeley EECS dept.
More informationMathematical finance and linear programming (optimization)
Mathematical finance and linear programming (optimization) Geir Dahl September 15, 2009 1 Introduction The purpose of this short note is to explain how linear programming (LP) (=linear optimization) may
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationBig Data - Lecture 1 Optimization reminders
Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Big Data - Lecture 1 Optimization reminders S. Gadat Toulouse, Octobre 2014 Schedule Introduction Major issues Examples Mathematics
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationThe Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method
The Steepest Descent Algorithm for Unconstrained Optimization and a Bisection Line-search Method Robert M. Freund February, 004 004 Massachusetts Institute of Technology. 1 1 The Algorithm The problem
More informationCONSTRAINED NONLINEAR PROGRAMMING
149 CONSTRAINED NONLINEAR PROGRAMMING We now turn to methods for general constrained nonlinear programming. These may be broadly classified into two categories: 1. TRANSFORMATION METHODS: In this approach
More informationRoots of Equations (Chapters 5 and 6)
Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have
More information1. Graphing Linear Inequalities
Notation. CHAPTER 4 Linear Programming 1. Graphing Linear Inequalities x apple y means x is less than or equal to y. x y means x is greater than or equal to y. x < y means x is less than y. x > y means
More informationChapter 2 Solving Linear Programs
Chapter 2 Solving Linear Programs Companion slides of Applied Mathematical Programming by Bradley, Hax, and Magnanti (Addison-Wesley, 1977) prepared by José Fernando Oliveira Maria Antónia Carravilla A
More informationLinear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.
1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationLinear Programming I
Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one
More informationConvex Programming Tools for Disjunctive Programs
Convex Programming Tools for Disjunctive Programs João Soares, Departamento de Matemática, Universidade de Coimbra, Portugal Abstract A Disjunctive Program (DP) is a mathematical program whose feasible
More information(Quasi-)Newton methods
(Quasi-)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable non-linear function g, x such that g(x) = 0, where g : R n R n. Given a starting
More information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of
More informationEdExcel Decision Mathematics 1
EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation
More informationChapter 5. Linear Inequalities and Linear Programming. Linear Programming in Two Dimensions: A Geometric Approach
Chapter 5 Linear Programming in Two Dimensions: A Geometric Approach Linear Inequalities and Linear Programming Section 3 Linear Programming gin Two Dimensions: A Geometric Approach In this section, we
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationSolutions Of Some Non-Linear Programming Problems BIJAN KUMAR PATEL. Master of Science in Mathematics. Prof. ANIL KUMAR
Solutions Of Some Non-Linear Programming Problems A PROJECT REPORT submitted by BIJAN KUMAR PATEL for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision
More informationChapter 6. Cuboids. and. vol(conv(p ))
Chapter 6 Cuboids We have already seen that we can efficiently find the bounding box Q(P ) and an arbitrarily good approximation to the smallest enclosing ball B(P ) of a set P R d. Unfortunately, both
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationChapter 13: Binary and Mixed-Integer Programming
Chapter 3: Binary and Mixed-Integer Programming The general branch and bound approach described in the previous chapter can be customized for special situations. This chapter addresses two special situations:
More informationA Simple Introduction to Support Vector Machines
A Simple Introduction to Support Vector Machines Martin Law Lecture for CSE 802 Department of Computer Science and Engineering Michigan State University Outline A brief history of SVM Large-margin linear
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
More informationChapter 3: Section 3-3 Solutions of Linear Programming Problems
Chapter 3: Section 3-3 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 3-3 Solutions of Linear Programming
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725
Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationDuality in Linear Programming
Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis, we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First, these shadow
More informationSpecial Situations in the Simplex Algorithm
Special Situations in the Simplex Algorithm Degeneracy Consider the linear program: Maximize 2x 1 +x 2 Subject to: 4x 1 +3x 2 12 (1) 4x 1 +x 2 8 (2) 4x 1 +2x 2 8 (3) x 1, x 2 0. We will first apply the
More informationInterior Point Methods and Linear Programming
Interior Point Methods and Linear Programming Robert Robere University of Toronto December 13, 2012 Abstract The linear programming problem is usually solved through the use of one of two algorithms: either
More informationUsing the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood
PERFORMANCE EXCELLENCE IN THE WOOD PRODUCTS INDUSTRY EM 8720-E October 1998 $3.00 Using the Simplex Method to Solve Linear Programming Maximization Problems J. Reeb and S. Leavengood A key problem faced
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationSeveral Views of Support Vector Machines
Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationLinear Programming. Widget Factory Example. Linear Programming: Standard Form. Widget Factory Example: Continued.
Linear Programming Widget Factory Example Learning Goals. Introduce Linear Programming Problems. Widget Example, Graphical Solution. Basic Theory:, Vertices, Existence of Solutions. Equivalent formulations.
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationNonlinear Algebraic Equations Example
Nonlinear Algebraic Equations Example Continuous Stirred Tank Reactor (CSTR). Look for steady state concentrations & temperature. s r (in) p,i (in) i In: N spieces with concentrations c, heat capacities
More informationCHAPTER 9. Integer Programming
CHAPTER 9 Integer Programming An integer linear program (ILP) is, by definition, a linear program with the additional constraint that all variables take integer values: (9.1) max c T x s t Ax b and x integral
More informationWe can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationLargest Fixed-Aspect, Axis-Aligned Rectangle
Largest Fixed-Aspect, Axis-Aligned Rectangle David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: February 21, 2004 Last Modified: February
More informationAdaptive Linear Programming Decoding
Adaptive Linear Programming Decoding Mohammad H. Taghavi and Paul H. Siegel ECE Department, University of California, San Diego Email: (mtaghavi, psiegel)@ucsd.edu ISIT 2006, Seattle, USA, July 9 14, 2006
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson
More informationDerivative Free Optimization
Department of Mathematics Derivative Free Optimization M.J.D. Powell LiTH-MAT-R--2014/02--SE Department of Mathematics Linköping University S-581 83 Linköping, Sweden. Three lectures 1 on Derivative Free
More informationSolving Linear Programs
Solving Linear Programs 2 In this chapter, we present a systematic procedure for solving linear programs. This procedure, called the simplex method, proceeds by moving from one feasible solution to another,
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More information! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationLinear Programming Problems
Linear Programming Problems Linear programming problems come up in many applications. In a linear programming problem, we have a function, called the objective function, which depends linearly on a number
More informationInternational Doctoral School Algorithmic Decision Theory: MCDA and MOO
International Doctoral School Algorithmic Decision Theory: MCDA and MOO Lecture 2: Multiobjective Linear Programming Department of Engineering Science, The University of Auckland, New Zealand Laboratoire
More informationconstraint. Let us penalize ourselves for making the constraint too big. We end up with a
Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationINTEGER PROGRAMMING. Integer Programming. Prototype example. BIP model. BIP models
Integer Programming INTEGER PROGRAMMING In many problems the decision variables must have integer values. Example: assign people, machines, and vehicles to activities in integer quantities. If this is
More informationEpipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.
Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations).
More informationLecture 2: The SVM classifier
Lecture 2: The SVM classifier C19 Machine Learning Hilary 2015 A. Zisserman Review of linear classifiers Linear separability Perceptron Support Vector Machine (SVM) classifier Wide margin Cost function
More informationIn this section, we will consider techniques for solving problems of this type.
Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationInterior-Point Algorithms for Quadratic Programming
Interior-Point Algorithms for Quadratic Programming Thomas Reslow Krüth Kongens Lyngby 2008 IMM-M.Sc-2008-19 Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800
More informationCHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS
Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationLINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL
Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables
More informationSupport Vector Machines Explained
March 1, 2009 Support Vector Machines Explained Tristan Fletcher www.cs.ucl.ac.uk/staff/t.fletcher/ Introduction This document has been written in an attempt to make the Support Vector Machines (SVM),
More information26 Linear Programming
The greatest flood has the soonest ebb; the sorest tempest the most sudden calm; the hottest love the coldest end; and from the deepest desire oftentimes ensues the deadliest hate. Th extremes of glory
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationChapter 10: Network Flow Programming
Chapter 10: Network Flow Programming Linear programming, that amazingly useful technique, is about to resurface: many network problems are actually just special forms of linear programs! This includes,
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More information1 Review of Least Squares Solutions to Overdetermined Systems
cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationLAKE ELSINORE UNIFIED SCHOOL DISTRICT
LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationPaper 2 Revision. (compiled in light of the contents of paper1) Higher Tier Edexcel
Paper 2 Revision (compiled in light of the contents of paper1) Higher Tier Edexcel 1 Topic Areas 1. Data Handling 2. Number 3. Shape, Space and Measure 4. Algebra 2 Data Handling Averages Two-way table
More informationNonlinear Programming
Nonlinear Programming 13 Numerous mathematical-programming applications, including many introduced in previous chapters, are cast naturally as linear programs. Linear programming assumptions or approximations
More informationGENERALIZED INTEGER PROGRAMMING
Professor S. S. CHADHA, PhD University of Wisconsin, Eau Claire, USA E-mail: schadha@uwec.edu Professor Veena CHADHA University of Wisconsin, Eau Claire, USA E-mail: chadhav@uwec.edu GENERALIZED INTEGER
More information17.3.1 Follow the Perturbed Leader
CS787: Advanced Algorithms Topic: Online Learning Presenters: David He, Chris Hopman 17.3.1 Follow the Perturbed Leader 17.3.1.1 Prediction Problem Recall the prediction problem that we discussed in class.
More informationc 2006 Society for Industrial and Applied Mathematics
SIAM J. OPTIM. Vol. 17, No. 4, pp. 943 968 c 2006 Society for Industrial and Applied Mathematics STATIONARITY RESULTS FOR GENERATING SET SEARCH FOR LINEARLY CONSTRAINED OPTIMIZATION TAMARA G. KOLDA, ROBERT
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationTOMLAB - For fast and robust largescale optimization in MATLAB
The TOMLAB Optimization Environment is a powerful optimization and modeling package for solving applied optimization problems in MATLAB. TOMLAB provides a wide range of features, tools and services for
More informationConstrained curve and surface fitting
Constrained curve and surface fitting Simon Flöry FSP-Meeting Strobl (June 20, 2006), floery@geoemtrie.tuwien.ac.at, Vienna University of Technology Overview Introduction Motivation, Overview, Problem
More informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationLecture 6: Logistic Regression
Lecture 6: CS 194-10, Fall 2011 Laurent El Ghaoui EECS Department UC Berkeley September 13, 2011 Outline Outline Classification task Data : X = [x 1,..., x m]: a n m matrix of data points in R n. y { 1,
More information10. Proximal point method
L. Vandenberghe EE236C Spring 2013-14) 10. Proximal point method proximal point method augmented Lagrangian method Moreau-Yosida smoothing 10-1 Proximal point method a conceptual algorithm for minimizing
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More information4.6 Linear Programming duality
4.6 Linear Programming duality To any minimization (maximization) LP we can associate a closely related maximization (minimization) LP. Different spaces and objective functions but in general same optimal
More informationLinear Programming. April 12, 2005
Linear Programming April 1, 005 Parts of this were adapted from Chapter 9 of i Introduction to Algorithms (Second Edition) /i by Cormen, Leiserson, Rivest and Stein. 1 What is linear programming? The first
More informationArrangements And Duality
Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,
More information