...Assignment 1–Advanced Operations Research - MATH 3010 Posted 23 August 2014 Due date: 19 September 2014, by 5pm In all the statements below, the notation, as well as references to page numbers, equations, etc, are as in the textbook Primal-dual interior-point methods, by Wright, Stephen, which is available online for UniSA staff and students. All relevant chapters of the textbook are also available in the webpage of the course. For solving this assignment, you need to read the handwritten Lecture Notes posted in the web and the material in the book up to Chapter 4, page 70. Question 1 (2+2+3+3+3+3=16 points) Fix A ∈ Rm×n , b ∈ Rm , and c ∈ Rn . (a) Write down the KKT conditions for the following problem, on the variable x ∈ Rn : min cT x Ax = b ; x ≥ 0. (1) (b) Write down the KKT conditions for the following problem, on the variable (λ, s) ∈ Rm+n AT λ max λT b + s = c; s ≥ 0, (2) Show that both the KKT conditions associated with both problems are identical. (c) Given x, s ∈ Rn , define the matrices X = diag(x1 , . . . , xn ), S = diag(s1 , . . . , sn ), and the vector e = (1, . . . , 1)T ∈ Rn . Let F : R2n+m → R2n+m be defined as T A λ+s−c . Ax − b F (x, λ, s) = XS e Show that a solution of F (x, λ, s) = 0 does not necessarily satisfy the KKT conditions of part (a) (or part (b)). Prove that, on the other hand, every vector (x, λ, s) that satisfies the KKT conditions must satisty F (x, λ, s) = 0. (d) Recall that the search direction (∆x, ∆λ, ∆s) generated by a Newton...
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...• Ch. 1 of Discrete and Combinatorial Mathematics o Supplementary Exercises 1, 2, 7, & 8 1. In the manufacture of a certain type of automobile, four kinds of major defects and seven kinds of minor defects can occur. For those situations in which defects do occur, in how many ways can there be twice as many minor defects as there are major ones? 2. A machine has nine different dials, each with five settings labeled 0, 1, 2, 3, and 4. a) In how many ways can all the dials on the machine be set? b) If the nine dials are arranged in a line at the top of the machine, how many of the machine settings have no two adjacent dials with the same setting? 7. There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the dance with eight of these 12 men? 8. In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels? • Ch. 2 of Discrete and Combinatorial Mathematics o Exercise 2.1, problems 2 o Exercise 2.2, problems 3 o Exercise 2.4, problems 1 o Exercise 2.5, problems 1 2. Identify the primitive statements in Exercise 1 below: Exercise 1. Determine whether each of the following sentences is a statement. a) In 2003 GeorgeW. Bush was the president of the United States. b) x + 3 is a positive integer. c) Fifteen is an even number. d) If Jennifer is late for the party, then her cousin Zachary will be quite angry...
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...In today’s world, there are a multitude of mathematical theorems and formulas. One theorem that is particularly renowned is the Pythagorean Theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides of any right triangle. While most people have heard of or even used the Pythagorean Theorem, many know little of the man who proved it. Pythagoras was born in 570 BC in Samos, Greece. His father, Mnesarchus, was a merchant from Tyre who traveled abroad. It is rumored that Pythagoras traveled with his father during his early years and was introduced to several influential teachers, including Thales who was a famous Greek philosopher. Several years and many countries later, Pythagoras found himself in Egypt. It was here that he studied at the temple of Diospolis and was also imprisoned during the Persian invasion. During the time he was imprisoned, Pythagoras began to study the religion called Zoroastrianism (Lauer/Schlager, 2001). It was because of these teachings and ideals that Pythagoras eventually moved to Italy. At age 52, while living in Croton, Italy, Pythagoras established the Pythagorean society. It was through this society and his positions in local government that Pythagoras recruited men and women in order to lead them to the pure life with his spiritual and mathematical teachings. Pythagoras believed that number was limiting and gave shape to all matter and he impressed this upon his followers (Gale, 1998)...
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...you a guarantee that the statement has to be true. But when you dissect a proof right down to its base axiom, there you will have to rely on an assumption- that our intuition deems valid. I am not here to argue on the basis of the correctness in our underlying assumption of the base axiom. Proofs are merely a way to deduce results from a given premise. And the premise here is the truth of the axiom. I am uncomfortable about the fact that people are willing to accept the intuition behind the base axiom but not the intuition behind the results that follow. True, in most cases it is easier to be aware of the former- and the latter may be hard to see as obvious. A case in point would be Fermat’s last theorem. Andrew Wiles did come up with an absolutely marvellous proof of the theorem- something that puzzled the greatest minds for three and a half centuries. But in those three and a half centuries, what if some person saw it as obvious. Does he/she have to prove it in Wiles’ way or any other way to actually believe in it? Indeed Fermat himself could have been one such person. ‘The truly marvellous proof that this margin is too small to contain’ may have very well referred to a product of intuition that words find hard to explain. Let a world in which everybody speaks...
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...education is the process of acquiring the values, the knowledge and developing the attitudes, skills, and behaviors to live in harmony with oneself, with others, and with the natural environment. i•de•ol•o•gy (ˌaɪ diˈɒl ə dʒi, ˌɪd i-) n., pl. -gies. 1. the body of doctrine or thought that guides an individual, social movement, institution, or group. 2. such a body forming a political or social program, along with the devices for putting it into operation. 3. theorizing of a visionary or impractical nature. 4. the study of the nature and origin of ideas. 5. a philosophical system that derives ideas exclusively from sensation Distance Formula The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points. The subscripts refer to the first and second points; it doesn't matter which points you call first or second. x2 and y2 are the x,y coordinates for one point x1and y1 are the x,y coordinates for the second point d is the distance between the two...
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...International Kangaroo Mathematics Contest 2012 – Cadet Level Cadet (Class 7 & 8) Time Allowed : 3 hours SECTION ONE - (3 points problems) 1. Four chocolate bars cost 6 EUR more than one chocolate bar. What is the cost of one chocolate bar? (A) 1 EUR (B) 2 EUR (C) 3 EUR (D) 4 EUR (E) 5 EUR 2. 11.11 − 1.111 = (A) 9.009 (B) 9.0909 (C) 9.99 (D) 9.999 (E) 10 3. A watch is placed face up on a table so that its minute hand points north-east. How many minutes pass before the minute hand points north-west for the first time? (A) 45 (B) 40 (C) 30 (D) 20 (E) 15 4. Mary has a pair of scissors and five cardboard letters. She cuts each letter exactly once (along a straight line) so that it falls apart in as many pieces as possible. Which letter falls apart into the most pieces? (A) (B) (C) (D) (E) 5. A dragon has five heads. Every time a head is chopped off, five new heads grow. If six heads are chopped off one by one, how many heads will the dragon finally have? (A) 25 (B) 28 (C) 29 (D) 30 (E) 35 6. In which of the following expressions can we replace each occurrence of the number 8 by the same positive number (other than 8) and obtain the same result? (A) (8 + 8) : 8 + 8 (D) (8 + 8 − 8) · 8 (B) 8 · (8 + 8) : 8 (E) (8 + 8 − 8) : 8 (C) 8 + 8 − 8 + 8 7. Each of the nine paths in a park is 100 m long. Ann wants to go from A to B without going along any path more than once. What is the length of the longest route she can choose? 1 of 7 International Kangaroo...
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...to do your math homework everyday, not for just a grade, but it also helps you when it comes time for quizzes and tests. She rarely checks homework, but when she does, she will not tell you. It is also a great review for tests and quizzes. Ms.Hull’s tests and quizzes are not the easiest things you will take. The quizzes take new concepts and apply to the quiz. Also, her tests are usually always hard. It is a good idea to practice new concepts and review old ones from previous units, so you can get a good grade on the tests. I also advise you to be organized throughout the year. Organization is the key to success especially in math class. Tool kits are an extremely helpful resource to use. There are going to be a lot of conjectures and theorems that will be new, and it would be hard to just memorize them. My overall geometry year was not exactly the way I hoped it would turn out. It was extremely had, and it moves at a very quick pace, so keeping up was hard for me personally. If I could have done something differently, it would have been practicing math more often. Each concept was hard, and I did not have anytime to review it, because I have a lot of honors classes which require a lot of work too. The key to being successful in this course is to pay attention, practice a lot outside of class, and do ALL of your homework. If you do that, I am sure your year...
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...FHMM1124 General Mathematics II (Past Year Paper Answer) September 2010 1 (a) Z=92 , x =20 , y =16 (b) Laspeyres price index = 108.43 2 (a) Year 2006 2007 2008 Total Deviation Average deviation Adjustment Adjusted seasonal factor,S Q1 -2.00 -3.13 -5.13 -2.57 0.06 -2.63 Deviation (Y-T) Q2 Q3 0.87 4.62 1.37 6.50 11.12 2.24 5.56 1.12 0.06 0.06 5.50 1.06 Q4 -3.50 -4.25 -7.75 -3.88 0.05 -3.93 (b)(i) (ii) 3 (a)(i) (ii) Limit is exist (iii) f(x) is continuous at x =-2 (b)(ii) Area = (c) 4 (a) Equation of normal : (b)(i) f(x) is increasing at f(x) is decreasing at (ii) Local maximum point = Local minimum point = 1 FHMM1124 General Mathematics II (Past Year Paper Answer) December 2010 1 (a) g=34 , u =2 , v =6 (b)(i) Aggregate price Index for 2008=90.48 Aggregate price Index for 2009=112.38 (ii) Laspeyres price index = 89.53 2 (a) Week/ Day 2006 2007 2008 Total Deviation Average deviation Adjustment Adjusted seasonal factor,S Mon -12.20 -13.80 -26.00 -13.00 0.06 -13.06 Deviation (Y-T) Tue Wed 2.00 14.80 1.40 17.20 0.80 32.00 4.20 16.00 1.40 0.06 0.06 15.94 1.34 Thu 27.6 29.00 56.60 28.30 0.06 28.24 Fri -31.60 -33.20 -64.80 32.40 0.06 -32.46 (a)(iv) Yp=98 (b)(i) (ii) 3 (a)(i) (ii) Limit does not exist. (iii) f(x) is discontinuous at x = 0 (iv) c = 8 (b)(i) f(x) is increasing at f(x) is decreasing at (ii) Local maximum point = Local minimum point = (iv) Absolute maximum point =155 Absolute minimum point = -53 2 FHMM1124 General...
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...My Most Important Day I had always been curious about all the things in the world. In fact, my parents used to always mention how inquisitive I was. My world got difficult when I turned five. My parents thought it would be a good idea to hire an intense tutor to prepare me for school. At first, I liked my tutor, things were going well, but soon I became so overwhelmed by the pressure of studying that I began to rebel. As it turned out, all that preparation was a big waste of time. When trying to enroll several months later, new enrollment rules further inhibited me because I was too young. The school would not accept me until age seven and as a result I spent one more year with my tutor preparing for school. When I finally entered school, I was so bored because I already knew most of the material. As the years passed, my dislike for school slowly dwindled. I always wondered why we study all that stuff in school. I knew that if I could understand the purpose, I would be more motivated to study, which finally happened one day. It was a sunny Sunday; my father had a plan to remodel his garage into a workshop. At that time, I was fourteen years old and in the seventh grade, and of course I was eager to help him. The day before, we had cleaned his jungle-like garage. We were going to start to equip the garage with table saw, jig saw, drill, and other machines, but we had a problem. There was no electricity for the light or to run all the machines which we wanted to install. There...
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...Let a , b , and B be known, and let Bbe acute. Using the Law of Sines,sin(A) = . Five different cases exist. 1. If the side opposite the given angle, b , is shorter than the other given side,a , and less than a certain length, then > 1 , and no solution exists, because there exists no angle whose sine is greater than one. Such a case arises when, for example, a = 4 , b = 3 , and B = 57 o . 2. If the side opposite the given angle is shorter than the other given side, there exists an exact length at which = 1 , and A = 90o . Exactly one solution exists, and a right triangle is determined. This occurs, for example, when a = 3 , b = 3 , and B = 45 o . 3. If the side opposite the given angle is shorter than the other given side, but longer than in case (2), then < 1 , and two triangles are determined, one in which A = x o , and one in which A = 180 o - x o . 4. If the side opposite the given angle is equal in length to the other given side, then A = B , and one isosceles triangle is determined. 5. If the side opposite the given angle is longer than the other given side, then < 1 , and one triangle is determined. Each of these five case is illustrated below. Figure %: Two sides of an oblique triangle and an angle opposite one of them are given, and the angle is acute. When the Angle is Obtuse Let a , b , and B be known, and let B be obtuse. Using the Law of Sines,sin(A) = . Three different cases exist. 1. If the side opposite the given angle...
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...Reasoning Under Uncertainty Most tasks requiring intelligent behavior have some degree of uncertainty associated with them. The type of uncertainty that can occur in knowledge-based systems may be caused by problems with the data. For example: 1. Data might be missing or unavailable 1. Data might be present but unreliable or ambiguous due to measurement errors. 1. The representation of the data may be imprecise or inconsistent. 1. Data may just be user’s best guess. 1. Data may be based on defaults and the defaults may have exceptions. The uncertainty may also be caused by the represented knowledge since it might 1. Represent best guesses of the experts that are based on plausible or statistical associations they have observed. 1. Not be appropriate in all situations (e.g., may have indeterminate applicability) Given these numerous sources of errors, most knowledge-based systems require the incorporation of some form of uncertainty management. When implementing some uncertainty scheme we must be concerned with three issues: 1. How to represent uncertain data 2. How to combine two or more pieces of uncertain data 3. How to draw inference using uncertain data We will introduce three ways of handling uncertainty: Probabilistic reasoning. Certainty factors Dempster-Shafer Theory 1. Classical Probability The oldest and best defined...
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...Empress Land Rules: Form four groups with four members in each. Therefore in total there should be 16. Take a piece of paper rip it into four pieces. Write one through four on the papers. Then fold them each into little pieces take a hat and shake it well and have one member from each group choose a number. Whatever number they choose is the number that their group will go in. there will be no yelling or shouting during another groups turn each team has 30 seconds to answer their question two minutes in total if it’s a problem that needs to be solved with pencil, and paper. Any question answered incorrectly will result in the team losing 100 points. At the end all the points will be added up and the winner should receive 2 points on any assignment of choice. 100 Section Questions: | Questions | Answer | By the given information determine which word it is best describing. | 1. Has no dimension, and it’s represented by a dot. | Point | 2. In geometry terms that can be described using known words such as point or line are called. | Defined terms | 3. To find the length of AB , with endpoints (-7,5) and B(4,-6) you can use the …. | Distance formula | For the following questions below find the coordinates of the midpoint of the segment with the given endpoints. | 4. C(3,5) and D(7,5) | (5,5) | 5. G(-4,4) and H(6,4) | (1,4) | 6. P(-8,-7) and Q (11,5) | (1 ½ ,-1) | Basic math calculators should not be needed | 7. 6*6 | 36 | 8....
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...1 Choice experiment: Identifying “niche” versus “change-of-pace” brands Albert C. Bemmaor, December 23, 2011 March 30, 2012 October 12, 2012 The use of the “REIBST2_vaa.xls” data file is restricted to the course MKGM31203 (Q1, 2012-2013) at ESSEC Business School. “Niche” brands can be defined as brands that benefit from an abnormally high repeat rate whereas “change-of-pace” brands can be defined as brands with an abnormally low repeat rate for a given penetration level. Key notions Here are some basic notions you need be familiar with prior to carrying out this exercise: (i) (ii) (iii) (iv) (v) (vi) (vii) What a “variable” is. What a mean (expected value) is and how to compute it; What the notion of independence between two events means and how to test for it; What a correlation between two variables is and how to measure it; What a market share is and how to interpret it; What a penetration is and how to interpret it; What the duplication between two brands is and how to interpret it. Analysis as a scientific process Analyzing data consists of a three-step procedure: (i) (ii) (iii) Defining expectations: What do you expect to find and why? If you have “no idea” about your expectations, you need to develop these ideas by discussing with colleagues, “experts”, reading textbooks and/or using other supporting material; Running the analysis; Comparing your expectations with the findings. Do they match? Did you obtain any surprising result...
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...How much weight did I just lift? You might ask yourself after a good set of pushups. Was it 90% of my body weight? No, maybe it was 50%? I will calculate the percentage of your body weight that you would expect to "push up" during both regular and inclined pushups. Before I begin with the math, lets define what a pushup is. More specifically, lets discuss proper form and technique. First, get onto the ground. Elevate your body using your arms and keep you back straight as a board. Don't let your gluteus maximus stick into the air or hang low. There should be a 90 degree angle between your arms and the floor. Your hands should be placed about one and a half times your shoulder width apart and pointed parallel to your body. Your body should be raised on the balls of your feet. Your feet should also be touching or no more than shoulder width apart. When you go downward, only bend your elbows. You can come back up once the elbows break the plane of your back. I will calculate the percentage of body weight resisted during a pushup for an average sized person. Since the resulting number will be a percentage, it will be correct for any person who has the same dimensions or ratio of dimensions as the average person calculated here. The characteristics of an average 25 year old American male are: Height: 70 inches (1.778 m) Palm to Shoulder length: 23 inches (0.5842 m) Shoulder to Hip Length: 24.75 inches (0.62865 m) Hip to Ankle Length: 31.5 inches (0...
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...cause your school is just a school. There’re lecturers and professors but they won’t promise you a job. All they did was pumping students in order to get you educated but not get you a job yet they’re still doing it. Yes, take or leave it. You won’t get better than this. 2. High Salary You have your own expertise. You might a statistician, a doctor or maybe an engineer. But remember this, don’t let high salary history derails you job search because you won’t get it, unless you’re a son of big boss. Are you? Most people don’t. 3. Be a boss Do you really want to be a boss? Well, your degrees won’t take you there, believe it or not, that’s the reality of life. You don’t just be a boss since you know the Pythagoras theorem. Fuck Pythagoras theorem. You won’t be a boss of such a big company because they’re already having their own targets. He might be someone who has a big leverage or clout so that your company will stand stronger and longer. 4. Hang out with your best pals Three things in life once you lost are hard to build-up; friends, respect and trust. You’ll face it because I’ve faced it too and I think I’ve made it. Sometimes in life, you will think that you do not need anyone. But sometimes, you will need someone to simply be there, just to be there, not to fix anything or do anything in particular things but just to be there, because that’s why we call them friends. And 8-5 work won’t get you the best way to keep your buddies alive besides you. Trust me...
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