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1189 results in Optimization, OR and risk

Page 12 of 60

  • 5 - Multiparametric Nonlinear Programming

  • from II - Multiparametric Programming
  • Francesco Borrelli, University of California, Berkeley, Alberto Bemporad, Manfred Morari
  • Book:
    Predictive Control for Linear and Hybrid Systems
    Published online:
    28 May 2018
    Print publication:
    22 June 2017, pp 95-106

  • Summary

    The operations research community has addressed parameter variations in mathematical programs at two levels: sensitivity analysis, which characterizes the change of the solution with respect to small perturbations of the parameters, and parametric programming, where the characterization of the solution for a full range of parameter values is studied. In this chapter we introduce the concept of multiparametric programming and recall the main results of nonlinear multiparametric programming. The main goal is to make the reader aware of the complexities of general multiparametric nonlinear programming. Later in this book we will use multiparametric programming to characterize and compute the state feedback solution of optimal control problems. There we will only make use of Corollary 5.1 for multiparametric linear programs and Corollary 5.2 for multiparametric quadratic programs, which show that these specific programs are “well behaved.”

    Introduction to Multiparametric Programs

    Consider the mathematical program

    where z is the optimization vector and x is a vector of parameters.We are interested in studying the behavior of the value function J*(x) and the optimizer z*(x) as we vary the parameter x. Mathematical programs where x is a scalar are referred to as parametric programs, while programs where x is a vector are referred to as multiparametric programs.

    There are several reasons to look for efficient solvers of multiparametric programs. Typically, mathematical programs are affected by uncertainties due to factors that are either unknown or that will be decided later. Parametric programming systematically subdivides the space of parameters into characteristic regions, which depict the feasibility and corresponding performance as a function of the uncertain parameters, and hence provide the decision maker with a complete map of various outcomes.

    Our interest in multiparametric programming arises from the field of system theory and optimal control. For example, for discrete-time dynamical systems, finite time constrained optimal control problems can be formulated as mathematical programs where the cost function and the constraints are functions of the initial state of the dynamical system. In particular, Zadeh and Whalen [293] appear to have been the first ones to express the optimal control problem for constrained discrete-time linear systems as a linear program. We can interpret the initial state as a parameter. By using multiparametric programming we can characterize and compute the solution of the optimal control problem explicitly as a function of the initial state.

  • 12 - Receding Horizon Control

  • from IV - Constrained Optimal Control of Linear Systems
  • Francesco Borrelli, University of California, Berkeley, Alberto Bemporad, Manfred Morari
  • Book:
    Predictive Control for Linear and Hybrid Systems
    Published online:
    28 May 2018
    Print publication:
    22 June 2017, pp 243-276

  • Summary

    In this chapter we review the basics of Receding Horizon Control (RHC). In the first part we discuss the stability and the feasibility of RHC and we provide guidelines for choosing the terminal weight so that closed-loop stability is achieved.

    The second part of the chapter focuses on the RHC implementation. Since RHC requires at each sampling time to solve an open-loop constrained finite time optimal control problem as a function of the current state, the results of the previous chapters imply two possible approaches for RHC implementation.

    In the first approach a mathematical program is solved at each time step for the current initial state. In the second approach the explicit piecewise affine feedback policy (that provides the optimal control for all states) is precomputed off-line. This reduces the on-line computation of the RHC law to a function evaluation, thus avoiding the on-line solution of a quadratic or linear program. This technique is attractive for a wide range of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides insight into the structure underlying optimization-based controllers, describing the behavior of the RHC controller in different regions of the state space. Moreover, for applications where safety is crucial, the correctness of a piecewise affine control law is easier to verify than that of a mathematical program solver.

    RHC Idea

    In the previous chapter we discussed the solution of constrained finite time and infinite time optimal control problems for linear systems. An infinite horizon suboptimal controller can be designed by repeatedly solving finite time optimal control problems in a receding horizon fashion as described next. At each sampling time, starting at the current state, an open-loop optimal control problem is solved over a finite horizon (top diagram in Figure 12.1). The computed optimal manipulated input signal is applied to the process only during the following sampling interval [t, t+1]. At the next time step t+1 a new optimal control problem based on new measurements of the state is solved over a shifted horizon (bottom diagram in Figure 12.1). The resulting controller is referred to as a Receding Horizon Controller (RHC). A receding horizon controller where the finite time optimal control law is computed by solving an optimization problem on-line is usually referred to as Model Predictive Control (MPC).

  • 12 - Game Theory II: Nonzero-sum and Cooperative Games

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 263-286

  • Summary

    In the final sections of Chapter 11 we showed that all (two-person) zero-sum games can be solved by determining a set of equilibrium strategies. From a purely theoretical perspective, there is little more to say about zero-sum games. However, nonzero-sum games are more interesting, and require further attention by game theorists. This chapter gives a brief overview of what we currently know about nonzero-sum games. First, we will give a proper introduction to the equilibrium concept tacitly taken for granted in the previous chapter. We will then go on to analyze a couple of well-known nonzero-sum games. The chapter ends with a discussion of whether game theory has any implications for ethics, biology and other subjects, which some scholars believe is the case.

    The Nash Equilibrium

    The prisoner's dilemma is an example of a nonzero-sum game. This is because the loss or gain made by each player is not the exact opposite of that made by the other player. If we sum up the utilities in each box they do not always equal zero. In the prisoner's dilemma each prisoner will have to spend some time in prison no matter which strategy he chooses, so the total sum for both of them is always negative. To see this point more clearly, it is helpful to take a second look at the game matrix of the prisoner's dilemma (Table 12.1).

    Obviously, R2 is dominated by R1, whereas C2 is dominated by C1. Hence, rational players will select the pair of strategies (R1, C1). It follows that (R1, C1) is in equilibrium, because if Row knew that Col was going to confess (play C1), then he would not be better off by switching to another strategy; and if Col knew that Row was going to confess (play R1), then he would also be best off by not switching his strategy. The notion of equilibrium tacitly employed here was originally proposed by John Nash, a Princeton mathematician who was awarded the Nobel Prize in Economics for his contributions to game theory. Nash's basic idea was very simple: Rational players will do whatever they can to insure that they do not feel unnecessarily unhappy about their decision.

  • 13 - Social Choice Theory

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 287-310

  • Summary

    Megan, Joe and Nick have decided to spend the summer holiday together. Unfortunately, they disagree on where to go. The set of alternatives includes Acapulco (a), Belize (b) and Cape Cod (c). Everyone in the group agrees that it would be fair to make the decision by voting, and their preferences are as follows.

    Megan prefers a to b, and b to c. (a ≻ b ≻ c)

    Joe prefers b to c, and c to a. (b ≻ c ≻ a)

    Nick prefers c to a, and a to b. (c ≻ a ≻ b)

    It seems plausible to maintain that, conceived of as a group, the friends prefer a to b, because in a pairwise choice a will get two votes (from Megan and Nick) whereas b will get just one (from Joe). Furthermore, the group prefers b to c, because b will get two votes (from Megan and Joe) and c one (from Nick). Finally, the group also prefers c to a, because c will get two votes (from Joe and Nick) and a one (from Megan). However, by now the voting rule has produced a cyclic preference ordering: a is preferred to b, and b to c, and c to a. This holds true although none of the individual preference orderings is cyclic.

    The observation that the majority rule can give rise to cyclic preference orderings is not a novel discovery. This point was extensively analyzed already by the French nobleman Marquis de Condorcet in the eighteenth century. Nowadays this result is known as the voting paradox. It serves as a classic illustration of the difficulties that arise if a group wishes to aggregate the preferences of its individual members into a joint preference ordering. Evidently, if group preferences are cyclic, they cannot be choice guiding. (But why not roll a die and let chance make the decision? Answer: This does not explain why one option is better than another. It can hardly be better to spend the summer in Acapulco because a die landed in a certain way on a particular occasion.)

  • 9 - Causal vs. Evidential Decision Theory

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 206-219

  • Summary

    The focus of this chapter is on the role of causal processes in decision making. In some decision problems, beliefs about causal processes play a significant role in determining what we intuitively think it is rational to do. However, it has turned out to be very difficult to give a convincing account of what role beliefs about causal process should be allowed to play. Much of the discussion has focused on a famous example known as Newcomb's problem. We shall begin by taking a look at this surprisingly deep problem.

    Newcomb's Problem

    Imagine a being who is very good at predicting other people's choices. Ninety-nine percent of all predictions made by the being so far have been correct. You are offered a choice between two boxes, B1 and B2. Box B1 contains $1,000 and you know this, because it is transparent and you can actually see the money inside. Box B2 contains either a million dollars or nothing. This box is not transparent, so you cannot see its content. You are now invited to make a choice between the following pair of alternatives: You either take what is in both boxes, or take only what is in the second box. You are told that the predictor will put $1 million in box B2 if and only if she predicts that you will take just box B2, and nothing in it otherwise. The predictor knows that you know this. Thus, in summary, the situation is as follows. First the being makes her prediction, then she puts either $1 million or nothing in the second box, according to her prediction, and then you make your choice. What should you do?

    Alternative 1 Take box B1 ($1,000) and box B2 (either $0 or $1 m).

    Alternative 2 Take only box B2 (either $0 or $1 m).

    This decision problem was first proposed by the physicist William Newcomb in the 1960s. It has become known as Newcomb's problem, or the predictor's paradox. Philosopher Robert Nozick was the first author to discuss it in print. (According to legend, he learned about the problem from a mutual friend of his and Newcomb's at a dinner party.) Nozick identified two different, but contradictory, ways of reasoning about the problem.

  • 6 - The Mathematics of Probability

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 124-140

  • Summary

    The following decision problem appeared some years ago in Marilyn vos Savant's Ask Marilyn column in Parade Magazine.

    You are a contestant in a game show hosted by Monty Hall. You have to choose one of three doors, and you win whatever is behind the door you pick. Behind one door is a brand new car. Behind the other two is a goat. Monty Hall, who knows what is behind the doors, now explains the rules of the game: “First you pick a door without opening it. Then I open one of the other doors. I will always pick a door to reveal a goat. After I have shown you the goat, I will give you the opportunity to make your final decision whether to stick with your initial door, or to switch to the remaining door.” Should you accept Monty Hall's offer to switch?

    The Monty Hall problem has a simple and undisputable solution, which all decision theorists agree on. If you prefer to win a brand new car rather than a goat, then you ought to switch. It is irrational not to switch. Yet, a considerable number of Marilyn vos Savant's readers argued that it makes no difference whether you switch or not, because the probability of winning the car must be 1/3 no matter how you behave. Although intuitively plausible, this conclusion is false.

    The point is that Monty Hall's behavior actually reveals some extra information that makes it easier for you to locate the car. To see this, it is helpful to distinguish between two cases. When you make your initial choice, the door you select either has a goat or the car behind it. In the first case, you pick a door with a goat behind it. This leaves two doors: one with a goat and the other with the car. Since only one of the remaining doors has a goat, Monty Hall must open that door.

  • 7 - The Philosophy of Probability

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 141-173

  • Summary

    Philosophers disagree about what probability is. Somewhat roughly put, there are three camps in this debate, objectivism, subjectivism and the epistemic approach. Objectivists maintain that probability statements refer to some facts in the external world. If you claim that the probability is fifty percent that the coin I hold in my hand will land heads up when tossed, then you are – according to the objectivist – referring to a property of the external world, such as the physical propensity of the coin to land heads up about every second time it is tossed. From an intuitive point of view, this seems to be a very plausible idea. However, subjectivists disagree with this picture. Subjectivists deny that probability statements can be understood as claims about the external world. What is, for instance, the probability that your suitor will ask you to marry him? It seems rather pointless to talk about some very complex propensity of that person, or count the number of marriage proposals that other people make, because this does not tell us anything about the probability that you will be faced with a marriage proposal. If it is true that the probability is, say, 1/2 then it is true because of someone's mental state.

    According to the subjective view, probability statements refer to the degree to which the speaker believes something. When you say that the probability is 1/2 that your suitor will make a marriage proposal, this merely means that you believe to a certain degree that this proposal will be made, and the strength of your belief can be represented by the number 1/2; and if you were to start thinking that it was absolutely certain that you would get the offer, your subjective degree of belief would be 1. The scope of the subjective analysis can easily be extended to all kinds of examples, because virtually any probability statement can be interpreted as a claim about the speaker's subjective degree of belief. For instance, when you say that the probability is 1/2 that the coin will land heads up next time it is tossed, you are essentially referring to your subjective degree of belief that that event will occur.

  • 3 - Decisions under Ignorance

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 41-64

  • Summary

    Jane is having a romantic dinner with her fiancé in a newly opened French bistro in Santa Barbara, California. After enjoying a vegetarian starter, Jane has to choose a main course. There are only two options on the menu, Hamburger and Lotte de mer. Jane recalls that Lotte de mer means monkfish, and she feels that this would be a nice option so long as it is cooked by a first-class chef. However, she has some vague suspicions that this may not be the case at this particular restaurant. The starter was rather poor and cooking monkfish is difficult. She knows that almost every chef can make a decent hamburger.

    Jane feels that she cannot assign any probability to the prospect of getting good monkfish. She simply knows too little about this newly opened restaurant. Because of this, she is in effect facing a decision under ignorance. In decision theory ignorance is a technical term with a very precise meaning. It refers to cases in which the decision maker (i) knows what her alternatives are and what outcomes they may result in, but (ii) is unable to assign any probabilities to the states corresponding to the outcomes (see Table 3.1). Sometimes the term “decision under uncertainty” is used synonymously.

    Jane feels that ordering a hamburger would be a safe option, and a hamburger would also be better than having no main course at all. Furthermore, she feels that good monkfish is better than an edible hamburger, but terrible monkfish is worse than having no main course. This ranking can be represented on an ordinal scale, as explained in Chapter 2. Since ordinal scales only preserve the order between objects, any set of numbers will do as long as better outcomes are represented by higher numbers. An example is given in Table 3.2.

    Which alternative should Jane choose? To answer this question the decision maker has either to apply some decision rule or provide some other kind of structured reasoning for choosing one alternative over another. This chapter surveys some of the most influential rules for making decisions under ignorance.

  • Preface

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp ix-x

  • Summary

    This book is an introduction to decision theory. My ambition is to present the subject in a way that is accessible to readers with a background in a wide range of disciplines, such as philosophy, economics, psychology, political science and computer science. That said, I am myself a philosopher, so it is hardly surprising that I have chosen to discuss philosophical and foundational aspects of decision theory in some detail. In my experience, readers interested in specific applications of the subject may find it helpful to start with a thorough discussion of the basic principles before moving on to their chosen field of specialization.

    My ambition is to explain everything in a way that is accessible to everyone, including readers with limited knowledge of mathematics. I therefore do my best to emphasize the intuitive ideas underlying the technical concepts and results before I state them in a more formal vocabulary. This means that some points are made twice, first in a nontechnical manner and thereafter in more rigorous ways. I think it is important that students of decision theory learn quite a bit about the technical results of the subject, but most of those results can no doubt be explained much better than what is usually offered in textbooks. I have tried to include only theorems and proofs that are absolutely essential, and I have made an effort to prove the theorems in ways I believe are accessible for beginners. In my experience, this sometimes comes into conflict with the ambition to present technical material in the minimalist style usually preferred by experts.

    Most of the technical results are presented in twenty “boxes” clearly separated from the main body of the text. In principle, it should be possible to read the book without reading the boxes, although they hopefully deepen the student's understanding of the subject. I have also included over 100 exercises (and solutions), most of which should be fairly straightforward. Unlike other textbooks, no exercise asks the reader to prove some theorem I did not bother to prove myself. Finally, the Appendix contains a glossary in which I try to briefly explain some of the key terms and concepts. I believe the glossary might be particularly useful for readers wishing to study only a selected number of the chapters.

  • 14 - Overview of Descriptive Decision Theory

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 311-322

  • Summary

    This chapter gives an overview of how people actually do make decisions. The headline news is that people frequently act in ways deemed to be irrational by decision theorists. This shows either that people should behave differently, or that there is something wrong with the normative theories discussed in the preceding chapters of this book. After reviewing the empirical findings, both conclusions will be considered.

    The interest in descriptive decision theory arose alongside the development of normative theories. Given the enormous influence axiomatic theories had in the academic community in the latter half of the twentieth century, it became natural to test the axioms in empirical studies. Since many decision theorists advocate (some version of) the expected utility principle, it is hardly surprising that the axioms of expected utility theory are the ones most researched. Early studies cast substantial doubt on the expected utility principle as an accurate description of how people actually choose. However, it was not until 1979 and the publication of a famous paper by Kahneman and Tversky that it finally became widely accepted that expected utility theory is a false descriptive hypothesis. Kahneman and Tversky's paper has become one of the most frequently quoted academic publications of all time. (Kahneman was awarded the Nobel Prize in Economics in 2002, but Tversky had died a few years earlier.) In what follows, we will summarize their findings, as well as some later observations.

    Observed Violations of the Expected Utility Principle

    Kahneman and Tversky asked a group of 72 students to state a preference between prospect A and B below. It turned out that 82% of the participants preferred prospect B to A.

    A: $2,500 with probability 0.33 B: $2,400 with certainty

    $2,400 with probability 0.66

    $0 with probability 0.01

    It can be easily verified that the expected monetary value of prospect A is slightly higher than that of prospect B. However, this is of course not sufficient for concluding that the students are irrational. It cannot be excluded that they preferred B to A simply because their marginal utility is decreasing in the interval between $2,400 and $2,500. To make a stronger case against the expected utility principle, Kahneman and Tversky also asked the same group of students to state a preference between prospects C and D below.

  • 2 - The Decision Matrix

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 17-40

  • Summary

    Before you make a decision you have to determine what to decide about. Or, to put it differently, you have to specify what the relevant acts, states and outcomes are. Suppose, for instance, that you are thinking of taking out fire insurance on your home. Perhaps it costs $100 to take out insurance on a house worth $100,000 and you ask, “Is it worth it?” Before you decide, you have to get the formalization of the decision problem right. In this case, it seems that you face a decision problem with two acts, two states, and four outcomes. It is helpful to visualize this information in a decision matrix; see Table 2.1.

    To model one's decision problem in a formal representation is essential in decision theory, because decision rules are only defined relative to a formal representation. For example, it makes no sense to say that the principle of maximizing expected value recommends one act rather than another unless there is a formal listing of the available acts, the possible states of the world and the corresponding outcomes. However, instead of visualizing information in a decision matrix it is sometimes more convenient to use a decision tree. The decision tree in Figure 2.1 is equivalent to the matrix in Table 2.1.

    The square represents a choice node, and the circles represent chance nodes. At the choice node the decision maker decides whether to go up or down in the tree. If there are more than two acts to choose from, we simply adds more lines. At the chance nodes nature decides which line to follow. The rightmost boxes represent the possible outcomes. Decision trees are often used for representing sequential decisions, i.e. decisions that are divided into several separate steps. (Example: In a restaurant, you can either order all three courses before you start to eat, or divide the decision-making process into three separate decisions taken at three points in time. If you opt for the latter approach, you face a sequential decision problem.) To represent a sequential decision problem in a tree, one simply adds new choice and chance nodes to the right of the existing leaves.

  • 5 - Utility

  • Martin Peterson, Texas A & M University
  • Book:
    An Introduction to Decision Theory
    Published online:
    28 May 2018
    Print publication:
    30 March 2017, pp 97-123

  • Summary

    It is hard to think of any minimally reasonable decision rule that totally ignores how much utility an act brings about. However, the concept of utility has many different technical meanings, and it is important to keep these different meanings apart. In Chapter 2 we distinguished three fundamentally different kinds of measurement scales. All scales are numerical, i.e. utility is represented by real numbers, but the information conveyed by the numbers depends on which type of scale is being used.

  • 1. Ordinal scales (“10 is better than 5”)

  • 2. Interval scales (“the difference between 10 and 5 equals that between 5 and 0”)

  • 3. Ratio scales (“10 is twice as valuable as 5”)

    • In ordinal scales, better objects are assigned higher numbers. However, the numbers do not reflect any information about differences or ratios between objects. If we wish to be entitled to say, for instance, that the difference between ten and five units is exactly as great as the difference between five and zero units, then utility has to be measured on an interval scale. Furthermore, to be entitled to say that ten units of utility is twice as much as five, utility must be measured on a ratio scale.

    Arguably, utility cannot be directly revealed by introspection. We could of course ask people to estimate their utilities, but answers gathered by this method would most certainly be arbitrary. Some more sophisticated method is needed. So how on earth is it possible to assign precise numbers to outcomes and acts that accurately reflect their value? And how can this process be a scientific one, rather than a process that is merely arbitrary and at best based on educated guesses?

    How to Construct an Ordinal Scale

    Let us first show how to construct an ordinal utility scale. To make the discussion concrete, imagine that you have a collection of crime novels, and that you wish to assign ordinal utilities to each book in your collection. To avoid a number of purely mathematical obstacles we assume that the number of books in the collection is finite.

Page 12 of 60