constrained optimization

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pages: 320 words: 24,110

Elements of Mathematics for Economics and Finance by Vassilis C. Mavron, Timothy N. Phillips

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Strictly, we should say a local (or relative) maximum or minimum: for they may not give the overall maximum or minimum of a function. Within the locality of a maximum point x = x0 , y = y0 (that is, for any x and y sufficiently close to these values), the value f (x, y) of f attains a maximum when x = x0 and y = y0 . Similarly for a minimum point. 9. Optimization 193 3. The problem stated in Example 9.5 above was initially a constrained optimization problem in three variables. But, upon substitution for z, it became an unconstrained optimization problem in the remaining two variables, x and y. 9.3 Constrained Optimization Optimization of a quantity in economic models, or indeed in many practical situations, is rarely unconstrained. Usually there are constraints involving some or all of the variables. For instance, in considering ways to maximize, say, output, there will be constraints due to costs or of the available labour.

Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.2 Functions of Two or More Variables . . . . . . . . . . . . . . . . . . . . . . . . 160 8.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 8.4 Higher Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.5 Partial Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8.6 The Chain Rule and Total Derivatives . . . . . . . . . . . . . . . . . . . . . . 168 8.7 Some Applications of Partial Derivatives . . . . . . . . . . . . . . . . . . . . 171 8.7.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.7.2 Elasticity of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 8.7.3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.7.4 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 8.7.5 Graphical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9. Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 9.2 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.3 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.1 Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.2 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.3.3 The Lagrange Multiplier λ: An Interpretation . . . . . . . . . . 201 9.4 Iso Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.

A government may try to minimize interest rates while trying to keep inflation at a certain level. A pilot may fly an aircraft so as to cover the maximum possible air miles when the total fuel cost is stipulated. Consumers try to maximize utility subject to a given budget. In this chapter, we will describe techniques of optimization when there are no constraints specified (unconstrained optimization) and subject to a constraint (constrained optimization). 185 186 Elements of Mathematics for Economics and Finance 9.2 Unconstrained Optimization The optimization of functions of one variable was discussed in Chapter 7. Optimization there meant finding the stationary (or critical or turning) points of the function and then testing to see whether they were maxima or minima. Maxima (or minima) are points where the function changes from being increasing to decreasing (or vice versa).

pages: 293 words: 88,490

The End of Theory: Financial Crises, the Failure of Economics, and the Sweep of Human Interaction by Richard Bookstaber

Well, not really optimizing—we don’t really want to rank all ten thousand possible Starbucks combinations every morning—so it must be that people are operating under cognitive constraints. People must act as if they are doing a constrained optimization, looking at the cost or limits to search or to available information. There’s an irony to adding constraints to reflect cognitive limits—constraints just make the problem worse. It is even more difficult to solve a constrained optimization. If people can’t optimize because that exercise is too difficult, then they certainly cannot work their way through a constrained optimization. It might be that we don’t fit into an optimization because we are constrained and cannot pull off the task, or it could be that we are in a world where things change suddenly and wholly unexpectedly, where the past does not give us a window into what is happening, and where what is happening could run off the rails and down the chasm, so the conditions required for following the mathematically rational path of optimization are no longer really optimal.

It might be that we don’t fit into an optimization because we are constrained and cannot pull off the task, or it could be that we are in a world where things change suddenly and wholly unexpectedly, where the past does not give us a window into what is happening, and where what is happening could run off the rails and down the chasm, so the conditions required for following the mathematically rational path of optimization are no longer really optimal. And so it becomes just an academic exercise, because people actually do not behave in a way that is “as if” they were optimizing or even executing a constrained optimization. They display various behavioral flaws that cannot be consistent with a constrained optimization. Economics has a roundabout answer for this, too. People now are said to behave as if they are doing a constrained optimization with a utility function that essentially zigs and zags to accommodate their irrational behavior, behaving as if they are both constrained and have a weird utility function that adds more and more adjustable parameters. The economist Thomas Sargent, though of the view that bounded rationality means optimization under constraints, wrote, “Ironically, when we economists make the people in our models more ‘bounded’ in their rationality … we must be smarter, because our models become larger and more demanding mathematically and econometrically.”5 The advantage of taking the route of simple heuristics doesn’t happen because the world is simple.

The mathematical approach is to assume that, absent constraints on cognitive ability, people will solve the same sort of problem a mathematician will solve in decision making: one of optimization. Then, recognizing that people cannot always do so, they step back to concede that people will solve the optimization problem subject to constraints, such as limited time, information, and computational ability. If computational ability is an issue, then moving into a constrained optimization is moving in the wrong direction, because a constrained optimization problem is generally more difficult to solve than an unconstrained one. But given the axioms, what else can you do? It doesn’t take much familiarity with humans—even human mathematicians—to realize that we don’t actually solve these complex, and often unsolvable, problems. So the optimization school moves into “as if” mode. “We don’t know how people really think (and we don’t care to know) but we will adjust our axioms to assume that they act as if they are optimizing.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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One method to solve such problems is by using Lagrange multipliers, as outlined below. Let U c ~n be an open set, and let j : U -----7 ~ be a smooth function, e.g., infinitely many times differentiable. We want to find the extrema of j (x) subject to m constrains given by g(x) = 0, where 9 : U -----7 ~m is a smooth function, i.e., Find Xo E U such that max g(x) = 0 xEU j(x) = j(xo) or min g(x) = 0 xEU j(x) j(xo). (8.1) Problem (8.1) is called a constrained optimization problem. For this problem to be well posed, a natural assumption is that the number of constraints is smaller than the number of degrees of freedom, i.e., m < n. Another way to formalize problem (8.1) is to introduce the set S of points satisfying the constraint g( x) = 0, and find extrema for the restriction of the function j (x) to the space S, i.e., for j Is. 235 236 CHAPTER 8. LAGRANGE MULTIPLIERS.

LAGRANGE MULTIPLIERS. NEWTON'S METHOD. 8.1. LAGRANGE MULTIPLIERS Definition 8.1. The point Xo E U c IRn is called a constrained extremum of the function f : U -----+ IR with respect to the constraint g( x) = 0 if and only if Xo is an extremum point of fls, where S = {x E U such that g(x) = o}. where Vf(x) and Vg(x), the gradients of f : U -----+ IR and 9 : U -----+ IRm are given by v f(x) To solve the constrained optimization problem (8.1), let A = (Ai)i=l:m be a vector of the same size, m, as the number of constraints; A is called a Lagrange multipliers vector. Let F : U x IRm -----+ IR given by F(x, A) = f(x) + be the Lagrangian function of f and g. If ::n (x, >.)) ; ~;~ (x) ~;~ (x) ... ~;~ (x) ) Vg(x) (8.2) At g(x) (:;, (x, >.) ... 237 Og2 (x) Og2 (x) '" OXl. OX2 . . .. .. .. ( ogm (x) ogm (x) OXl OX2 Og2 (x) OXn. .. ogm (x) OXn ; cf. (1.36) and (1.38), respectively.

The reason is that if a problem is known to have a unique constrained extremum, and if the Lagrangian has exactly one critical point, then that critical point must be the constrained extremum point. For this line of thinking to be sound we must know that the constrained extremum problem has a unique solution. In general, showing that a problem has a unique solution is not straightforward. The steps required to solve a constrained optimization problem using Lagrange multipliers can be summarized as follows: Step 1: Check that rank(\lg(x)) = m, for all xES. q( v) = (V2 - 2V3? + 2v~ + 3v~ - 2(V2 - 2V3)V2 + 4V2V3 = v~ + 4V2V3 + 7v~. Note that Vred = ( ~~ ). Step 2: Find (xo, Ao) E U X ffi.m such that \l(x,)..)F(xo, Ao) Step 3.1: Compute q(v) + 4V2V3 + 7v~. 0 (8.15) Whether the point Xo is a constrained extremum for f (x) will depend on the quadratic form qred being either positive definite2 , i.e., = v D2 Fo(xo) v, where Fo(x) = f(x) + Ab9(x).

pages: 320 words: 33,385

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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Then examining the Hessian matrix for positive definiteness, or evaluating the function at and around these points, will tell us whether a given stationary point is a local maximum, a local minimum or neither. Once we have found all possible local maxima and minima, the global maximum (if it exists) is the one of the local maxima where the function takes the highest value and the global minimum (if it exists) is the one of the local minima where the function takes the lowest value. More generally, a constrained optimization problem takes the form max fx such that x hx ≤ 0 (I.1.47) where hx ≤ 0 is a set of linear or non-linear equality or inequality constraints on x. Examples of constrained optimization in finance include the traditional portfolio allocation problem, i.e. how to allocate funds to different types of investments when the investor has constraints such as no short sales and/or at least 50% of his capital must be in UK bonds. To find a constrained optimum we must introduce new variables, which are called Lagrange multipliers and denoted = 1 k , where k is the number of constraints on the objective.

This result forms the basis of portfolio theory, and will be used extensively in Chapter I.6. Another application of differentiation is to the optimal allocation problem for an investor who faces certain constraints, such as no short sales and/or at least 30% of his funds must be invested in US equities. The investor’s problem is to choose his portfolio weights to optimize his objective whilst respecting his constraints. This falls into the class of constrained optimization problems, problems that are solved using differentiation. Risk is the uncertainty about an expected value, and a risk-averse investor wants to achieve the maximum possible return with the minimum possible risk. Standard measures of portfolio risk are the variance of a portfolio and its square root which is called the portfolio volatility.6 The portfolio variance is a quadratic function of the portfolio weights.

pages: 396 words: 117,149

The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World by Pedro Domingos

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Or, equivalently, we can minimize the weights under the constraint that all examples have a given margin, which could be one—the precise value is arbitrary. This is what SVMs usually do. Constrained optimization is the problem of maximizing or minimizing a function subject to constraints. The universe maximizes entropy subject to keeping energy constant. Problems of this type are widespread in business and technology. For example, we may want to maximize the number of widgets a factory produces, subject to the number of machine tools available, the widgets’ specs, and so on. With SVMs, constrained optimization became crucial for machine learning as well. Unconstrained optimization is getting to the top of the mountain, and that’s what gradient descent (or, in this case, ascent) does. Constrained optimization is going as high as you can while staying on the road. If the road goes up to the very top, the constrained and unconstrained problems have the same solution.

You know you’ve reached the highest point on the road when you can’t go any higher without driving off the road; in other words, when the path to the top is at right angles to the road. If the road and the path to the top form an oblique angle, you can always get higher by driving farther along the road, even if that doesn’t get you higher as quickly as aiming straight for the top of the mountain. So the way to solve a constrained optimization problem is to follow not the gradient but the part of it that’s parallel to the constraint surface—in this case the road—and stop when that part is zero. In general, we have to deal with many constraints at once (one per example, in the case of SVMs). Suppose you wanted to get as close as possible to the North Pole but couldn’t leave your room. Each of the room’s four walls is a constraint, and the solution is to follow the compass until you bump into the corner where the northeast and northwest walls meet.

The connectionists’ is gradient descent. The evolutionaries’ is genetic search, including crossover and mutation. The Bayesians are unusual in this regard: they don’t just look for the best model, but average over all models, weighted by how probable they are. To do the weighting efficiently, they use probabilistic inference algorithms like MCMC. The analogizers (or more precisely, the SVM mavens) use constrained optimization to find the best model. After a long day’s journey, the sun is rapidly nearing the horizon, and you need to hurry before it gets dark. The city’s outer wall has five massive gates, each controlled by one of the tribes and leading to its district in Optimization Town. Let us enter through the Gradient Descent Gate, after whispering the watchword—“deep learning”—to the guard, and spiral in toward the Towers of Representation.

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Optimization Methods in Finance by Gerard Cornuejols, Reha Tutuncu

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A typical optimization model addresses the allocation of scarce resources among a set of alternative activities in order to maximize an objective function–a measure of the modeler’s satisfaction with the solution, for example, the total profit. Decision variables, the objective function, and constraints are three essential elements of any optimization problem. Some problems may lack constraints so that any set of decision variables (of appropriate dimension) are acceptable as alternative solutions. Such problems are called unconstrained optimization problems, while others are often referred to as constrained optimization problems. There are problem instances with no objective functions–the so-called feasibility problems, and others with multiple objective functions. Such problems are often addressed by reduction to a single or a sequence of single-objective optimization problems. If the decision variables in an optimization problem are restricted to integers, or to a discrete set of possibilities, we have an integer or discrete optimization problem.

(5.6) Thus, we proved the following result: Proposition 5.1 Given a set X of feasible portfolios with the property that eT x = 1, ∀x ∈ X , the portfolio x∗ with the maximum Sharpe ratio in this set can be found by solving the following problem with a convex quadratic objective function min xT Qx s.t. (x, κ) ∈ X + , (µ − rf e)T x = 1, (5.7) with X + as in (5.4). If (x̂, κ̂) is the solution to (5.7), then x∗ = κ̂x̂ . This last problem can be solved using the techniques we discussed for convex quadratic programming problems. 5.3. RETURNS-BASED STYLE ANALYSIS 5.3 63 Returns-Based Style Analysis In two ground-breaking articles, Sharpe described how constrained optimization techniques can be used to determine the effective asset mix of a fund using only the return time series for the fund and a number of carefully chosen asset classes [13, 14]. Often, passive indices or index funds are used to represent the chosen asset classes and one tries to determine a portfolio of these funds/indices whose returns provide the best match for the returns of the fund being analyzed.

pages: 252 words: 73,131

The Inner Lives of Markets: How People Shape Them—And They Shape Us by Tim Sullivan

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Suppose you want to help people commute safely and efficiently across the East River, from Brooklyn to Manhattan. Traditional economics is the equivalent of assuming that the only two ways of doing so are bridge or ferry. Mechanism design imagines the broad set of possibilities—zip line, catapult, people mover—then figures out which will work best. It’s what, in technical terms, is called constrained optimization (the same technique Vickrey deployed to determine that the best way to get to work at Columbia was on roller skates). Mechanism designers consider the restrictions imposed by laws, human nature, our sense of right and wrong, and the strategizing that kidney patients, school applicants, and others may engage in to get a better organ or education. And they design a mechanism to best to fulfill society’s needs and wants within those constraints.

., 169–170, 172 Camp, Garrett, 170 candle auctions, 82 capitalism, free-market, 172–173 car service platform, 169–171 cash-back bonus, 116 cash-for-sludge transactions, 167–169 See also Summers, Larry centralized clearinghouses, 140–141 Champagne fairs, 105–106, 126–128 Changi POW camp, 175–177 Le Chatelier, Henry Louis, 29 Le Chatelier’s principle, 29 cheap talk, 62–66, 69 chess, difference between Cold War and, 26 See also poker, bluffing in child labor, 180 cigarettes, as currency in German POW camp, 8–9 Clarke, Edward, 93 Clavell, James, 175 clerkship offers, with federal judges, 140 coat hook, 151–152, 174 Codes of the Underworld (Gambetta), 68 Cold War, difference between chess and, 26 See also poker, bluffing in Collectible Supplies, 128–129 “College Admissions and the Stability of Marriage” (Gale and Shapley), 137 commitment, signs of, 62–63, 69–71, 72–75 community game, 178–179 competition models of, 35, 166, 172–173 platform, 124–126 unethical conduct with, 180–181 “Competition is for Losers” (Thiel), 173 competitive equilibrium, existence of, 29, 31–34, 36–37, 40, 45, 76 competitive markets, 35, 124–126, 172–174, 180–181 See also platforms competitive signaling, 70–71 congestion pricing model, 86, 94 constrained optimization, 85–86, 133 contractorsfromhell.com, 120 copycat competitors, 172–173 corporate philanthropy, 72–75 Cowles, Alfred, 25, 27 Cowles Commission for Research in Economics, 25, 27, 31, 134 “creative destruction,” 50 credit card platforms, 113–116, 123–124 criminal organizations, informational challenges of, 68 currency, at Stalag VII-A POW camp, 8–9 customer feedback, 52, 74–75 Davis, Harry, 154, 157 Debreu, Gérard, 20, 24, 25, 32–33, 36–37 decentralized match, 139–140 deferred acceptance algorithm, 137–141, 145–149 Delmonico, Frank, 164 descending price auctions, 81–82 design, auction, 14, 101–102 Digital Dealing (Hall), 94 Discover card, 115–116 distribution of income, 22 Domar, Evsey, 36–37 Dorosin, Neil, 142–144 Douglas Aircraft Company, 25 Dow, Bob, 1–2 Dow, Edna, 1–2 Drèze, Jacques, 85–86 dumping toxic waste, transactions for, 167–169 Dutch auctions, 81–82 dysfunction, market, 36, 75–77, 143 eBay adverse selection on, 51–55, 57 auction listings, 94–97 concerns on model for, 43, 46, 48 on seller motivation for giving to charities, 73–75 start of, 39–41 as two-sided market, 109, 119 e-commerce, 41–43, 52–55 “The Economic Organization of a P.O.W.

pages: 500 words: 145,005

Misbehaving: The Making of Behavioral Economics by Richard H. Thaler

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Of all the goods and services a family could buy, the family chooses the best one that it can afford. Furthermore, the beliefs upon which Econs make choices are assumed to be unbiased. That is, we choose on the basis of what economists call “rational expectations.” If people starting new businesses on average believe that their chance of succeeding is 75%, then that should be a good estimate of the actual number that do succeed. Econs are not overconfident. This premise of constrained optimization, that is, choosing the best from a limited budget, is combined with the other major workhorse of economic theory, that of equilibrium. In competitive markets where prices are free to move up and down, those prices fluctuate in such a way that supply equals demand. To simplify somewhat, we can say that Optimization + Equilibrium = Economics. This is a powerful combination, nothing that other social sciences can match.

(Lamont and Thaler), 250 capital asset pricing model (CAPM), 226–29, 348 “CAPM is Wanted, Dead or Alive, The” (Fama and French), 228 Car Talk, 32 Case, Chip, 235 Case-Shiller Home Price Index, 235 cashews, 21, 24, 42, 85–86, 92, 100, 102–3, 107n casinos, 49n cautious paternalism, 323 Census Bureau, 47 Center for Research in Security Prices (CRSP), 208, 221 charity, 66, 129 cheap stocks, 219–21 Checklist Manifesto, The (Gawande), 356 Chen, Nai-fu, 243 Chetty, Raj, 320, 357–58 Chicago, University of, 255–56 behavioral economics conference at, 159–64, 167–68, 169, 170, 205 conference on 1987 crash at, 237 debate on behavioral economics at, 159–63, 167–68, 169, 170, 205 finance studied at, 208 offices at, 270–76, 278 Chicago Bulls, 19 Chicago police department, 260 chicken (game of), 183 choice: number of, 21, 85, 99–103 preferences revealed by, 86 choice architecture, 276, 326–27, 357 Choices, Values, and Frames, xiv Chrysler, 121, 123, 363 Cialdini, Robert, 180, 335, 336 Clegg, Nick, 333 Clinton, Hillary, 22 closed-end funds, 238–39, 239, 240 puzzles of, 240–43, 244, 250 coaches, 292–93 Coase, Ronald, 261 Coase theorem, 261–62, 264–65, 264, 267–68 Cobb, David, 115 Cobb, Michael, 115, 116, 117, 118n, 119, 120, 123 Coca-Cola, 134–35 cognitive dissonance, 178 commitment strategies, 100, 102–3, 106–7 compliance (medical), 189–90 COMPUSTAT, 221 computing power, 208 concert tickets, 18–19, 66 conditional cooperators, 146, 182, 335n “Conference Handbook, The” (Stigler), 162–63 confirmation bias, 171–72 Conservative Party, U.K., 330–33 constrained optimization, 5–6, 8, 27, 43, 161, 207, 365 “Consumer Choice: A Theory of Economists’ Behavior” (Thaler), 35 consumers, optimization problem faced by, 5–6, 8, 27, 43, 161, 207, 365 consumer sovereignty, 268–69 consumer surplus, 59 consumption function, 94–98, 106, 309 “Contrarian Investment, Extrapolation, and Risk” (Lakonishok, Shleifer and Vishny), 228 cooperation, 143–47 conditional, 146, 182, 335n Prisoner’s Dilemma and, 143–44, 145, 301–5, 302 Copernican revolution, 169 Cornell University, 42, 43, 115, 140–43, 153–55, 157 Costco, 63, 71–72 Council of Economic Advisors, 352 coupons, 62, 63, 67–68, 120 credit cards, 18, 74, 76–77 late fees for, 360 crime, 265 Daily Mail, 135 Daily Show, The, 352 Dallas Cowboys, 281 data: financial, 208 collection and recording of, 355–56 Dawes, Robyn, 146 Deal or No Deal, 296–301, 297, 303 path dependence on, 298–300 deals, 61–62 De Bondt, Werner, 216–18, 221, 222–24, 226n, 233, 278 debt, 78 default investment portfolio, 316 default option, 313–16, 327 default saving rate, 312, 316, 319, 357 delayed gratification, 100–102 De Long, Brad, 240 Demos, 330 Denmark, 320, 357–58 descriptive, 25, 30, 45, 89 Design of Everyday Things, The (Norman), 326 Diamond, Doug, 273, 276 Diamond, Peter, 323 Dictator Game, 140–41, 142, 160, 182, 301 diets, 342 diminishing marginal utility, 106 of wealth, 28, 30 diminishing sensitivity, 30–34 discount, surcharge vs., 18 discounts, returns and, 242–43 discounted utility model, 89–94, 99, 110, 362 discretion, 106 Ditka, Mike, 279, 280 dividends, 164–67, 365 present value of, 231–33, 231, 237 Dodd, David, 219 doers, planners vs., 104–9 Donoghue, John, 265n “Do Stock Prices Move Too Much to be Justified by Subsequent Changes in Dividends?”

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Algorithms to Live By: The Computer Science of Human Decisions by Brian Christian, Tom Griffiths

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Being a circuit lawyer meant literally making a circuit—moving through towns in fourteen different counties to try cases, riding hundreds of miles over many weeks. Planning these circuits raised a natural challenge: how to visit all the necessary towns while covering as few miles as possible and without going to any town twice. This is an instance of what’s known to mathematicians and computer scientists as a “constrained optimization” problem: how to find the single best arrangement of a set of variables, given particular rules and a scorekeeping measure. In fact, it’s the most famous optimization problem of them all. If it had been studied in the nineteenth century it might have become forever known as “the prairie lawyer problem,” and if it had first come up in the twenty-first century it might have been nicknamed “the delivery drone problem.”

See also uncertainty charity Cheshire, Stuart chess childhood Chomsky, Noam Churchill, Winston circuit switching clairvoyance clinical trials closet, organizing Cobham, Alan Cobham-Edmonds thesis Cockcroft, George (Luke Rhinehart) coconut water cognitive decline coincidences coins denominations two-headed tosses collators commitment problem communications. See also language; networking; storytelling confirmation priors and community-supported agriculture (CSA) Comparison Counting Sort comparison-shopping websites complexity penalizing computation, defined by Turing computational kindness confidence interval confirmation congestion avoidance of price of anarchy and Connection Machine constant-time (O(1)) constrained optimization problems constrained problem, preferences for Constraint Relaxation construction projects content distribution networks (CDNs) context switching continuous optimization problems Continuous Relaxation control without hierarchy Cooper, Martin cooperation Copernican Principle Copernicus, Nicolaus corporate marketing cost-benefit analysis Cramer, Jim Cravath system creativity crêpe stand queue Cross-Validation cryptography customer service hold times Darwin, Charles data.

pages: 257 words: 13,443

Statistical Arbitrage: Algorithmic Trading Insights and Techniques by Andrew Pole

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A single pair spread expected to ‘‘revert to its local mean’’ may continue to increase beyond the point at which stop loss limits force exit from the position. This new element, risk, complicates the goal, which now becomes twofold: Maximize expected return and maintain the risk of achieving that return below a certain tolerance. So far so good. Going from foresight to forecast we exchange certainty for uncertainty; we move from guaranteed optimization to constrained optimization of a best guess. However, in practice matters are not quite as straightforward as that sentence seems to imply. The first obstacle is precisely specifying the notion of risk—or, at least, its practical implementation. Risk arises because there is no guarantee that a particular forecast will be borne out in reality. Indeed, the truth is that it would be an extraordinary event if a forecast turned out to be 100 percent accurate.

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Lean In: Women, Work, and the Will to Lead by Sheryl Sandberg

Because no matter what any of us has—and how grateful we are for what we have—no one has it all. Nor can we. The very concept of having it all flies in the face of the basic laws of economics and common sense. As Sharon Poczter, professor of economics at Cornell, explains, “The antiquated rhetoric of ‘having it all’ disregards the basis of every economic relationship: the idea of trade-offs. All of us are dealing with the constrained optimization that is life, attempting to maximize our utility based on parameters like career, kids, relationships, etc., doing our best to allocate the resource of time. Due to the scarcity of this resource, therefore, none of us can ‘have it all,’ and those who claim to are most likely lying.”1 “Having it all” is best regarded as a myth. And like many myths, it can deliver a helpful cautionary message.

pages: 408 words: 85,118

Python for Finance by Yuxing Yan

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Our major function would start from Step 3 as shown in the following code: # Step 3: generate a return matrix (annul return) n=len(ticker) # number of stocks x2=ret_annual(ticker[0],begdate,enddate) for i in range(1,n): x_=ret_annual(ticker[i],begdate,enddate) x2=pd.merge(x2,x_,left_index=True,right_index=True) # using scipy array format R = sp.array(x2) print('Efficient porfolio (mean-variance) :ticker used') print(ticker) [ 216 ] Chapter 8 print('Sharpe ratio for an equal-weighted portfolio') equal_w=sp.ones(n, dtype=float) * 1.0 /n print(equal_w) print(sharpe(R,equal_w)) # for n stocks, we could only choose n-1 weights w0= sp.ones(n-1, dtype=float) * 1.0 /n w1 = fmin(negative_sharpe_n_minus_1_stock,w0) final_w = sp.append(w1, 1 - sum(w1)) final_sharpe = sharpe(R,final_w) print ('Optimal weights are ') print (final_w) print ('final Sharpe ratio is ') print(final_sharpe) From the following output, we know that if we use a naïve equal-weighted strategy, the Sharpe ratio is 0.63. However, the Sharpe ratio for our optimal portfolio is 0.67: Constructing an efficient frontier with n stocks Constructing an efficient frontier is always one of the most difficult tasks for finance instructors since the task involves matrix manipulation and a constrained optimization procedure. One efficient frontier could vividly explain the Markowitz Portfolio theory. The following Python program uses five stocks to construct an efficient frontier: from matplotlib.finance import quotes_historical_yahoo import numpy as np import pandas as pd [ 217 ] Statistical Analysis of Time Series from numpy.linalg import inv, pinv # Step 1: input area begYear,endYear = 2001,2013 stocks=['IBM','WMT','AAPL','C','MSFT'] # Step 2: define a few functions # function 1 def ret_monthly(ticker): x = quotes_historical_yahoo(ticker,(begYear,1,1),(endYear,12,31),asob ject=True,adjusted=True) logret=log(x.aclose[1:]/x.aclose[:-1]) date=[] d0=x.date for i in range(0,size(logret)): date.append(''.join([d0[i].strftime("%Y"),d0[i].strftime("%m")])) y=pd.DataFrame(logret,date,columns=[ticker]) return y.groupby(y.index).sum() # function 2: objective function def objFunction(W, R, target_ret): stock_mean=np.mean(R,axis=0) port_mean=np.dot(W,stock_mean) # portfolio mean cov=np.cov(R.T) # var-cov matrix port_var=np.dot(np.dot(W,cov),W.T) # portfolio variance penalty = 2000*abs(port_mean-target_ret)# penalty 4 deviation return np.sqrt(port_var) + penalty # objective function # Step 3: Generate a return matrix R R0=ret_monthly(stocks[0]) # starting from 1st stock n_stock=len(stocks) # number of stocks for i in xrange(1,n_stock): # then merge with other stocks x=ret_monthly(stocks[i]) R0=pd.merge(R0,x,left_index=True,right_index=True) R=np.array(R0) # Step 4: estimate optimal portfolio for a given return out_mean,out_std,out_weight=[],[],[] stockMean=np.mean(R,axis=0) for r in np.linspace(np.min(stockMean), np.max(stockMean), num=100): [ 218 ] Chapter 8 W = ones([n_stock])/n_stock # starting from equal weights b_ = [(0,1) for i in range(n_stock)] # bounds, here no short c_ = ({'type':'eq', 'fun': lambda W: sum(W)-1. })#constraint result=sp.optimize.minimize(objFunction,W,(R,r),method='SLSQP',constr aints=c_, bounds=b_) if not result.success: # handle error raise BaseException(result.message) out_mean.append(round(r,4)) # 4 decimal places std_=round(np.std(np.sum(R*result.x,axis=1)),6) out_std.append(std_) out_weight.append(result.x) # Step 4: plot the efficient frontier title('Efficient Frontier') xlabel('Standard Deviation of the porfolio (Risk))') ylabel('Return of the portfolio') figtext(0.5,0.75,str(n_stock)+' stock are used: ') figtext(0.5,0.7,' '+str(stocks)) figtext(0.5,0.65,'Time period: '+str(begYear)+' ------ '+str(endYear)) plot(out_std,out_mean,'--') The output graph is presented as follows: [ 219 ] Statistical Analysis of Time Series Understanding the interpolation technique Interpolation is a technique used quite frequently in finance.

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What to Think About Machines That Think: Today's Leading Thinkers on the Age of Machine Intelligence by John Brockman

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He even predicted the tendency to see computer-intensive hill climbing as something cognitively special: “Perhaps what amounts to straightforward hill-climbing on one level may sometimes appear (on a lower level) as the sudden jumps of ‘insight.’” There are other AI algorithms, but most fall into categories Minsky wrote about. One example is running Bayesian probability algorithms on search trees or graphs. They have to grapple with exponential branching or some related form of the curse of dimensionality. Another example is convex or other nonlinear constrained optimization for pattern classification. Italian mathematician Joseph-Louis Lagrange found the general solution algorithm we still use today. He came up with it in 1811. Clever tricks and tweaks will always help. But progress here depends crucially on running these algorithms on ever faster computers. The algorithms themselves consist mainly of vast numbers of additions and multiplications, so they’re not likely to suddenly wake up one day and take over the world.

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

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MSCI Global −0.52 −1.25 17.23 12.56 0.21 1.57% 4.32% 0.63 0.82 0.87 3.24 1.51% 0.30% 28 MSCI Global 1.18% 4.15% 28 Futures Fund Index 50–60% 0.64 2.07 0.19 1.05 3.90% 0.41% 3.88% 28 Futures Fund Index −0.30 −1.32 28.80 13.37 −0.01 2.49% 2.42% 0.32% 28 MSCI Global 60–70% −0.48 0.18 −0.49 −1.23 3.33% 0.12 −1.45 34.47 14.29 0.31 3.66% 28 3.57% 0.43% 28 MSCI Global −0.29% 3.31% Futures Fund Index 70–80% 1.61 4.08 1.98 6.28 5.88% 2.77% 5.15% 28 Futures Fund Index 0.47 −1.16 50.82 15.51 0.38 5.04% 4.94% 0.44% 28 MSCI Global 80–90% 0.93 0.44 0.65 2.47 5.71% 1.25% 5.56% 28 Futures Fund Index 0.82 −0.34 24.41 17.87 0.26 8.02% 7.71% 1.68% 28 MSCI Global 90–100% 356 0/100 10/90 20/80 30/70 40/60 50/50 60/40 70/30 80/20 90/10 100/0 Combination Returns −3.36% −2.86% −2.36% −1.85% −1.35% −0.85% −1.02% 0.16% 0.66% 1.16% 1.67% −7.57% −6.57% −5.57% −4.57% −3.57% −2.57% −3.08% −0.57% 0.43% 1.44% 2.44% 10– 20% Returns 0– 10% −1.86% −1.71% −1.56% −1.42% −1.27% −1.12% −1.34% −0.83% −0.68% −0.53% −0.38% Returns 20– 30% −0.58% −0.49% −0.40% −0.31% −0.22% −0.14% −0.16% 0.04% 0.13% 0.22% 0.31% Returns 30– 40% 0.43% 0.43% 0.44% 0.45% 0.45% 0.46% 0.55% 0.47% 0.48% 0.49% 0.50% Returns 40– 50% 1.51% 1.48% 1.44% 1.41% 1.38% 1.34% 1.61% 1.28% 1.25% 1.21% 1.18% Returns 50– 60% TABLE 19.5 Combining Futures and Equity Indices in Different Proportions 2.42% 2.22% 2.02% 1.82% 1.62% 1.42% 1.70% 1.01% 0.81% 0.61% 0.41% Returns 60– 70% 3.57% 3.18% 2.79% 2.41% 2.02% 1.64% 1.96% 0.87% 0.48% 0.09% −0.29% Returns 70– 80% 90– 100% 4.94% 4.72% 4.50% 4.29% 4.07% 3.85% 4.62% 3.42% 3.20% 2.99% 2.77% 7.71% 7.06% 6.41% 5.77% 5.12% 4.48% 5.37% 3.18% 2.54% 1.89% 1.25% Returns Returns 80– 90% 7.19% 7.45% 7.72% 7.98% 8.25% 8.51% 10.22% 9.04% 9.31% 9.58% 9.84% Portfolio Returns CTA Strategies for Returns-Enhancing Diversification 357 were to look for the least number of negative periods, then the combination of 90/10 would almost ensure that there would only be a 1 in 10 chance of negative returns. The results illustrate a useful idea: If we are concerned about event risk, we may wish to define our objective function as one that has the least number of negative returns during the investment horizon, with the constraint that the correlation at first decile should be the lowest. This could be a useful framework to carry out constrained optimization of portfolio returns. CONCLUSION Our results indicate that the risk-adjusted returns as measured by Sharpe and Sortino ratios are always higher in CTA strategies than in most traditional asset classes for the entire sample period under study. Unlike hedge funds, the correlation coefficients of the CTAs with the equity markets are negative during bad times (worst performance period of the equity markets).

Culture and Prosperity: The Truth About Markets - Why Some Nations Are Rich but Most Remain Poor by John Kay

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He should do just the amount of calculation needed to find the best strategy in the light of his knowledge that every second devoted to calculation increases the chances ofbeing caught by the bear. 18 Borrowing Herbert Simon's term (but for a very different concept), Oliver Williamson calls this optimization under constraints-bounded rationality. 19 In this vein, transactions costs economics often degenerates into a Panglossian view of the world: institutions that exist must be the solution to some constrained-optimization problem. Economists even have a word-recoverability-for deducing the maximization problem to which observed behavior is the answer. But this version of bounded rationality confronts a fundamental problem. How could the economist know when to stop calculating { 220} John Kay when he cannot know the benefits of further calculation? If we knew enough to be boundedly rational, we would know enough to be completely rational.

The Blockchain Alternative: Rethinking Macroeconomic Policy and Economic Theory by Kariappa Bheemaiah

Each generation consists of a population of character strings that are analogous to the chromosome that we see in our DNA. Each individual represents a point in a search space and a possible solution. The individuals in the population are then made to go through a process of evolution. GAs are used for searching through large and complex data sets as they are capable of finding reasonable solutions to complex issues by solving unconstrained and constrained optimization issues. They are used to solve problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. GAs also work particularly well on mixed (continuous and discrete) combinatorial problems, as they are less susceptible to getting ‘stuck’ at local optima than classical gradient search methods.

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Never Let a Serious Crisis Go to Waste: How Neoliberalism Survived the Financial Meltdown by Philip Mirowski

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While the focus on procyclical movement of leverage and asset prices is an important departure from much of orthodox models, the commitment to Walrasian general equilibrium so hampers the formalism that good intentions ended up in Bedlam. Geanakoplos likes to phrase his commentary in terms of “cycles,” but in fact, general equilibrium forces him to posit separate and unconnected sequential states of equilibrium, driven in a Rube Goldberg fashion by the arbitrarily sequenced arrival of asset issuance→“news”→valuation. Each state along the way is in “equilibrium,” as defined by constrained optimization and market clearing. Because everything is always already in equilibrium, the only reason anything changes is the deus ex machina imposed from outside by the model builder. Worse, the sequence itself is completely arbitrary, dictated more by the math than by anything that happens in the world: for instance, each player has only one chance to issue assets, is proscribed from trading them on secondary markets, and consumption can happen only at the initial issuance and at the end of time.108 It seems that, however proud he may be of hewing to Walrasian heuristics, the suite of models he has constructed rarely do much to actually illuminate the actual crisis and aftermath.