Outline Formulation of the . . . Because of NP- . . . Enter Randomness How Design Quality Which Random Variable? Let Us Use Max- . . . Improves with Increasing Resulting Formula: . . . Computational Abilities: How to Apply Our . . . Caution General Formulas and Home Page Case Study of Title Page ◭◭ ◮◮ Aircraft Fuel Efficiency ◭ ◮ Joe Lorkowski 1 , Olga Kosheleva 1 , Page 1 of 15 Vladik Kreinovich 1 , and Sergei Soloviev 2 Go Back 1 University of Texas at El Paso, El Paso, TX 79968, USA lorkowski@computer.org, olgak@utep.edu, vladik@utep.edu Full Screen 2 Institut de Recherche en Informatique de Toulouse (IRIT) Toulouse, France, sergei.soloviev@irit.fr Close Quit
Outline Formulation of the . . . 1. Outline Because of NP- . . . • It is known that the problems of optimal design are Enter Randomness NP-hard. Which Random Variable? Let Us Use Max- . . . • This means that, in general, a feasible algorithm can Resulting Formula: . . . only produce close-to-optimal designs. How to Apply Our . . . • The more computations we perform, the better design Caution we can produce. Home Page • In this paper, we theoretically derive the dependence Title Page of design quality on computation time. ◭◭ ◮◮ • We then empirically confirm this dependence on the ◭ ◮ example of aircraft fuel efficiency. Page 2 of 15 Go Back Full Screen Close Quit
Outline Formulation of the . . . 2. Formulation of the Problem Because of NP- . . . • Since 1980s, computer-aided design (CAD) has become Enter Randomness ubiquitous in engineering; example: Boeing 777. Which Random Variable? Let Us Use Max- . . . • The main objective of CAD is to find a design which Resulting Formula: . . . optimizes the corresponding objective function. How to Apply Our . . . • Example: we optimize fuel efficiency of an aircraft. Caution Home Page • The corresponding optimization problems are non-linear, and such problems are, in general, NP-hard. Title Page • So – unless P = NP – a feasible algorithm cannot al- ◭◭ ◮◮ ways find the exact optimum, only an approximate one. ◭ ◮ • The more computations we perform, the better the de- Page 3 of 15 sign. Go Back • It is desirable to quantitatively describe how increasing Full Screen computational abilities improve the design quality. Close Quit
Outline Formulation of the . . . 3. Because of NP-Hardness, More Computations Because of NP- . . . Simply Means More Test Cases Enter Randomness • In principle, each design optimization problem can be Which Random Variable? solved by exhaustive search. Let Us Use Max- . . . Resulting Formula: . . . • Let d denote the number of parameters. How to Apply Our . . . • Let C denote the average number of possible values of Caution a parameter. Home Page • Then, we need to analyze C d test cases. Title Page • For large systems (e.g., for an aircraft), we can only ◭◭ ◮◮ test some combinations. ◭ ◮ • NP-hardness means that optimization algorithms to be Page 4 of 15 significantly faster than exponential time C d . Go Back • This means that, in effect, all possible optimization al- Full Screen gorithms boil down to trying many possible test cases. Close Quit
Outline Formulation of the . . . 4. Enter Randomness Because of NP- . . . • Increasing computational abilities mean that we can Enter Randomness test more cases. Which Random Variable? Let Us Use Max- . . . • Thus, by increasing the scope of our search, we will Resulting Formula: . . . hopefully find a better design. How to Apply Our . . . • Since we cannot do significantly better than with a Caution simple search, Home Page – we cannot meaningfully predict whether the next Title Page test case will be better or worse, ◭◭ ◮◮ – because if we could, we would be able to signifi- ◭ ◮ cantly decrease the search time. Page 5 of 15 • The quality of the next test case cannot be predicted Go Back and is, in this sense, a random variable. Full Screen Close Quit
Outline Formulation of the . . . 5. Which Random Variable? Because of NP- . . . • Many different factors affect the quality of each indi- Enter Randomness vidual design. Which Random Variable? Let Us Use Max- . . . • Usually, the distribution of the resulting effect of sev- Resulting Formula: . . . eral independent random factors is close to Gaussian. How to Apply Our . . . • This fact is known as the Central Limit Theorem . Caution Home Page • Thus, the quality of a (randomly selected) individual design is normally distributed, with some µ and σ . Title Page • After we test n designs, the quality of the best-so-far ◭◭ ◮◮ design is x = max( x 1 , . . . , x n ). ◭ ◮ • We can reduce the case of y i with µ = 0 and σ = 1: Page 6 of 15 namely, x i = µ + σ · y i hence x = µ + σ · y, where Go Back def = max( y 1 , . . . , y n ) . y Full Screen Close Quit
Outline Formulation of the . . . 6. Let Us Use Max-Central Limit Theorem Because of NP- . . . � y − µ n � Enter Randomness • For large n , y ’s cdf is F ( y ) ≈ F EV , where: σ n Which Random Variable? def Let Us Use Max- . . . • F EV ( y ) = exp( − exp( − y )) ( Gumbel distribution ), Resulting Formula: . . . � 1 − 1 � def = Φ − 1 , where Φ( y ) is cdf of N (0 , 1), • µ n How to Apply Our . . . n Caution � 1 − 1 � � 1 − 1 � def n · e − 1 = Φ − 1 − Φ − 1 • σ n . Home Page n Title Page • Thus, y = µ n + σ n · ξ , where ξ is distributed according to the Gumbel distribution. ◭◭ ◮◮ • The mean of ξ is the Euler’s constant γ ≈ 0 . 5772. ◭ ◮ • Thus, the mean value m n of y is equal to µ n + γ · σ n . Page 7 of 15 � • For large n , we get asymptotically m n ∼ γ · 2 ln( n ) . Go Back • Hence the mean value e n of x = µ + σ · y is asymptot- Full Screen � ically equal to e n ∼ µ + σ · γ · 2 ln( n ) . Close Quit
Outline Formulation of the . . . 7. Resulting Formula: Let Us Test It Because of NP- . . . • Situation: we test n different cases to find the optimal Enter Randomness design. Which Random Variable? Let Us Use Max- . . . • Conclusion: the quality e n of the resulting design in- Resulting Formula: . . . creases with n as How to Apply Our . . . � e n ∼ µ + σ · γ · 2 ln( n ) . Caution Home Page • We test this formula: on the example of the average Title Page fuel efficiency E of commercial aircraft. ◭◭ ◮◮ • Empirical fact: E changes with time T as ◭ ◮ E = exp( a + b · ln( T )) = C · T b , for b ≈ 0 . 5 . Page 8 of 15 � • Question: can our formula e n ∼ µ + σ · γ · 2 ln( n ) Go Back explain this empirical dependence? Full Screen Close Quit
Outline Formulation of the . . . 8. How to Apply Our Theoretical Formula to This Because of NP- . . . Case? Enter Randomness � • The formula q ∼ µ + σ · γ · 2 ln( n ) describes how the Which Random Variable? quality changes with the # of computational steps n . Let Us Use Max- . . . Resulting Formula: . . . • In the case study, we know how it changes with time T . How to Apply Our . . . • According to Moore’s law , the computational speed Caution grows exponentially with time T : n ≈ exp( c · T ). Home Page • Crudely speaking, the computational speed doubles ev- Title Page ery two years. ◭◭ ◮◮ • When n ≈ exp( c · T ), we have ln( n ) ∼ T ; thus, ◭ ◮ √ q ≈ a + b · T. Page 9 of 15 Go Back • This is exactly the empirical dependence that we actu- ally observe. Full Screen Close Quit
Outline Formulation of the . . . 9. Caution Because of NP- . . . • Idea: cars also improve their fuel efficiency. Enter Randomness Which Random Variable? • Fact: the dependence of their fuel efficiency on time is Let Us Use Max- . . . piece-wise constant. Resulting Formula: . . . • Explanation: for cars, changes are driven mostly by How to Apply Our . . . federal and state regulations. Caution Home Page • Result: these changes have little to do with efficiency of Computer-Aided design. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 15 Go Back Full Screen Close Quit
Outline Formulation of the . . . 10. Acknowledgments Because of NP- . . . This work was supported in part: Enter Randomness Which Random Variable? • by the National Science Foundation grants: Let Us Use Max- . . . – HRD-0734825 and HRD-1242122 Resulting Formula: . . . (Cyber-ShARE Center of Excellence) and How to Apply Our . . . – DUE-0926721, Caution Home Page • by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and Title Page ◭◭ ◮◮ • by grant N62909-12-1-7039 from the Office of Naval Research. ◭ ◮ Page 11 of 15 Go Back Full Screen Close Quit
Outline Formulation of the . . . 11. Bibliography: CAD Because of NP- . . . • D. A. Madsen and D. P. Madsen, Engineering Drawing Enter Randomness and Design , Delmar, Cengage Learning, Clifton Park, Which Random Variable? New York, 2012. Let Us Use Max- . . . Resulting Formula: . . . • K. Sabbagh, Twenty-First-Century Jet: The Making How to Apply Our . . . and Marketing of the Boeing 777 , Scribner, New York, Caution 1996. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 15 Go Back Full Screen Close Quit
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