Commonsense . . . Sometimes . . . Analysis of the Problem Related Known . . . Adding Constraints – A Uniqueness Implies . . . (Seemingly Counterintuitive Resulting . . . Resulting . . . but) Useful Heuristic in Caution Acknowledgments Solving Difficult Problems Home Page Title Page Olga Kosheleva, Martine Ceberio, and Vladik Kreinovich ◭◭ ◮◮ ◭ ◮ University of Texas at El Paso El Paso, TX 79968, USA Page 1 of 12 olgak@utep.edu, mceberio@utep.edu vladik@utep.edu Go Back Full Screen Close Quit
Commonsense . . . Sometimes . . . 1. Commonsense Intuition: The More Constraints, Analysis of the Problem The More Difficult The Problem Related Known . . . • If we want to hire a lecturer in Computer Science, this Uniqueness Implies . . . is reasonably easy. Resulting . . . Resulting . . . • However, once we impose constraints on research record Caution etc., hiring becomes complicated. Acknowledgments • If a person coming to a conference is looking for a hotel Home Page to stay, this is usually an easy problem to solve. Title Page • But once you add constraints on how far this hotel is ◭◭ ◮◮ from the conference site, the problem becomes difficult. ◭ ◮ • Similarly, in numerical computations, Page 2 of 12 – unconstrained optimization problems are usually Go Back reasonably straightforward to solve, but Full Screen – once we add constraints, the problems often be- come much more difficult. Close Quit
Commonsense . . . Sometimes . . . 2. Sometimes Constraints Help: A Seemingly Coun- Analysis of the Problem terintuitive Phenomenon Related Known . . . • Mathematicians often aim for an optimal control or an Uniqueness Implies . . . optimal design. Resulting . . . Resulting . . . • To a practitioner, this may seem like a waste of time: Caution once we are within ε of the maximum, we can stop. Acknowledgments • However, algorithmically, it is often easier to find x Home Page s.t. f ( x ) ≥ f 0 by finding x max s.t. f ′ ( x max ) = 0. Title Page • A challenging theorem often becomes proven when we ◭◭ ◮◮ look for proofs of a more general result. ◭ ◮ • In physics, equations were found when additional beauty constraints were imposed (Einstein, Bolzmann). Page 3 of 12 • In art, many great objects were designed within strict Go Back requirements on shape, form, etc. Full Screen • How to explain this counter-intuitive phenomenon? Close Quit
Commonsense . . . Sometimes . . . 3. Analysis of the Problem Analysis of the Problem • By definition: Related Known . . . Uniqueness Implies . . . – when we impose an additional constraint, Resulting . . . – some alternatives which were originally solutions Resulting . . . stop being solutions – Caution – since we impose extra constraints, constraints that Acknowledgments are not always satisfied by all original solutions. Home Page • Thus, the effect of adding a constraint is that the num- Title Page ber of solution decreases. ◭◭ ◮◮ • At the extreme, when we have added the largest pos- ◭ ◮ sible number of constraints, we get a unique solution. Page 4 of 12 • It turns out that this indeed explains why adding con- Go Back straints can make the problems easier. Full Screen Close Quit
Commonsense . . . Sometimes . . . 4. Related Known Results: The Fewer Solutions, Analysis of the Problem the Easier to Solve the Problem Related Known . . . • Many numerical problems are, in general, algorithmi- Uniqueness Implies . . . cally undecidable: Resulting . . . Resulting . . . – no algorithm can always find a solution to an algo- Caution rithmically defined system of equation; Acknowledgments – no algorithm can always find a location of the max- Home Page imum of an algorithmically defined function, etc. Title Page • The proofs of most algorithmic non-computability re- ◭◭ ◮◮ sults essentially use: ◭ ◮ – functions which have several maxima, Page 5 of 12 – equations which have several solutions, etc. Go Back • It turned out that this is not an accident: uniqueness actually implies algorithmic computability. Full Screen Close Quit
Commonsense . . . Sometimes . . . 5. Uniqueness Implies Algorithmic Computability Analysis of the Problem • This result was applied to design many algorithms: Related Known . . . Uniqueness Implies . . . – optimal approximation of functions; Resulting . . . – reconstructing a convex body from its internal met- Resulting . . . ric; Caution – constructing a shortest path in a curved space, etc. Acknowledgments • On the other hand, it was proven that: Home Page – a general algorithm is not possible for functions Title Page that have exactly two global maxima; ◭◭ ◮◮ – a general algorithm is not possible for systems that ◭ ◮ have exactly two solutions. Page 6 of 12 • Moreover, there are results showing that for every m : Go Back – problems with exactly m solutions are, in general, more computationally difficult Full Screen – than problems with m − 1 solutions. Close Quit
Commonsense . . . Sometimes . . . 6. Resulting Recommendation: Applied Math Analysis of the Problem • The above discussion leads to the following seemingly Related Known . . . counter-intuitive recommendation: Uniqueness Implies . . . Resulting . . . – if a problem turns out to be too complex to solve, Resulting . . . – maybe a good heuristic is to add constraints and Caution make it more complex. Acknowledgments • For example: Home Page – if it is difficult to solve an applied mathematical Title Page problem, ◭◭ ◮◮ – maybe a good idea is not to simplify this problem ◭ ◮ but rather to make it more realistic. Page 7 of 12 • Indeed, applied mathematicians know that often, Go Back – learning more about the physical or engineering problem Full Screen – helps to solve this problem. Close Quit
Commonsense . . . Sometimes . . . 7. Resulting Recommendation: Education Analysis of the Problem • This can also be applied to education: Related Known . . . Uniqueness Implies . . . – if students have a hard time solving a class of prob- Resulting . . . lems, Resulting . . . – maybe a good idea is not to make these problems Caution easier, but to make them more complex. Acknowledgments • This may sound counter-intuitive. Home Page • However, in pedagogy, it is a known fact: Title Page – if a school is failing, ◭◭ ◮◮ – the solution is usually not to make classes easier – ◭ ◮ this will lead to a further decline in knowledge; Page 8 of 12 – a turnaround often happens when a new teacher Go Back starts giving challenging problems to students. Full Screen • This is in line with a general American idea – that to be satisfying, the job must be a challenge. Close Quit
Commonsense . . . Sometimes . . . 8. Caution Analysis of the Problem • Of course, it is important: Related Known . . . Uniqueness Implies . . . – not to introduce so many constraints – Resulting . . . – because then, the problem simply stops having so- Resulting . . . lutions at all. Caution • It is difficult to guess which level of constraints will Acknowledgments lead to inconsistency. Home Page • Thus, it may be a good idea: Title Page ◭◭ ◮◮ – to simultaneously try to solve several different ver- sions of the original problem, ◭ ◮ – with different number of constraints added. Page 9 of 12 • This way, we will hopefully be able to successfully solve Go Back one of these versions. Full Screen Close Quit
Commonsense . . . Sometimes . . . 9. Acknowledgments Analysis of the Problem This work was supported in part: Related Known . . . Uniqueness Implies . . . • by National Science Foundation grants HRD-0734825, Resulting . . . EAR-0225670, and DMS-0532645 and Resulting . . . • by Grant 1 T36 GM078000-01 from the National Insti- Caution tutes of Health. Acknowledgments Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 12 Go Back Full Screen Close Quit
Commonsense . . . Sometimes . . . 10. Proof: Main Idea Analysis of the Problem • To compute x 0 s.t. f ( x 0 ) = 0 with accuracy ε > 0, take Related Known . . . an ( ε/ 4)-net { x 1 , . . . , x n } ⊆ K . Uniqueness Implies . . . • For each i , we can compute ε ′ ∈ ( ε/ 4 , ε/ 2) for which Resulting . . . Resulting . . . def = { x : d ( x, x i ) ≤ ε ′ } is a computable compact set. B i Caution def • Thus, we can compute m i = min {| f ( x ) | : x ∈ B i } . Acknowledgments f ( x ) = 0 & d ( x, x i ) < ε Home Page � � • If m i = 0, then ∃ x , hence 2 Title Page d ( x i , x 0 ) ≤ ε 2. ◭◭ ◮◮ • So, if m i = 0 and m j = 0 then ◭ ◮ d ( x i , x j ) ≤ d ( x i , x 0 ) + d ( x 0 , x j ) ≤ ε 2 + ε Page 11 of 12 2 = ε. Go Back • Vice versa, if d ( x i , x 0 ) > ε > 0, we get m i > 0. Full Screen Close Quit
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