Need for . . . First Step: Computing . . . Interval Computation . . . Formulation of the . . . Constraint Optimization: Main Result From Efficient Computation Additional Result Comparison to Interval . . . of What Can Be Achieved to Algorithm: General . . . Proof that Our . . . Efficient Computation of Home Page How to Achieve The Title Page Corresponding Optimum ◭◭ ◮◮ ◭ ◮ Ali Jalal-Kamali, Martine Ceberio, Page 1 of 17 and Vladik Kreinovich Go Back Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA Full Screen ajalalkamai@miners.utep.edu, mceberio@utep.edu, vladik@utep.edu Close Quit
Need for . . . First Step: Computing . . . 1. Need for Optimization: General Reminder Interval Computation . . . • In many practical situations, we need to select the best Formulation of the . . . alternative: Main Result Additional Result – a location of a plant, Comparison to Interval . . . – values of the control to apply to a system, etc. Algorithm: General . . . • Let n be the total number of parameters x 1 , . . . , x n Proof that Our . . . needed to uniquely determine an alternative. Home Page • For each parameter x i , we know the range x i = [ x i , x i ] Title Page of its possible values. ◭◭ ◮◮ • The “best” alternative is defined as the one for which ◭ ◮ an appropriate objective function f ( x 1 , . . . , x n ) is max. Page 2 of 17 • It is reasonable to assume that the objective function Go Back is feasibly computable. Full Screen • Then, the problem is to find the best values x 1 , . . . , x n for which f ( x 1 , . . . , x n ) → max. Close Quit
Need for . . . First Step: Computing . . . 2. First Step: Computing the Largest Possible Interval Computation . . . Value of the Objective Function Formulation of the . . . • It often makes sense to first check what we can, in Main Result principle, achieve within the given setting. Additional Result Comparison to Interval . . . • Example: if min possible pollution of a coal-burning Algorithm: General . . . steam engine is too high, look for different engines. Proof that Our . . . • So, we need to compute the max y (or min y ) of a given Home Page function f ( x 1 , . . . , x n ) over given intervals x i . Title Page • The problem of computing the range [ y, y ] of the func- ◭◭ ◮◮ tion under x i ∈ x i is known as interval computations . ◭ ◮ • The values y and y are, in general, irrational and thus, Page 3 of 17 cannot be exactly computer represented. Go Back • So, what we need is, given any rational number ε > 0, compute r and r s.t. | r − y | ≤ ε and | r − y | ≤ ε . Full Screen Close Quit
Need for . . . First Step: Computing . . . 3. Interval Computation Is, in General, NP-hard Interval Computation . . . • It is known that in general, the problem of computing Formulation of the . . . the corresponding range is NP-hard. Main Result Additional Result • This means, crudely speaking, that it is not possible Comparison to Interval . . . to have: Algorithm: General . . . – a feasible algorithm Proof that Our . . . – that would always compute the desired range. Home Page • Because of this, it is important to find: Title Page ◭◭ ◮◮ – practically useful classes of problems – for which it is feasibly possible to compute this ◭ ◮ range. Page 4 of 17 • Many such classes are known. Go Back Full Screen Close Quit
Need for . . . First Step: Computing . . . 4. Formulation of the Problem Interval Computation . . . • In practice, we often have additional constraints of Formulation of the . . . equality or inequality type. Main Result Additional Result • In such situations, it is necessary to restrict ourselves Comparison to Interval . . . only to values ( x 1 , . . . , x n ) which satisfy these constraints. Algorithm: General . . . • Once we know the largest value, we need to find the Proof that Our . . . values x 1 , . . . , x n that lead to this largest value. Home Page • At present: Title Page – once we have developed an algorithm for computing ◭◭ ◮◮ the max of a given function f ( x 1 , . . . , x n ), ◭ ◮ – we need to develop a second algorithm – for locating Page 5 of 17 this largest value. Go Back • In this talk, we describe a general technique for gener- Full Screen ating the second algorithm once the first one is known. Close Quit
Need for . . . First Step: Computing . . . 5. Main Result Interval Computation . . . • Let F be a class of functions, and let C be a class of Formulation of the . . . constraints. Main Result Additional Result • We consider two problems, in both we are given: Comparison to Interval . . . – a f-n f ( x 1 , . . . , x n ) ∈ F and constraints C ∈ C , Algorithm: General . . . – rational-valued intervals [ x 1 , x 1 ] , . . . , [ x n , x n ], and Proof that Our . . . – a rational number ε > 0, Home Page • Problem 1: compute rational values r and r which are Title Page ε -close to the endpoints y and y of the range ◭◭ ◮◮ [ y, y ] = { f ( x 1 , . . . , x n ) : x i ∈ [ x i , x i ] , ( x 1 , . . . , x n ) ∈ C } . ◭ ◮ Page 6 of 17 • Problem 2: compute rational r 1 , . . . , r n s.t. f ( x 1 , . . . , x n ) ≥ y − ε for some x i which are ε -close to r i and satisfy C . Go Back • Main Result: once we have a feasible algorithm for Full Screen solving Problem 1, we can feasible solve Problem 2. Close Quit
Need for . . . First Step: Computing . . . 6. Additional Result Interval Computation . . . • Reminder: we compute rat. r 1 , . . . , r n s.t. f ( x 1 , . . . , x n ) ≥ Formulation of the . . . y − ε for some x i which are ε -close to r i and satisfy C . Main Result Additional Result • Important case: Comparison to Interval . . . – there are no additional constraints, only interval Algorithm: General . . . bounds x i ≤ x i ≤ x i , and Proof that Our . . . – we can also feasibly compute the bound M on all Home Page partial derivatives of a function f . Title Page • In this case, we can also feasibly produce: ◭◭ ◮◮ – given a rational number ε > 0, ◭ ◮ – rational values r 1 , . . . , r n for which already for these Page 7 of 17 values r i , we have Go Back f ( r 1 , . . . , r n ) ≥ y − ε. Full Screen Close Quit
Need for . . . First Step: Computing . . . 7. Comparison to Interval Computations Interval Computation . . . • Locating maxima is one of the main applications of Formulation of the . . . interval computations in optimization; main idea: Main Result Additional Result – use interval computations to find the enclosure of Comparison to Interval . . . a function on subboxes; Algorithm: General . . . – compute values in the subboxes’ midpoints; Proof that Our . . . – compute maximum-so-far as the maximum of all Home Page midpoint values; Title Page – and then dismiss the subboxes for which the upper bound is smaller than the maximum-so-far; ◭◭ ◮◮ – bisect remaining boxes. ◭ ◮ • What is new: Page 8 of 17 – the above idea can take exponential time – by re- Go Back quiring us to consider 2 n sub-boxes, while Full Screen – the computation time for our algorithm is always Close feasible (polynomial). Quit
Need for . . . First Step: Computing . . . 8. Constraints-Based Intuitive Explanation of Our Interval Computation . . . Result Formulation of the . . . • There are two different constraint problems: Main Result Additional Result – constraint satisfaction – finding values that satisfy Comparison to Interval . . . given constraints, and Algorithm: General . . . – constraint optimization – among all values that sat- Proof that Our . . . isfy constraints, find the ones for which f → max. Home Page • It is clear that constraint optimization is harder than Title Page constraint satisfaction. ◭◭ ◮◮ • Once we know y = max f , locating max becomes a ◭ ◮ constraint satisfaction problem: just add a constraint Page 9 of 17 f ( x 1 , . . . , x n ) ≥ y − ε. Go Back • Thus, to locate the maximum, it is sufficient to solve Full Screen an easier-to-solve constraint satisfaction problem. Close Quit
Need for . . . First Step: Computing . . . 9. Algorithm: General Overview Interval Computation . . . • At each stage of this algorithm, we will have a box B k . Formulation of the . . . Main Result • We start with the original box B 0 = B . Additional Result • Then, we repeatedly decrease the x 1 -size of this box in Comparison to Interval . . . half until its size is smaller than or equal to 2 ε . Algorithm: General . . . • After this, we decrease the x 2 -size of this box in half, Proof that Our . . . etc., until all n sizes are bounded by 2 ε . Home Page Title Page • For each side, we start with the interval [ x i , x i ] of width w i = x i − x i . ◭◭ ◮◮ • After s i bisection steps, the width decreases to w i · 2 − s i . ◭ ◮ � w i � �� Page 10 of 17 • One can see that we need ln steps to reach the 2 ε desired size ( ≤ 2 ε ) of the i -th side. Go Back n �� w i Full Screen �� def � • Overall, we need s = ln bisection steps. 2 ε i =1 Close Quit
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