Design and Control . . . How to Describe . . . Constraint . . . What Are Soft . . . Priority Approach to . . . A Computational . . . Towards an Optimal Priority Approach to . . . When Is a Method . . . Approach to Soft Constraint Main Result Proof (cont-d) Problems Acknowledgments What If Constraints . . . Proof of NP-hardness Martine Ceberio and Vladik Kreinovich Constraint . . . Department of Computer Science New Idea University of Texas at El Paso, 500 W. University New Algorithm El Paso, TX 79968, USA, { mceberio,vladik } @cs.utep.edu Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 1. Design and Control Problems Constraint . . . What Are Soft . . . • In many areas of science and engineering, we are interested in solving design Priority Approach to . . . and control problems. A Computational . . . Priority Approach to . . . • In mathematical terms: a design or a control can be usually represented by When Is a Method . . . the values of the relevant numerical parameters x = ( x 1 , . . . , x n ). Main Result • Usually, in these problems, the users describe several constraints that the Proof (cont-d) desired design or control must satisfy. Acknowledgments What If Constraints . . . • Objective: find a design (corr., a control) that satisfies all these constraints. Proof of NP-hardness Constraint . . . New Idea New Algorithm Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 2. How to Describe Constraints? Constraint . . . What Are Soft . . . • Example: an airplane design can be described in terms of: Priority Approach to . . . A Computational . . . – the geometric parameters of the plane, Priority Approach to . . . – the thickness of the plates that form the airplane’s skin, When Is a Method . . . – the weight and power of the engine, etc. Main Result Proof (cont-d) • Typical constraint: a limitation on some characteristics y = f ( x 1 , . . . , x n ) of Acknowledgments this design. What If Constraints . . . • Examples Proof of NP-hardness Constraint . . . – the airplane’s speed must exceed some y 0 , New Idea – its fuel use must not exceed a certain amount, New Algorithm – the overall cost must be within given limits. Title Page • So, constraints are of the type f ( x 1 , . . . , x n ) ≤ y 0 or f ( x 1 , . . . , x n ) ≥ y 0 (or ◭◭ ◮◮ f ( x 1 , . . . , x n ) = y 0 ). ◭ ◮ Page 3 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 3. Constraint Satisfaction vs. Constrained Optimiza- Constraint . . . What Are Soft . . . tion Priority Approach to . . . A Computational . . . • Constraint satisfaction: find a design that satisfies given constraints. Priority Approach to . . . • Problem: When Is a Method . . . Main Result – different designs that satisfy the given constraints; Proof (cont-d) – we must select one of these designs. Acknowledgments What If Constraints . . . • Users can often describe their preference in terms of an objective function Proof of NP-hardness g ( x 1 , . . . , x n ) (whose value should be made as large as possible). Constraint . . . • Constrained optimization: maximizing g ( x 1 , . . . , x n ) under the given con- New Idea straints. New Algorithm • In general: both problem are NP-hard. Title Page • In practice: there are many efficient tools for solving them. ◭◭ ◮◮ ◭ ◮ Page 4 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 4. What Are Soft Constraints? Constraint . . . What Are Soft . . . • Problem: sometimes, the users constraints are inconsistent. Priority Approach to . . . A Computational . . . • Example: design a plane that is: Priority Approach to . . . When Is a Method . . . – as fast and as fuel-efficient as the existing Airbus or Boeing planes, Main Result – but with 0 noise level. Proof (cont-d) • Reasons for inconsistency: Acknowledgments What If Constraints . . . – some constraints are absolute (e.g., safety constraints), Proof of NP-hardness – others are desires – they can be dismissed if not possible. Constraint . . . New Idea • Such “not required” constraints are called soft constraints . New Algorithm • Comment: soft constraints are an important research topic, with annual con- Title Page ferences. ◭◭ ◮◮ ◭ ◮ Page 5 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 5. Priority Approach to Soft Constraints: A Brief De- Constraint . . . What Are Soft . . . scription Priority Approach to . . . A Computational . . . • Idea: Priority Approach to . . . – when we cannot satisfy all the constraints, When Is a Method . . . Main Result – we should satisfy as many constraints as possible. Proof (cont-d) • Natural idea: Acknowledgments What If Constraints . . . – ask the user to prioritize their constraints C i , from the absolutely re- Proof of NP-hardness quired to the less required: Constraint . . . C 1 ≻ C 2 ≻ . . . ≻ C n ; New Idea New Algorithm – find the largest possible value k = k opt for which all the constraints Title Page C 1 , C 2 , . . . , C k are still consistent. ◭◭ ◮◮ ◭ ◮ Page 6 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 6. A Computational Question Constraint . . . What Are Soft . . . • Constraint satisfaction tools can check consistency. Priority Approach to . . . A Computational . . . • Possibility: apply a tool to { C 1 } , { C 1 , C 2 } , . . . , until we get inconsistency. Priority Approach to . . . When Is a Method . . . • Problem: for large number of constraints (1 ≪ k opt ≪ n ), we need too many iterations. Main Result Proof (cont-d) • Alternative idea: use iterative bisection: Acknowledgments – at each stage, we have an interval [ k − , k + ] ∋ k opt ; What If Constraints . . . – initially, k − = 0 and k + = n ; Proof of NP-hardness Constraint . . . – at each stage, we check consistency for the midpoint New Idea New Algorithm = ⌊ ( k − + k + ) / 2 ⌋ , def k m Title Page and replace the interval with a half-size one: [ k − , k m ] or [ k m , k + ]. ◭◭ ◮◮ • Problem: on some stages, too many ( n ≫ k opt ) constraints, takes too long. ◭ ◮ • Question: which method for finding k opt is optimal? Page 7 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 7. Priority Approach to Soft Constraints: Toward For- Constraint . . . What Are Soft . . . malizing the Computational Question Priority Approach to . . . A Computational . . . • Definition: A method is a mapping that maps each pair ( I, s ), where: Priority Approach to . . . – I is an integer-valued interval [ k − , k + ], where 0 ≤ k − < k + ≤ n and When Is a Method . . . k + > k − + 1, and Main Result Proof (cont-d) – s is a positive integer (= number of step) Acknowledgments into an integer k next from the open interval ( k − , k + ). What If Constraints . . . • Meaning: first k − constraints are consistent, but first k + constraints are not. Proof of NP-hardness Constraint . . . • Examples: New Idea – sequential search: k next = k − + 1; New Algorithm – bisection: k next = ⌊ ( k − + k + ) / 2 ⌋ . Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 8. When Is a Method Optimal? Constraint . . . What Are Soft . . . • Known fact: constraint satisfaction is NP-hard. Priority Approach to . . . A Computational . . . • Meaning: crudely speaking, computational complexity of a system of k con- Priority Approach to . . . straints is ∼ 2 k . When Is a Method . . . • More precisely: t ∼ p k for p ≥ 2. Main Result Proof (cont-d) • Assumption: t = C · p k . Acknowledgments What If Constraints . . . • Definition: for a method M , T M ( k ) is defined as the worst-case overall time this method spends on checking when k opt = k . Proof of NP-hardness Constraint . . . • Ideal case: we only check that k are consistent and k + 1 are not, with time New Idea p k + p k +1 . New Algorithm T M ( k ) def Title Page • Overhead: O p ( M ) = max p k + p k +1 . k ◭◭ ◮◮ • Objective: we want to find a method M opt with the smallest overhead, i.e., for which O p ( M opt ) = min M O p ( M ). ◭ ◮ Page 9 of 17 Go Back Full Screen Close Quit
Design and Control . . . How to Describe . . . 9. Main Result Constraint . . . What Are Soft . . . • Theorem. For every p ≥ 2, the sequential search method S is optimal. Priority Approach to . . . A Computational . . . • Proof. For sequential search S , Priority Approach to . . . T S ( k ) = p + p 2 + . . . + p k +1 = When Is a Method . . . Main Result p k +1 · (1 + p − 1 + p − 2 + . . . + p − k ) < Proof (cont-d) Acknowledgments p k +1 p k +1 · (1 + p − 1 + p − 2 + . . . ) = (1 − p − 1 ) . What If Constraints . . . Proof of NP-hardness Since p k + p k +1 = p k +1 · (1 + p − 1 ), we conclude that Constraint . . . New Idea 1 O p ( S ) < (1 − p − 1 ) · (1 + p − 1 ) . New Algorithm Title Page • In a method M � = S , there exists an interval in which k next ≥ k − + 2. ◭◭ ◮◮ • So, if k opt = k − , we check both k and ≥ k + 2; we also check k + 1 – to make sure that k is indeed the largest. ◭ ◮ Page 10 of 17 Go Back Full Screen Close Quit
Recommend
More recommend
