Massimo Clementi University of Trento 26 May 2020 Introduction to Multi-Objective Optimization Chapter 40 - LIONBook
Brief recap Goal: find the optimal point(s) of a model, for which no other point is better Ω is the set of possible points f ( x ) , x * : f ( x *) ≤ f ( x ) ∀ x ∈ Ω (or ) ≥ More generally: f ( x *) minimize/maximize f ( x ) x * x subject to x ∈ Ω , 2
Brief recap Example f ( x ) : R n → R and x 1 , x 2 ∈ Ω Suppose: - minimization task - , f ( x 1 ) = 10 f ( x 2 ) = 30 Can we determine which is the better solution between the two? therefore trivial f ( x 1 ) < f ( x 2 ) x 1 → 3
Multi-Objective optimization Example f ( x ) = { f 1 ( x ), f 2 ( x )} : R n → R 2 and x 1 , x 2 ∈ Ω (objectives) Suppose: - minimize and maximize at the same time f 1 ( x ) f 2 ( x ) - [15, 15] and [30, 30] f ( x 1 ) = f ( x 2 ) = Can we determine which is the better solution between the two? > Not trivial 4
Multi-Objective optimization Mathematical formulation Statement: minimize f ( x ) = { f 1 ( x ), . . . , f m ( x )} subject to x ∈ Ω where: x ∈ R n are the variables and x ∈ Ω feasible region f : Ω → R m is made of objective functions m As anticipated before, the problem is ill-posed when the objective functions are conflicting, it is not possible to optimize the objectives independently 5
Multi-Objective optimization Mathematical formulation f ( x k ) = { f 1 ( x k ), . . . , f m ( x k )} x k ∈ Ω region of feasible region objective points Input space Objective space 6
Multi-Objective optimization For a non-trivial multi-objective optimization problem, objectives are conflicting and it is not possible to find a solution that optimize all objectives at the same time. What we have to do is to evaluate tradeoffs 7
̂ Pareto Optimality • Define the objective vector : z = f ( x ) = { f 1 ( x ), . . . , f m ( x )} • Consider a minimization task, an objective vector is said to z dominate if and z ′ z k ≤ z ′ k ∀ k ∃ h such as z k < z ′ k • A point is Pareto-optimal if there x is no other such that f ( x ) x ∈ Ω dominates f ( ̂ x ) 8
Pareto Optimality Pareto frontier Pareto frontier • The Pareto frontier is made by the set of all the Pareto-optimal solutions • Only on the Pareto frontier it makes sense to consider tradeo ff s , because for points outside of it the solution would be suboptimal 9
Pareto Optimality Example Problem: find best airplane tickets (minimize price and maximize comfort) Price Comfort Price Comfort Price Comfort A A 70 € 10 70 € 10 A 70 € 10 B 50 € 7 B 50 € 7 B 50 € 7 C 65 € 6 C 65 € 6 40 € 5 D D 40 € 5 D 40 € 5 Pareto frontier B dominates C but B price ≤ C price B comfort ≥ C comfort 10
Pareto Optimality We can explicit tradeoffs between objectives and find the optimal points in the Pareto frontier applying a combination of the objectives. g ( x , w ) = w 1 f 1 ( x ) + w 2 f 2 ( x ) and then minimize/maximize g ( x ) subject to x ∈ Ω , of the linear combination are unknown Problem: weights w 1 , w 2 11
Multi-Objective optimization To sum up • MOOP consist in multiple objectives to optimize • No univocal optima solution, need tradeoffs • Pareto Optimality helps distinguish solutions which behave better than others • Consider tradeoffs on the Pareto Frontier only, undominated solutions 12
Following: main Pareto optimization techniques
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