Multicriteria Optimization Some continuous and discrete dynamics - PowerPoint PPT Presentation

Multicriteria Optimization Some continuous and discrete dynamics Guillaume Garrigos Institut de Mathématiques et de Modélisation de Montpellier Universidad Tecnica Federico Santa Maria Sestri-Levante: Franco/Italian workshop 8-12 September 2014 Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 1/35

Context H is an Hilbert space, f i : H → R are Lipschitz continuous on bounded sets. K ⊂ H is a closed convex non empty set of constraints, One of the objective functions is bounded from below. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 2/35

Context H is an Hilbert space, f i : H → R are Lipschitz continuous on bounded sets. K ⊂ H is a closed convex non empty set of constraints, One of the objective functions is bounded from below. One approach, the scalarization method : q q chose 0 ≤ θ i ≤ 1, � θ i = 1, and minimize � θ i f i . i = 1 i = 1 Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 2/35

Context H is an Hilbert space, f i : H → R are Lipschitz continuous on bounded sets. K ⊂ H is a closed convex non empty set of constraints, One of the objective functions is bounded from below. One approach, the scalarization method : q q chose 0 ≤ θ i ≤ 1, � θ i = 1, and minimize � θ i f i . i = 1 i = 1 We are looking for the simultaneous minimization of the f i ’s. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 2/35

Contents Multicriteria analysis 1 Continuous steepest descent dynamic 2 Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 3/35

Contents Multicriteria analysis 1 Continuous steepest descent dynamic 2 Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 4/35

Nonsmooth analysis tools Directional derivative (of Clarke) f ( x ′ + td ) − f ( x ′ ) df ( x ; d ) := lim sup . t t ↓ 0 x ′→ x Subdifferential (of Clarke) ∂ f ( x ) := { p ∈ H | � p , d � ≤ df ( x ; d ) ∀ d ∈ H } . Remark If f is of class C 1 , then ∂ f ( x ) = {∇ f ( x ) } and df ( x ; d ) = �∇ f ( x ) , d � . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 5/35

Nonsmooth analysis tools Tangent and normal cones T K ( x ) := cl { d ∈ H | ∃ ε > 0 , ∀ t ∈ ] 0 , ε [ , x + td ∈ K } . N K ( x ) := { p ∈ H | � p , d � ≤ 0 ∀ d ∈ T K ( x ) } . K x T K ( x ) N K ( x ) Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 6/35

Multicriteria analysis Descent direction We say that d ∈ H is a descent direction at x if df i ( x ; d ) < 0 holds for all i = 1 .. q . We say that it is an admissible descent direction if moreover d ∈ T K ( x ) . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 7/35

b Example ∇ f 1 ( x ) x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 8/35

b Example ∇ f 2 ( x ) ∇ f 1 ( x ) x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 9/35

Multicriteria analysis Armijo direction We say that a descent direction d ∈ H is an Armijo direction if ∃ ε > 0 s.t. for all t ∈ ] 0 , ε [ : ∀ i , f i ( x + td ) < f i ( x ) + t 2 df i ( x ; d ) . We say that it is an admissible Armijo direction if moreover x + td ∈ K . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 10/35

Multicriteria analysis Pareto equilibrium(s) Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 11/35

Multicriteria analysis Pareto equilibrium(s) We say that x ∈ K is a Pareto if there is no y ∈ K such that ∀ i f i ( y ) ≤ f i ( x ) and ∃ I f I ( y ) < f I ( x ) . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 11/35

Multicriteria analysis Pareto equilibrium(s) We say that x ∈ K is a Pareto if there is no y ∈ K such that ∀ i f i ( y ) ≤ f i ( x ) and ∃ I f I ( y ) < f I ( x ) . We say that x ∈ K is a weak Pareto if there is no y ∈ K s.t. ∀ i f i ( y ) < f i ( x ) . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 11/35

b Example ∇ f 2 ( x ) ∇ f 1 ( x ) x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 12/35

Multicriteria analysis Pareto equilibrium(s) We say that x ∈ K is a Pareto if there is no y ∈ K such that ∀ i f i ( y ) ≤ f i ( x ) and ∃ I f I ( y ) < f I ( x ) . We say that x ∈ K is a weak Pareto if there is no y ∈ K s.t. ∀ i f i ( y ) < f i ( x ) . We say that x ∈ K is a critical Pareto if 0 ∈ N K ( x ) + Conv { ∂ f i ( x ) } . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 13/35

b Example Conv {∇ f i ( x ) } x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 14/35

Multicriteria analysis Pareto equilibrium(s) We say that x ∈ K is a Pareto if there is no y ∈ K such that ∀ i f i ( y ) ≤ f i ( x ) and ∃ I f I ( y ) < f I ( x ) . We say that x ∈ K is a weak Pareto if there is no y ∈ K s.t. ∀ i f i ( y ) < f i ( x ) . We say that x ∈ K is a critical Pareto if 0 ∈ N K ( x ) + Conv { ∂ f i ( x ) } . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 15/35

Multicriteria analysis Pareto equilibrium(s) We say that x ∈ K is a Pareto if there is no y ∈ K such that ∀ i f i ( y ) ≤ f i ( x ) and ∃ I f I ( y ) < f I ( x ) . We say that x ∈ K is a weak Pareto if there is no y ∈ K s.t. ∀ i f i ( y ) < f i ( x ) . We say that x ∈ K is a critical Pareto if 0 ∈ N K ( x ) + Conv { ∂ f i ( x ) } . Properties Pareto ⇒ weak Pareto ⇒ critical Pareto. If the f i are convex, then weak Pareto ⇔ critical Pareto. If the f i are strictly convex, then the 3 notions both coincide. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 15/35

Link between descent direction and Pareto equilibrium Proposition The following statements are equivalent : x is a critical Pareto point, There is no admissible descent direction at x , There is no admissible Armijo direction at x . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 16/35

Objectif We will consider 1 a continuous dynamic ˙ u ( t ) = s ( u ( t )) , where s : K → H verify s ( u ) = 0 if u is a critical Pareto point, s ( u ) is an admissible descent direction else. 2 an algorithm u n + 1 = u n + t n d n where d n is an admissible Armijo direction. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 17/35

Contents Multicriteria analysis 1 Continuous steepest descent dynamic 2 Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 18/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 19/35

b b Example − s ( x ) Conv {∇ f i ( x ) } x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 20/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 21/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Obviously, x is a Pareto critical iff s ( x ) = 0. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 21/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Obviously, x is a Pareto critical iff s ( x ) = 0. In a sense, s ( x ) selects itself a different convex combination of the functions at each x . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 21/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Obviously, x is a Pareto critical iff s ( x ) = 0. In a sense, s ( x ) selects itself a different convex combination of the functions at each x . Example If q = 1, then s ( x ) = proj T K ( x ) ( −∇ f ( x )) . Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 21/35

The multiobjective steepest descent direction Definition Given x ∈ K , the multiobjective steepest descent direction is s ( x ) := − ( N K ( x ) + Conv { ∂ f i ( x ) } ) 0 . Obviously, x is a Pareto critical iff s ( x ) = 0. In a sense, s ( x ) selects itself a different convex combination of the functions at each x . Example If q = 1, then s ( x ) = proj T K ( x ) ( −∇ f ( x )) . Property s ( x ) is an admissible descent direction at x , whenever s ( x ) � = 0. Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 21/35

b b Example − s ( x ) Conv {∇ f i ( x ) } x Multicriteria optimization - Guillaume Garrigos - Franco/Italian workshop 22/35