Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Approximating Pareto Curves using Semidefinite Relaxations Victor MAGRON Postdoc LAAS-CNRS (Joint work with Didier Henrion and Jean-Bernard Lasserre) Optimisation Non Linéaire en Variables Continues et Discrètes 18 June 2014 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 1 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Multiobjective Polynomial Optimization Optimization Problems with several criteria in engineering, economics, applied mathematics. Design of a beam of length l , heigth x 1 and width x 2 : light construction: minimize the volume lx 1 x 2 1 cheap construction: minimize the sectional area π /4 ( x 2 1 + x 2 2 ) 2 under stress and nonnegativity constraints 3 x 2 x 1 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 2 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Multiobjective Polynomial Optimization Let f 1 , f 2 ∈ R d [ x ] two conflicting criteria Let S : = { x ∈ R n : g 1 ( x ) � 0, . . . , g m ( x ) � 0 } a semialgebraic set � � x ∈ S ( f 1 ( x ) f 2 ( x )) ⊤ ( P ) min Assumption The image space R 2 is partially ordered in a natural way ( R 2 + is the ordering cone). Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 3 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Multiobjective Polynomial Optimization Definition Let the previous assumption be satisfied. A point x ∈ S is called an Edgeworth-Pareto (EP) optimal point of Problem P , when there is no x ∈ S such that f j ( x ) � f j ( x ) , j = 1, 2 and f ( x ) � = f ( x ) . A point x ∈ S is called a weakly (EP) optimal point of Problem P , when there is no x ∈ S such that f j ( x ) < f j ( x ) , j = 1, 2. y 2 f 1 ( x ) : = x 1 , 1 f 2 ( x ) : = x 2 , f ( S ) S : = { x ∈ R 2 : 0 � x 1 � 1, 0 � x 2 � 1 } . y 1 0 1 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 4 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Pareto Curve Definition The image set of weakly Edgeworth-Pareto optimal points is called the Pareto curve . Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 5 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Some Examples: f ( S ) + R 2 + is convex g 1 : = − x 2 1 + x 2 , f 1 : = − x 1 , f 2 : = x 1 + x 2 g 2 : = − x 1 − 2 x 2 + 3 , 2 . S : = { x ∈ R 2 : g 1 � 0, g 2 � 0 } . Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 6 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Some Examples: f ( S ) + R 2 + is not convex f 1 : = ( x 1 + x 2 − 7.5 ) 2 /4 + ( − x 1 + x 2 + 3 ) 2 , g 1 : = − ( x 1 − 2 ) 3 /2 − x 2 + 2.5 , g 2 : = − x 1 − x 2 + 8 ( − x 1 + x 2 + 0.65 ) 2 + 3.85 , f 2 : = ( x 1 − 1 ) 2 /4 + ( x 2 − 4 ) 2 /4 . S : = { x ∈ R 2 : g 1 � 0, g 2 � 0 } . Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 7 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Scalarization Techniques Common workaround by reducing P to a scalar POP : � � x ∈ S f p ( λ , x ) : = (( λ | f 1 ( x ) − µ 1 | ) p + (( 1 − λ ) | f 2 ( x ) − µ 2 | ) p ) 1 ( P p λ ) min , p with the weight λ ∈ [ 0, 1 ] and the goals µ 1 , µ 2 ∈ R . x ∈ S f j ( x ) , j = 1, 2. Possible choice: µ j < min Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 8 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Weighted convex sum approximation: method (a) ( P 1 f 1 ( λ ) : = min x ∈ S f 1 ( λ , x ) λ ) : f 1 ( λ , x ) : = λ f 1 ( x ) + ( 1 − λ ) f 2 ( x ) Theorem ([Borwein 77], [Arrow-Barankin-Blackwell 53]) Assume that f ( S ) + R 2 + is convex. A point x ∈ S is an EP optimal point of Problem P ⇐ ⇒ ∃ λ such that x is an image unique solution of Problem P 1 λ . 2 1.5 1 y 2 0.5 0 −0.5 −1 −0.5 0 0.5 1 y 1 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 9 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Weighted convex sum approximation: method (a) ( P 1 f 1 ( λ ) : = min x ∈ S f 1 ( λ , x ) λ ) : f 1 ( λ , x ) : = λ f 1 ( x ) + ( 1 − λ ) f 2 ( x ) Theorem ([Borwein 77], [Arrow-Barankin-Blackwell 53]) Assume that f ( S ) + R 2 + is convex. A point x ∈ S is an EP optimal point of Problem P ⇐ ⇒ ∃ λ such that x is an image unique solution of Problem P 1 λ . 2.5 2 1.5 y 2 1 0.5 0 6 8 10 12 14 16 18 20 y 1 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 9 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Weigthed Chebyshev approximation: method (b) ( P ∞ f ∞ ( λ ) : = min x ∈ S f ∞ ( λ , x ) λ ) : f ∞ ( λ , x ) : = max { λ ( f 1 ( x ) − µ 1 ) , ( 1 − λ )( f 2 ( x ) − µ 2 ) } Theorem ([Jahn 10, Corollary 11.21 (a)], [Bowman 76], [Steuer-Choo 83]) Suppose that ∀ x ∈ S , µ j < f j ( x ) , j = 1, 2. A point x ∈ S is an EP optimal point of Problem P ⇐ ⇒ ∃ λ ∈ ( 0, 1 ) such that x is an image unique solution of Problem P ∞ λ . 2.5 2 1.5 y 2 1 0.5 0 6 8 10 12 14 16 18 20 y 1 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 10 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Parametric sublevel set approximation: method (c) Inspired by previous research on multiobjective linear optimization [1] For each λ ∈ [ a 1 , b 1 ] , consider the following parametric POP ( P u f u ( λ ) : = min λ ) : x ∈ S { f 2 ( x ) : f 1 ( x ) � λ } , with a 1 : = min x ∈ S f 1 ( x ) , b 1 : = f 1 ( x ) and x a solution of min x ∈ S f 2 ( x ) . Lemma Suppose that x ∈ S is an optimal solution of Problem P u λ , with λ ∈ [ a 1 , b 1 ] . Then x belongs to the set of weakly EP points of Problem P . 1 B. Gorissen, D. den Hertog. Approximating the pareto set of multiobjective linear programs via robust optimization. (2012) Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 11 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Questions Is it mandatory to use discretization schemes? Can we approximate the Pareto curve in a relatively strong sense? Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 12 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Contributions Yes! We provide two approaches together with numerical schemes that avoid computing finitely many points . Parametric POP: for methods (a) and (b) (resp. method (c) ), we 1 approximate the Pareto curve with polynomials so that convergence in L 2 -norm (resp. L 1 -norm) holds Hierarchy of outer approximation: we provide certified 2 underestimators of the Pareto curve with strong convergence to f ( S ) in L 1 -norm Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 13 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Contributions Yes! We provide two approaches together with numerical schemes that avoid computing finitely many points . Parametric POP: for methods (a) and (b) (resp. method (c) ), we 1 approximate the Pareto curve with polynomials so that convergence in L 2 -norm (resp. L 1 -norm) holds Hierarchy of outer approximation: we provide certified 2 underestimators of the Pareto curve with strong convergence to f ( S ) in L 1 -norm Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 13 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Contributions Yes! We provide two approaches together with numerical schemes that avoid computing finitely many points . Parametric POP: for methods (a) and (b) (resp. method (c) ), we 1 approximate the Pareto curve with polynomials so that convergence in L 2 -norm (resp. L 1 -norm) holds Hierarchy of outer approximation: we provide certified 2 underestimators of the Pareto curve with strong convergence to f ( S ) in L 1 -norm Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 13 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Outline Parametric POP 1 Outer Approximations of f ( S ) 2 Perspectives 3 Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 14 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Preliminaries: method (a) λ ) : f ∗ ( λ ) : = f 1 ( λ ) = min Parametric POP ( P 1 x ∈ S f ( λ , x ) Assumption For almost all λ ∈ [ 0, 1 ] , the solution x ∗ ( λ ) of the scalarized problem ( P 1 λ ) is unique. Non-uniqueness may be tolerated on a Borel set B ⊂ [ 0, 1 ] , in which case one assumes image uniqueness of the solution. Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 15 / 43
Pareto Curves Parametric POP Outer Approximations of f ( S ) Perspectives Preliminaries: method (a) Parametric POP ( P 1 λ ) : f ∗ ( λ ) : = f 1 ( λ ) = min x ∈ S f ( λ , x ) Let K : = [ 0, 1 ] × S Let M ( K ) the set of probability measures supported on K � ρ : = min K f ( λ , x ) d µ ( λ , x ) µ ∈M ( K ) ( P ) � K λ k d µ ( λ , x ) = 1/ ( 1 + k ) , k ∈ N . s.t. Victor MAGRON (GTPM) Approximating Pareto Curves using SDP 16 / 43
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