Consistent Approximations in Optimization Johannes O. Royset - PowerPoint PPT Presentation

Consistent Approximations in Optimization Johannes O. Royset Professor of Operations Research Naval Postgraduate School, Monterey, California Supported in part by AFOSR, ONR, and DARPA Linz, Austria, November 2019 1 / 36

Searcher Trajectory t = 0 − t = 37.5 10 Target Trajectories 5 0 −5 −10 −20 −15 −10 −5 0 5 10 15 20 Searcher Trajectory t = 37.5 − t =75 10 Target Trajectories 5 0 −5 −10 −20 −15 −10 −5 0 5 10 15 20 Phelps, Royset & Gong, “Optimal Control of Uncertain Systems using Sample Average Approximations,” SIAM J. Control and Optimization, 2016 Stone, Royset & Washburn, Optimal Search for Moving Targets, Springer, 2016 2 / 36

� � � � � � Maximize probability of HVU survival hvu attackers defenders Walton, Lambrianides, Kaminer, Royset & Gong, “Optimal Motion Planning in Rapid-Fire Combat Situations with Attacker Uncertainty,” Naval Research Logistics, 2018 3 / 36

Seven defenders vs 100 attackers 4 / 36

� � � � � � � � Modeling probability of detection � � � , � ( � ) target � � searcher r ( x ( t ) , y ( t ))∆ t : probability of detection during [ t , t + ∆ t ) q ( t ): probability of no detection during [0 , t ] q ( t + ∆ t ) = q ( t )(1 − r ( x ( t ) , y ( t ))∆ t ) q ( t ) = − q ( t ) r ( x ( t ) , y ( t )), ˙ q (0) = 1 5 / 36

� � � � � � Target uncertainty � � , � ? � � target searcher { y ( t , ξ ) , t ∈ [0 , 1] } uncertain track of target; ξ random vector q ( t , ξ ): prob. of no detection during [0 , t ] given ξ q ( t , ξ ) = − q ( t , ξ ) r ( x ( t ) , y ( t , ξ ) , ξ ), q (0 , ξ ) = 1 ˙ � � E q (1 , ξ ) probability of no detection during [0 , 1] Combine q ( t , ξ ) with searcher state x ( t ) to get state x ( t , ξ ) ϕ ( x u (1 , ξ ) , ξ ) � � minimize E u ∈ U with x u ( · , ξ ) solving ˙ � � x ( t , ξ ) = f x ( t , ξ ) , u ( t ) , ξ ; x (0 , ξ ) = x 0 ( ξ ) a.s. 6 / 36

� � � � � � Attacker-Defender attackers defenders p 0 ( t , ξ ) = − r ( x ( t ) , y ( t , ξ ) , ξ ) p 0 ( t , ξ ) Q ( t ) ˙ � � p 1 ( t , ξ ) = − r ( x ( t ) , y ( t , ξ ) , ξ ) ˙ p 1 ( t , ξ ) − p 0 ( t , ξ ) Q ( t ) . . . � � p N − 1 ( t , ξ ) = − r ( x ( t ) , y ( t , ξ ) , ξ ) ˙ p N − 1 ( t , ξ ) − p N − 2 ( t , ξ ) Q ( t ) q 0 ( t , ξ ) = − s ( x ( t ) , y ( t , ξ ) , ξ ) q 0 ( t , ξ ) P ( t ) ˙ � � q 1 ( t , ξ ) = − s ( x ( t ) , y ( t , ξ ) , ξ ) ˙ q 1 ( t , ξ ) − q 0 ( t , ξ ) P ( t ) . . . � � q N − 1 ( t , ξ ) = − s ( x ( t ) , y ( t , ξ ) , ξ ) ˙ q N − 1 ( t , ξ ) − q N − 2 ( t , ξ ) P ( t ) � N − 1 � N − 1 P ( t ) = n =0 p n ( t ) Q ( t ) = n =0 q n ( t ) 7 / 36

Setting for presentation ( X , d ) metric space f ν , f : X → [ −∞ , ∞ ], usually lower semicontinuous (lsc) Actual problem: minimize f ( x ) x ∈ X f ν ( x ) Approximating problem: minimize x ∈ X Constraints often handled abstractly: Setting objective function to ∞ if x infeasible (wlog) 8 / 36

Setting for presentation ( X , d ) metric space f ν , f : X → [ −∞ , ∞ ], usually lower semicontinuous (lsc) Actual problem: minimize f ( x ) x ∈ X f ν ( x ) Approximating problem: minimize x ∈ X Constraints often handled abstractly: Setting objective function to ∞ if x infeasible (wlog) What constitutes a consistent approximation? Level 0: convergence of minimizers, minima Level 1: convergence of first-order stationary points 8 / 36

� � � � � Would pointwise convergence suffice? � � = � = � X 1 2 Pointwise convergence not sufficient for convergence of minimizers 9 / 36

� � What about uniform convergence? � ( � ) � ( � ) f argmin { � � � | � ( � ) � � } 10 / 36

� � � What about uniform convergence? � ( � ) � ( � ) � ( � ) f 11 / 36

� � � Uniform “approximation,” but large error in argmin � ( � ) � ( � ) f argmin { � � � | � � ( � ) � 0 } 12 / 36

� � � � Passing to epigraphs of the effective functions epi � epi � � � if � � ( � ) � 0 � � if � ( � ) � 0 � � = �� � � = �� otherwise otherwise 13 / 36

� � � � � Epi-convergence epi � epi � epi � � epi � � X f ν epi-converges to f ⇐ ⇒ epi f ν set-converges to epi f Main consequence: f ν epi-converges to f and x ν ∈ argmin f ν → ¯ x = ⇒ ¯ x ∈ argmin f 14 / 36

� � � � � � � Approximation of constraints � = � � 1 X � 1 15 / 36

� � � � � � � Approximation of constraints � = � � 1 X � 1 If C ν set-converges to C and f 0 continuous, then � � f 0 ( x ) if x ∈ C ν f 0 ( x ) if x ∈ C f ν ( x ) = epi-conv to f ( x ) = ∞ otherwise ∞ otherwise 15 / 36

� � � � � � � Approximation of constraints � = � � 1 X � 1 If C ν set-converges to C and f 0 continuous, then � � f 0 ( x ) if x ∈ C ν f 0 ( x ) if x ∈ C f ν ( x ) = epi-conv to f ( x ) = ∞ otherwise ∞ otherwise ⇒ C ν set-converges to C Example: C 1 , C 2 , . . . dense in C = X = 15 / 36

� � � Recall failure under uniform convergence What can be done in this case? � ( � ) � ( � ) � ( � ) f 16 / 36

Constraint softening minimize f 0 ( x ) subject to g i ( x ) ≤ 0 , i = 1 , . . . , q x ∈ X 0 ( x ) − f 0 ( x ) | ≤ α ν and sup | f ν sup i =1 ,..., q | g ν max i ( x ) − g i ( x ) | ≤ α ν x ∈ X x ∈ X 17 / 36

Constraint softening minimize f 0 ( x ) subject to g i ( x ) ≤ 0 , i = 1 , . . . , q x ∈ X 0 ( x ) − f 0 ( x ) | ≤ α ν and sup | f ν sup i =1 ,..., q | g ν max i ( x ) − g i ( x ) | ≤ α ν x ∈ X x ∈ X q � minimize x ∈ X , y ∈ R q f ν 0 ( x )+ θ ν y i subject to g ν i ( x ) ≤ y i , 0 ≤ y i , i = 1 , . . . , q i =1 17 / 36

Constraint softening minimize f 0 ( x ) subject to g i ( x ) ≤ 0 , i = 1 , . . . , q x ∈ X 0 ( x ) − f 0 ( x ) | ≤ α ν and sup | f ν sup i =1 ,..., q | g ν max i ( x ) − g i ( x ) | ≤ α ν x ∈ X x ∈ X q � minimize x ∈ X , y ∈ R q f ν 0 ( x )+ θ ν y i subject to g ν i ( x ) ≤ y i , 0 ≤ y i , i = 1 , . . . , q i =1 f 0 continuous g i lsc, i = 1 , . . . , q θ ν → ∞ , α ν → 0, θ ν α ν → 0 Then, approximation epi-converges to actual 17 / 36

Epi-convergence under sampling and forward Euler Searcher Trajectory t = 0 − t = 37.5 Target Trajectories 10 5 0 −5 −10 −20 −15 −10 −5 0 5 10 15 20 Searcher Trajectory t = 37.5 − t =75 10 Target Trajectories 5 0 −5 −10 −20 −15 −10 −5 0 5 10 15 20 ϕ ( x u (1 , ξ ) , ξ ) � � minimize E u ∈ U with x u ( · , ξ ) solving ˙ � � x ( t , ξ ) = f x ( t , ξ ) , u ( t ) , ξ ; x (0 , ξ ) = x 0 ( ξ ) a.s. Sampling and Forward Euler result in epi-convergence Phelps, Royset & Gong, “Optimal Control of Uncertain Systems using Sample Average Approximations,” SIAM J. Control and Optimization, 2016 18 / 36

Truncated Hausdorff distance between sets For C , D ⊂ X (metric space) C exs D B � ( � ) � �� d ˆ � � � l ρ ( C , D ) = max exs C ∩ B X ( ρ ); D , exs D ∩ B X ( ρ ); C 19 / 36

Consequence for minima and near-minimizers For f , g : X → [ −∞ , ∞ ], | inf f − inf g | ≤ d ˆ l ρ (epi f , epi g ) ≤ d ˆ � � exs ε - argmin g ∩ B X ( ρ ); δ - argmin f l ρ (epi f , epi g ) if δ > ε + 2 d ˆ l ρ (epi f , epi g ) (product metric is used on X × R and ρ large enough) Replace > by ≥ when f and g lsc and X has compact balls 20 / 36

Bounds are sharp ≤ d ˆ � � exs ε - argmin g ∩ B X ( ρ ); δ - argmin f l ρ (epi f , epi g ) if δ ≥ ε + 2 d ˆ l ρ (epi f , epi g ) epi � 1 epi � 1 2 � 1 21 / 36

What about minimizers? When f ( x ) − inf f ≥ g (dist( x , argmin f )) ∀ x ∈ X for incr g argmin f ν ∩ B X ( ρ ) , argmin f ≤ d ˆ � � exs l ρ (epi f , epi f ν ) + g − 1 � 2 d ˆ � l ρ (epi f , epi f ν ) � ( � � ) � inf � � ( � ) t 22 / 36

� � � Sharpness of bound on minimizers d ˆ l ρ (epi f , epi f ν ) = η = 1 / 2; f has growth g ( t ) = t 2 � � = � � � + 2 � argmin f ν ∩ B X ( ρ ) , argmin f ≤ η + g − 1 � � � � exs 2 η 23 / 36

Computing distances for compositions For κ -Lipschitz f : Y → R and F , G : X → Y , d ˆ ≤ max { 1 , κ } d ˆ � � l ρ epi( f ◦ F ) , epi( f ◦ G ) l ¯ ρ (gph F , gph G ) provided that ¯ ρ large enough 24 / 36

Distances for sums f i , g i : X → [ −∞ , ∞ ], i = 1 , 2, f 1 , g 1 are Lipschitz continuous with common modulus κ d ˆ � � epi( f 1 + f 2 ) , epi( g 1 + g 2 ) ≤ sup A ρ | f 1 − g 1 | l ρ d ˆ � � + 1 + κ ρ (epi f 2 , epi g 2 ) l ¯ provided that epi( f 1 + f 2 ) and epi( g 1 + g 2 ) are nonempty, A ρ = { f 1 + f 2 ≤ ρ } ∪ { g 1 + g 2 ≤ ρ } ∩ B X ( ρ ), ρ ≥ ρ + max { 0 , − inf B X ( ρ ) f 1 , − inf B X ( ρ ) g 1 } ¯ 25 / 36

Convergence of stationary points First-order conditions for minimize x ∈ X f ( x ): Oresme Rule: df ( x ; w ) ≥ 0 ∀ w ∈ X Fermat Rule: 0 ∈ ∂ f ( x ) 26 / 36