Dualité sans contraintes en optimisation globale Simon Boulmier www.localsolver.com 1 | 28
Global optimization and lower bounds Mixed-integer nonlinear problem : Dual search Primal search v v f 2 | 28 min s.c. x ∈ R n f ( x ) l ≤ x ≤ u , v ∗ g ( x ) = 0 , h ( x ) ≤ 0 , x i ∈ Z , i ∈ I .
Table of content Global optimization in a nutshell I - Reformulation II - Bound tightening techniques III - Convex relaxations IV - Branch-and-bound Reliability of lower bounds I - Solving convex relaxations II - Unconstrained Wolfe duality 3 | 28
Global optimization in a nutshell I - Reformulation 4 | 28
The optimal bucket s.c. h 5 | 28 max r R π ( x 2 1 + x 1 x 2 + x 2 ) 3 x 3 2 x ∈ R 3 0 ≤ x 1 ≤ 1 0 ≤ x 2 ≤ 1 0 ≤ x 3 ≤ 1 ( x 1 − x 2 ) 2 + x 2 x 2 √ 1 + ( x 1 + x 2 ) 3 ≤ 1
Factorisation of the model 6 | 28 w
The main idea [7] 7 | 28
Factorisation of the model 8 | 28
Global optimization in a nutshell II - Bound tightening techniques 9 | 28
The box constraint 10 | 28 z y x
Feasibility based bound tightening (FBBT) y l y u y u x l x y x l y u y u x l x x x l y u y u x l x y x l y u y u x l x y 11 | 28
Global optimization in a nutshell III - Convex relaxations 12 | 28
Convex relaxations [8, 2] 13 | 28
Convex relaxations [4, 3, 1] 14 | 28
Global optimization in a nutshell IV - Branch-and-bound 15 | 28
The branch-and-reduce algorithm [6] 16 | 28 z y x
The branch-and-reduce algorithm [6] 16 | 28 z y x
Summary Reformulation Convex relaxations Bound tightening Branch-and-bound 17 | 28
Solving the convex relaxations Reliability of lower bounds 18 | 28
g i x i g i x g i x i g i x Some issues with the solvers statio compl Complementarity feas Feasibility x x x x Stationarity Approximate Exact Condition 19 | 28 ◮ Smooth convex problem : min { f ( x ) | x ∈ R n , g ( x ) ≤ 0 ∈ R m } ◮ KKT certifjcate depends on L ( x , µ ) = f ( x ) + µ T g ( x ) :
g i x i g i x Some issues with the solvers x compl Complementarity feas Feasibility statio x Stationarity Approximate Exact Condition 19 | 28 ◮ Smooth convex problem : min { f ( x ) | x ∈ R n , g ( x ) ≤ 0 ∈ R m } ◮ KKT certifjcate depends on L ( x , µ ) = f ( x ) + µ T g ( x ) : ∇ x L ( x , µ ) = 0 g i ( x ) ≤ 0 µ i g i ( x ) = 0
Some issues with the solvers Condition Exact Approximate Stationarity Feasibility Complementarity 19 | 28 ◮ Smooth convex problem : min { f ( x ) | x ∈ R n , g ( x ) ≤ 0 ∈ R m } ◮ KKT certifjcate depends on L ( x , µ ) = f ( x ) + µ T g ( x ) : ∇ x L ( x , µ ) = 0 ∥∇ x L ( x , µ ) ∥ ≤ ϵ statio g i ( x ) ≤ 0 g i ( x ) ≤ ϵ feas µ i g i ( x ) = 0 | µ i g i ( x ) | ≤ ϵ compl
Some issues with the solvers Condition Exact Approximate Stationarity Feasibility Complementarity 19 | 28 ◮ Smooth convex problem : min { f ( x ) | x ∈ R n , g ( x ) ≤ 0 ∈ R m } ◮ KKT certifjcate depends on L ( x , µ ) = f ( x ) + µ T g ( x ) : ∇ x L ( x , µ ) = 0 ∥∇ x L ( x , µ ) ∥ ≤ ϵ statio g i ( x ) ≤ 0 g i ( x ) ≤ ϵ feas µ i g i ( x ) = 0 | µ i g i ( x ) | ≤ ϵ compl
The solution trick (Neumaier [5]) 20 | 28
The solution trick (Neumaier [5]) 20 | 28
The solution trick (Neumaier [5]) 20 | 28
The solution trick (Neumaier [5]) 20 | 28
x T x T A duality result i f x u f x f x i f x x i u i i f x x i i f x Theorem x wih the dual function : x x D max is the unconstrained maximisation problem s.c. The Wolfe dual of the convex differentiable problem 21 | 28 ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u
x T x T A duality result i f x u f x f x i f x x i u i i f x x i i f x Theorem x wih the dual function : x is the unconstrained maximisation problem s.c. The Wolfe dual of the convex differentiable problem 21 | 28 ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u ( D ) max θ ( x )
A duality result is the unconstrained maximisation problem i wih the dual function : Theorem x 21 | 28 The Wolfe dual of the convex differentiable problem s.c. ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u ( D ) max θ ( x ) ( ℓ i − x i ) ∇ i f ( x ) + + ( u i − x i ) ∇ i f ( x ) − ∑ θ ( x ) = f ( x ) + f ( x ) + ( ℓ − x ) T ∇ f ( x ) + + ( u − x ) T ∇ f ( x ) − =
Interpretation 22 | 28
x T x T A duality result x x u x x x wih the dual function : x max Theorem D is the bound constrained maximisation problem s.c. The Wolfe dual of the convex differentiable problem 23 | 28 ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u , g ( x ) ≤ 0
x T x T A duality result wih the dual function : x u x x x 23 | 28 Theorem is the bound constrained maximisation problem s.c. The Wolfe dual of the convex differentiable problem ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u , g ( x ) ≤ 0 ( D ) max x ,µ ≥ 0 θ ( x , µ )
A duality result Theorem The Wolfe dual of the convex differentiable problem s.c. is the bound constrained maximisation problem wih the dual function : 23 | 28 ( P ) min x ∈ R n f ( x ) { ℓ ≤ x ≤ u , g ( x ) ≤ 0 ( D ) max x ,µ ≥ 0 θ ( x , µ ) θ ( x , µ ) = L ( x , µ ) + ( ℓ − x ) T ∇L ( x , µ ) + + ( u − x ) T ∇L ( x , µ ) −
Applications min % newton systems solved % function evaluations, % solver fails, x k x max and x k x Defjne Approximate KKT : not enough, also use u x x Sequence of box-constrained subproblems min with augmented lagrangian g x f x Solve min Performance of some bound tightening techniques 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates
Applications min % newton systems solved % function evaluations, % solver fails, x k x max and x k x Defjne Approximate KKT : not enough, also use u x x Sequence of box-constrained subproblems min with augmented lagrangian g x f x Solve min 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques
Applications x % newton systems solved % function evaluations, % solver fails, x k x max and x k min Defjne Approximate KKT : not enough, also use u x x Sequence of box-constrained subproblems min 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques ◮ Solve min { f ( x ) | g ( x ) ≤ 0 } with augmented lagrangian
Applications and % newton systems solved % function evaluations, % solver fails, x k x max x k x min Defjne Approximate KKT : not enough, also use 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques ◮ Solve min { f ( x ) | g ( x ) ≤ 0 } with augmented lagrangian ◮ Sequence of box-constrained subproblems min { L ρ ( x ) | ℓ ≤ x ≤ u }
Applications and % newton systems solved % function evaluations, % solver fails, x k x max x k x min Defjne 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques ◮ Solve min { f ( x ) | g ( x ) ≤ 0 } with augmented lagrangian ◮ Sequence of box-constrained subproblems min { L ρ ( x ) | ℓ ≤ x ≤ u } ◮ Approximate KKT : not enough, also use L ρ − L ρ ≤ ϵ
Applications % solver fails, % function evaluations, % newton systems solved 24 | 28 ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques ◮ Solve min { f ( x ) | g ( x ) ≤ 0 } with augmented lagrangian ◮ Sequence of box-constrained subproblems min { L ρ ( x ) | ℓ ≤ x ≤ u } ◮ Approximate KKT : not enough, also use L ρ − L ρ ≤ ϵ ◮ Defjne L ρ = min ( L ρ ( x 1 ) , · · · , L ρ ( x k )) and L ρ = max ( θ ( x 1 ) , · · · , θ ( x k ))
24 | 28 Applications ◮ Robustness of dual bounds and inconsistency certifjcates ◮ Performance of some bound tightening techniques ◮ Solve min { f ( x ) | g ( x ) ≤ 0 } with augmented lagrangian ◮ Sequence of box-constrained subproblems min { L ρ ( x ) | ℓ ≤ x ≤ u } ◮ Approximate KKT : not enough, also use L ρ − L ρ ≤ ϵ ◮ Defjne L ρ = min ( L ρ ( x 1 ) , · · · , L ρ ( x k )) and L ρ = max ( θ ( x 1 ) , · · · , θ ( x k )) ◮ − 68 % solver fails, − 30 % function evaluations, − 15 % newton systems solved
The end Questions 25 | 28
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