Background Main contribution New framework for constrained global optimization Experimental evaluation Inner Regions and Interval Linearizations for Global Optimization G. Trombettoni, I. Araya, B. Neveu, G. Chabert INRIA, I3S, Univ. Nice-Sophia (France); UTFSM (Chile); Imagine, LIGM, Univ. Paris-Est (France); LINA, EMN (France) SWIM, Bourges, June 14 th , 2011 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Plan Background 1 Main contribution 2 New framework for constrained global optimization 3 Experimental evaluation 4 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Plan Background 1 Main contribution 2 New framework for constrained global optimization 3 Experimental evaluation 4 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Handled Problem Reliable continuous constrained global optimization : Constrained optimization: argmin x ∈ [ x ] ⊂ R n f ( x ) s . t . g ( x ) ≤ 0 ∧ h ( x ) = 0 Reliability: the best solution is guaranteed with a bounded error on the cost. Constraints Objective function Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Example (Coconut: ex 6 1 1) Variables x2, x3, x4, x5 in [1e-7,0.5]; x6 in [0,0.901]; x7 in [0,0.274]; x8 in [0,0.69]; x9 in [0,0.998]; Minimize x2*(log(x2) - log(x2 + x4)) + x4*(log(x4) - log(x2 + x4)) + x3*(log(x3) - log(x3 + x5)) + x5*(log(x5) - log(x3 + x5)) + 0.92*x2*x8 + 0.746*x4*x6 + 0.92*x3*x9 + 0.746*x5*x7; Subject to x6*(x2 + 0.159*x4) - x2 = 0; x7*(x3 + 0.159*x5) - x3 = 0; x8*(0.308*x2 + x4) - x4 = 0; x9*(0.308*x3 + x5) - x5 = 0; x2 + x3 = 0.5; x4 + x5 = 0.5; Operators ( + , − , ∗ , / , power , sinus , exp ,etc) are piecewise continuous and differentiable ⇒ non convex optimization Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Application domains Chemistry Robotics: design, proof of properties Signal processing: source separation ... Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Intervals and boxes Intervals Interval [ x ] = [ x , x ] { x ∈ R , x ≤ x ≤ x } x et x Floating-point bounds Set of all the intervals IR m ([ x ]) Midpoint of [ x ] w ([ x ]) := x − x Width or size of [ x ] Boxes Box [ x ] [ x 1 ] × ... × [ x i ] × ... × [ x n ] . w ([ x ]) max n ( w ([ x i ])) Outer box search space containing sols & non sols Inner box search space containing only solutions Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Interval Branch & Bound Constrained global optimization: Vocabulary Minimize an objective function f in a given box under inequality ( g ) and equality ( h ) constraints. A vector/point x satisfying the constraints is called feasible . Interval Branch & Bound Explore a box [ x ] in subdividing it in sub-boxes (of width greater than ǫ sol ). During the search, for every node/box: contract the current box, without loss of solution update a (generally non feasible) lower bound lb of the objective function cost, and update a feasible upper bound ub of the cost, until ub − lb < ǫ obj . Store the minimum cost associated to every box/node of the search tree. Perform a best-first search: select first the box with the lowest cost (possible memory overflow, utilization of a heap) Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Illustration Upper bound Lower bound Objective function Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Plan Background 1 Main contribution 2 New framework for constrained global optimization 3 Experimental evaluation 4 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Upper Bounding and reliability Constraints Objective function Upper bounding and reliability: issue A point obtained during local search is rarely feasible: it must be often corrected and certified by costly interval analysis methods. Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Alternative: Upper Bounding in inner regions Principles For each node, management of an outer box [ x ] out and of an inner region r in extracted from [ x ] out . Inner region ≡ box or polytope in which all points are feasible. Upper Bounding essentially in r in . Equations are relaxed: h ( x ) ∈ [ − ǫ eq , + ǫ eq ] Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Plan Background 1 Main contribution 2 New framework for constrained global optimization 3 Experimental evaluation 4 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation General schema System: S := { g ( x ) ≤ 0 , h ( x ) ∈ [ − ǫ eq , + ǫ eq ] , y = f ( x ) , y ≤ ub } Branching heuristic Principle: impact of a variable x i on a function f j � � � � ∂ f j Kearfott 1990: smear ( x i , f j ) = N ([ x ]) � ∗ w ([ x i ]) � � ∂ x i � smear ( x i , f j ) Variant: smearRelative ( x i , f j ) = xk ∈ x smear ( x k , f j ) ∈ [ 0 , 1 ] P At every node of the search tree y ← ub 1 // Contract [ x ] and improve y 2 OuterContractLB(...) // Upper bounding in inner regions 3 InnerExtractUB(...) Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation OuterContractLB Mohc Contract [ x ] : interval constraint propagation algorithm Mohc . Mohc exploits monotonicity of functions Side effect: improve y Improve y with safe linearization (lower bounding) Main difficulty: Compute a safe outer linearization (round-off errors) Existing approaches: Reliable reformulation-linearization-technique (Quad by Lebbah et al.), affine arithmetic Proposition: interval first-order taylor-based linearization (by Lin & Stadtherr) Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Convex interval Taylor f(m([x]))+f'.(x-m([x])) f(x)+f'.(x-x) f’ f(x) f(x) f’ f’ f(m([x]))+f'.(x-m([x])) f’ f(x)+f'.(x-x) x x Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation Convex interval Taylor ∀ x ∈ [ x ] , f ( x ) + a 1 ∗ x l 1 + ... + a n ∗ x l n ≤ f ( x ) with x l i = x i − x i Linear program LP lb = min f ( x ) + a 1 ∗ x l 1 + ... + a n ∗ x l n ∀ j g j ( x ) + a j 1 + ... + a j 1 ∗ x l n ∗ x l subject to : n ≤ 0 ∀ i 0 ≤ x l i , x l i ≤ w ([ x i ]) x l where : i = x i − x i Using two corners x and x ⇒ tighter polytope f ( x ) + � i a i ( x i − x i ) = f ( x ) + � i a i x i − a i x i = � i a i x i + f ( x ) − � i a i x i f ( x ) + � i a i ( x i − x i ) = f ( x ) + � i a i x i − a i x i = � i a i x i + f ( x ) − � i a i x i Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation InnerExtractUB Inner region extraction based on InHC4 (inner box) and InnerLinearization (inner polytope). Using InHC4 [ x ] in ← InHC4 ( S , [ x ] out ) /* See [Chabert, Beldiceanu, CP 2010] */ if [ x ] in � = ∅ then [ x ] in ← MonotonicityAnalysis ( f , [ x ] in ) x ← RandomProbing ( [ x ] in ) else x ← RandomProbing ( [ x ] out ) end if Update the upper bound Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
Background Main contribution New framework for constrained global optimization Experimental evaluation One word about InnerLinearization Inner interval linearization based on corner-based Taylor ∀ x ∈ [ x ] , f ( x ) ≤ f l ( x ) = f ( x ) + � i a i ∗ ( x i − x i ) ≤ 0 Trombettoni, Araya, Neveu, Chabert Inner Regions for Global Optimization
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