A class of randomly generated semi-infinite programming test - PowerPoint PPT Presentation

A class of randomly generated semi-infinite programming test problems A. Ismael F. Vaz and Edite M.G.P. Fernandes Production and Systems Department - Engineering School Minho University - Braga - Portugal {aivaz,emgpf}@dps.uminho.pt Optimization 2004 - FCUL - Lisbon - Portugal 25-28 July

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 1 Outline • Semi-Infinite Programming (SIP) • Motivation • Terminology/Optimality conditions • Signomials and extended signomials • Randomly generated constraints • Objective function of the randomly generated problem • The algorithm • Example (NSIPS output) and conclusions

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 2 Semi-Infinite Programming (SIP) x ∈ R n f ( x ) min s.t. g i ( x, t ) ≤ 0 , i = 1 , ..., m h i ( x ) ≤ 0 , i = 1 , ..., o h i ( x ) = 0 , i = o + 1 , ..., q ∀ t ∈ T ⊂ R p f ( x ) is the objective function, h i ( x ) are the finite constraint functions, g i ( x, t ) are the infinite constraint functions and T is, usually, a cartesian product of intervals ( [ α 1 , β 1 ] × [ α 2 , β 2 ] × · · · × [ α p , β p ] )

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 3 Motivation For most SIP problems, the exact solutions are not known a priori. This makes the selection of the best algorithm for SIP a difficult task (NSIPS solver). The existence of randomly generated SIP test problems (with known solutions) provides a way to evaluate accuracy, efficiency and reliability of known SIP algorithms.

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 4 Terminology For the remaining of the talk, we denote x ∈ R n f ( x ) min s.t. x ∈ X , with X = { x ∈ R n | g u ( x, t ) ≤ 0 , u = 1 , . . . , m, ∀ t ∈ T, h v ( x ) = 0 , v = 1 , . . . , o, h v ( x ) ≤ 0 , v = o + 1 , . . . , q } as the upper level problem .

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 5 Terminology - cont. and max t ∈ T g u ( x, t ) , u = 1 , . . . , m , as the lower level subproblems . Let ς u be the number of global maxima of the lower level subproblem u , which make the infinite constraint g u ( x, t ) ≤ 0 active. u κ t ∗ , for u = 1 , . . . , m and κ = 1 , . . . , ς u are the solutions to lower level subproblems.

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 6 Optimality conditions - lower level problem � α j − t j ≤ 0 j = 1 , . . . , p ∀ t ∈ T ≡ [ α 1 , β 1 ] × · · · × [ α p , β p ] ⇔ t j − β j ≤ 0 j = 1 , . . . , p ( t = ( t 1 , . . . , t p ))

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 6 Optimality conditions - lower level problem � α j − t j ≤ 0 j = 1 , . . . , p ∀ t ∈ T ≡ [ α 1 , β 1 ] × · · · × [ α p , β p ] ⇔ t j − β j ≤ 0 j = 1 , . . . , p ( t = ( t 1 , . . . , t p )) Given ¯ x (approximation to the upper level problem solution). The Lagrangian of the lower level subproblem u is p p � � L u ( t, u γ lb , u γ ub ) = g u ( x, t ) + u γ lb u γ ub j ( α j − t j ) + j ( t j − β j ) , j =1 j =1 u γ lb , u γ ub ∈ R p are the Lagrange multipliers vectors.

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 7 Lower level first order KKT conditions The first order KKT conditions for a local maximum: ∇ t L u ( u t ∗ , u γ lb ∗ , u γ ub ∗ ) = 0

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 7 Lower level first order KKT conditions The first order KKT conditions for a local maximum: ∇ t L u ( u t ∗ , u γ lb ∗ , u γ ub ∗ ) = 0 � α j − u t ∗ j ≤ 0 , j = 1 , . . . , p feasibility u t ∗ j − β j ≤ 0 , j = 1 , . . . , p

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 7 Lower level first order KKT conditions The first order KKT conditions for a local maximum: ∇ t L u ( u t ∗ , u γ lb ∗ , u γ ub ∗ ) = 0 � α j − u t ∗ j ≤ 0 , j = 1 , . . . , p feasibility u t ∗ j − β j ≤ 0 , j = 1 , . . . , p � u γ lb ∗ j ( α j − u t ∗ j ) = 0 , j = 1 , . . . , p complementarity u γ ub ∗ ( u t ∗ j − β j ) = 0 , j = 1 , . . . , p j

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 7 Lower level first order KKT conditions The first order KKT conditions for a local maximum: ∇ t L u ( u t ∗ , u γ lb ∗ , u γ ub ∗ ) = 0 � α j − u t ∗ j ≤ 0 , j = 1 , . . . , p feasibility u t ∗ j − β j ≤ 0 , j = 1 , . . . , p � u γ lb ∗ j ( α j − u t ∗ j ) = 0 , j = 1 , . . . , p complementarity u γ ub ∗ ( u t ∗ j − β j ) = 0 , j = 1 , . . . , p j u γ lb ∗ j , u γ ub ∗ Lagrange multipli- ≥ 0 , j = 1 , . . . , p j ers positiveness

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 8 Lower level second order KKT conditions The second order sufficient condition: tt L u ( u t ∗ , u γ lb ∗ , u γ ub ∗ ) Z ≺ 0 , Z T ∇ 2 Z is a basis for the null space of the active constraints Jacobian at u t ∗ .

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 9 Upper level Lagrangian The upper level Lagrangian function is q ς u m � � � κ δg u ( x, u u κ t ∗ ) , L ( x, λ, δ ) = f ( x ) + λ v h v ( x ) + v =1 u =1 κ =1 λ = ( λ 1 , . . . , λ q ) T is the Lagrange multipliers vector (finite constraints). ς u δ ) T is the multipliers vector (infinite constraint u δ = ( u 1 δ, . . . , u g u ( x, t ) ≤ 0 ( u = 1 , . . . , m )).

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 10 Upper level first order KKT conditions The first order KKT conditions for a local SIP minimum: ∇ x L ( x ∗ , λ ∗ , δ ∗ ) = 0

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 10 Upper level first order KKT conditions The first order KKT conditions for a local SIP minimum: ∇ x L ( x ∗ , λ ∗ , δ ∗ ) = 0 � λ ∗ v h v ( x ∗ ) = 0 , v = 1 , . . . , q complementarity u κ δ ∗ g u ( x ∗ , u κ t ∗ ) = 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 10 Upper level first order KKT conditions The first order KKT conditions for a local SIP minimum: ∇ x L ( x ∗ , λ ∗ , δ ∗ ) = 0 � λ ∗ v h v ( x ∗ ) = 0 , v = 1 , . . . , q complementarity u κ δ ∗ g u ( x ∗ , u κ t ∗ ) = 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u � λ ∗ v ≥ 0 , v = o + 1 , . . . , q Lagrange multipli- κ δ ∗ ≥ 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u u ers positiveness

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 10 Upper level first order KKT conditions The first order KKT conditions for a local SIP minimum: ∇ x L ( x ∗ , λ ∗ , δ ∗ ) = 0 � λ ∗ v h v ( x ∗ ) = 0 , v = 1 , . . . , q complementarity u κ δ ∗ g u ( x ∗ , u κ t ∗ ) = 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u � λ ∗ v ≥ 0 , v = o + 1 , . . . , q Lagrange multipli- κ δ ∗ ≥ 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u u ers positiveness � h v ( x ∗ ) = 0 , v = 1 , . . . , o feasibility h v ( x ∗ ) ≤ 0 , v = o + 1 , . . . , q

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 10 Upper level first order KKT conditions The first order KKT conditions for a local SIP minimum: ∇ x L ( x ∗ , λ ∗ , δ ∗ ) = 0 � λ ∗ v h v ( x ∗ ) = 0 , v = 1 , . . . , q complementarity u κ δ ∗ g u ( x ∗ , u κ t ∗ ) = 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u � λ ∗ v ≥ 0 , v = o + 1 , . . . , q Lagrange multipli- κ δ ∗ ≥ 0 , u = 1 , . . . , m, κ = 1 , . . . , ς u u ers positiveness � h v ( x ∗ ) = 0 , v = 1 , . . . , o feasibility h v ( x ∗ ) ≤ 0 , v = o + 1 , . . . , q κ t ∗ satisfies the KKT conditions of the lower level subproblem u . Each u

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 11 Upper level second order sufficient condition The second order sufficient condition for a minimum: ∇ 2 xx L ( x ∗ , λ ∗ , δ ∗ ) ≻ 0 .

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 12 Signomial and extended signomials Signomials are generalized polynomials of the form k n a ζη � � s ( x ) = , x > 0 , c η x ζ η =1 ζ =1

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 12 Signomial and extended signomials Signomials are generalized polynomials of the form k n a ζη � � s ( x ) = , x > 0 , c η x ζ η =1 ζ =1 and the extended signomials   p k n a e � � � sin 2 ( t l b l π ) , x, c e s e ( x, t ) = c e ζη η , b l > 0 , x   η ζ η =1 ζ =1 l =1 where c η , a ζη , c e η , a e ζη and b l are real numbers.

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 13 Randomly generated constraints m extended signomials s e 1 , . . . , s e m and q + 1 signomials s 0 , . . . , s q are randomly generated.

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 13 Randomly generated constraints m extended signomials s e 1 , . . . , s e m and q + 1 signomials s 0 , . . . , s q are randomly generated. g u ( x, t ) = s e u ( x, t ) − s e u ( x ∗ , u t ∗ ) , u = 1 , . . . , m a

Optimization 2004 - A.I.F. Vaz and E.M.G.P. Fernandes 13 Randomly generated constraints m extended signomials s e 1 , . . . , s e m and q + 1 signomials s 0 , . . . , s q are randomly generated. g u ( x, t ) = s e u ( x, t ) − s e u ( x ∗ , u t ∗ ) , u = 1 , . . . , m a g u ( x, t ) = s e u ( x, t ) − s e u ( x ∗ , u t ∗ ) − µ e u , u = m a + 1 , . . . , m