A constrained optimization problem under uncertainty Raluca Andrei, - PowerPoint PPT Presentation

A constrained optimization problem under uncertainty Raluca Andrei, Gert de Cooman, Erik Quaeghebeur & Keivan Shariatmadar CMU Games & Decision Meeting March 3rd, 2010

Toy problem: two-component massless rod L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL

Toy problem: two-component massless rod, tensile load L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL d 1 d 2 Y 1 Y 2 F

Toy problem: two-component massless rod, tensile load L Y 1 Y 2 a l 1 = ( 1 − x ) L l 2 = xL d 1 d 2 Y 1 Y 2 F Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions � c 1 + c 2 − c 2 �� d 1 � � 0 � c i = Y i a = , l i . − c 2 c 2 d 2 F d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions � c 1 + c 2 − c 2 �� d 1 � � 0 � c i = Y i a = , l i . − c 2 c 2 d 2 F Solution solving the system (analytically) gives d 1 = FL 1 − x d 2 = d 1 + FL x Y 1 , Y 2 . a a d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component massless rod, tensile load: FE analysis FE analysis 3 nodes, boundary conditions � c 1 + c 2 − c 2 �� d 1 � � 0 � c i = Y i a = , l i . − c 2 c 2 d 2 F Solution solving the system (analytically) gives d 1 = FL 1 − x d 2 = d 1 + FL x Y 1 , Y 2 . a a Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL . d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that d 2 < D .

Two-component rod, tensile load: design optimization d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ◮ a classical constrained optimization problem; ◮ considered ‘solved’. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ◮ a classical constrained optimization problem; ◮ considered ‘solved’. Uncertainty about elastic moduli Y 1 and Y 2 This problem is ◮ a constrained optimization problem under uncertainty; ◮ not well-posed as such. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Two-component rod, tensile load: design optimization Precisely known elastic moduli Y 1 and Y 2 This problem is ◮ a classical constrained optimization problem; ◮ considered ‘solved’. Uncertainty about elastic moduli Y 1 and Y 2 This problem is ◮ a constrained optimization problem under uncertainty; ◮ not well-posed as such. Approach: ◮ reformulate as a well-posed decision problem; ◮ solve the decision problem, i.e., derive a classical constrained optimization problem. d 1 d 2 L Y 1 Y 2 a Y 1 Y 2 F l 1 = ( 1 − x ) L l 2 = xL Goal Maximize x under the constraint that 1 − x Y 1 + x Y 2 < Da FL .

Overview Toy problem General problem formulation Uncertainty models Optimality criteria Probabilistic and indeterminacy aspects of uncertainty Objective Results Application: bridge design for vehicle-pillar collisions

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to R ) Y random variable (realizations y in Y ) R relation on X × Y .

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to R ) Y random variable (realizations y in Y ) R relation on X × Y . Decision problem Find the optimal decisions x : ◮ associate a utility function with every decision z : � f ( z ) , zRy , G z ( y ) = f ( z ) I zR + LI z � R = with penalty value L < inf f ; L , z � Ry , f ( x ) G z ( y ) f ( z ) f ( z ) L z x y zR z � R

A constrained optimization problem under uncertainty Goal Maximize f ( x ) under the constraint that xRY . x optimization variable (values in X ) f objective function (from X to R ) Y random variable (realizations y in Y ) R relation on X × Y . Decision problem Find the optimal decisions x : ◮ associate a utility function with every decision z : � f ( z ) , zRy , G z ( y ) = f ( z ) I zR + LI z � R = with penalty value L < inf f ; L , z � Ry , f ( x ) G z ( y ) f ( z ) f ( z ) L z x y zR z � R ◮ choose an optimality criterion, e.g., maximinity, maximality.

Uncertainty models Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Uncertainty models Random variable Y Formal model for the uncertainty about y in Y . Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Uncertainty models Random variable Y Formal model for the uncertainty about y in Y . Lower and upper expectation With (almost) all typical uncertainty models correspond lower and upper expectation operators ( E and E ), or (almost) equivalently, a set of linear expectation operators M : E M ( G ) : = inf E ∈ M E ( G ) , E M ( G ) : = sup E ∈ M E ( G ) , M E : = { E : E ≥ E } . Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Uncertainty models Random variable Y Formal model for the uncertainty about y in Y . Lower and upper expectation With (almost) all typical uncertainty models correspond lower and upper expectation operators ( E and E ), or (almost) equivalently, a set of linear expectation operators M : E M ( G ) : = inf E ∈ M E ( G ) , E M ( G ) : = sup E ∈ M E ( G ) , M E : = { E : E ≥ E } . ◮ probabilities (measures, PMF, PDF, CDF); Examples ◮ upper and/or lower of the above (inner/outer measures, Choquet capacities, p-boxes); ◮ intervals, vacuous expectations: E A ( G ) : = inf y ∈ A G ( y ) ; ◮ possibility distributions, belief functions, . . . ◮ convex mixtures of the lot (e.g., contamination models). Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Optimality criteria: maximizing expected utility generalized Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Optimality criteria: maximizing expected utility generalized Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility ( P ( A ) : = E ( I A ) ): E ( G x ) = sup z ∈ X E ( G z ) � � � � = sup z ∈ X E f ( z ) I zR + LI z � R = L + sup z ∈ X f ( z ) − L P ( zR ) . Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Optimality criteria: maximizing expected utility generalized Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility ( P ( A ) : = E ( I A ) ): E ( G x ) = sup z ∈ X E ( G z ) � � � � = sup z ∈ X E f ( z ) I zR + LI z � R = L + sup z ∈ X f ( z ) − L P ( zR ) . Maximality Optimal x are undominated in pairwise comparisons with all other decisions: 0 ≤ inf z ∈ X E ( G x − G z ) �� � � � � � � = inf z ∈ X E f ( x ) − f ( z ) I xR ∩ zR + f ( x ) − L I xR ∩ z � R + L − f ( z ) . I x � R ∩ zR Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Optimality criteria: maximizing expected utility generalized Maximinity Worst-case reasoning; optimal x maximize the lower (minimal) expected utility ( P ( A ) : = E ( I A ) ): E ( G x ) = sup z ∈ X E ( G z ) � � � � = sup z ∈ X E f ( z ) I zR + LI z � R = L + sup z ∈ X f ( z ) − L P ( zR ) . Maximality Optimal x are undominated in pairwise comparisons with all other decisions: 0 ≤ inf z ∈ X E ( G x − G z ) �� � � � � � � = inf z ∈ X E f ( x ) − f ( z ) I xR ∩ zR + f ( x ) − L I xR ∩ z � R + L − f ( z ) . I x � R ∩ zR Others Maximaxity, E -admissibility, interval dominance Goal Faced with uncertainty about y in Y , find optimal x in X given an optimality criterion and utility functions G z on Y for all z in X .

Probabilistic and indeterminacy aspects of uncertainty Example X = Y : = R , R : = ≤ . f ( x ) g y ( x ) = G x ( y ) sup g y sup f | Ry L y x y x Ry � Ry