Explore the formal definitions of Mathematical Programming with Equilibrium Constraints (MPEC) and Extended Math Programming with Equilibrium Constraints (EPEC). Delve into simple and complex examples, along with insight into two-level problems and feasible regions within the overall problem space.
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Presentation Transcript
Optimizacin Complementaria: MPEC and EPEC Felipe Feijoo, Ph.D. Oficina: EII, IBC 6-7. Email: Felipe.Feijoo@pucv.cl
Summary Overview of two-level problems Two simple examples More complicated examples
Formal definition of MPEC Problem Statement min x y f x y st x y ( ) , , ( ) ( ) . . , , Z y S x where f x y ( ) n m + , : is overall objective function R R n m + is nonempty closed subset of is the solution set of lowe S x upper-level variables are fixed Lower-level problem can be an optimization, MCP, or VI representing joint constraints for , r-level problem given that Z R x y ( ) x
Formal definition of MPEC: complementarity problems Simple Example (Luo, Pang, Ralph) , : f x y R R ( ) 2 ( ) min , f x y , x y . . st x y 0 argmin : ( ) y y C x where + + : 2 + 10, 2 2 16,2 21, y x R x y x y x y + = ( ) C x 2 38, 18 y x y
Simple Example (Luo, Pang, Ralph) , : f x y R R ( ) 2 ( ) min , f x y Simple Example (Luo, Pang, Ralph) , : f x y R R , x y . . st x y 0 ( ) 2 argmin : ( ) y y C x Example of MPEC ( , min , . . 0 argmin where y C x x ) f x y x y where st x y + + : 2 + 10, 2 2 16,2 21, y x R x y x y x y : ( ) + y y C x = ( ) C x 2 38, 18 y x y + + : 2 + 10, 2 2 16,2 21, R x y x y x y + = ( ) 2 38, 18 y x y
Example of MPEC Each inner problem is an LP, can solve for optimal for each range of y x if x = + + x 5 0,8 2 x ( ) S x 3 8,12 if x 2 21 2 12,16 x if x
Example of MPEC ( ) Feasible region to overall problem, set of F = + but closed and connected The solution set is a singleton for each (not always the case) , s.t. x y x ,5 , 0,8 x x 2 x ( ) , 21 2 + , 3 , 8,12 , 12,16 x x x x x 2 2 Union of three noncollinear line segments in the plane so is nonconvex, F R x
Formal definition of MPEC: complementarity problems MPEC MPEC (Reformulated) CP min ( , ) f x y 0 z min ( , ) f x y s.t. y ( , ) ( ) 0 x y g z s.t. y ( x , ) x y = T ( ) 0 z g z ( ) S x 0 y ( , ) 0 g y MPEC (Disjunctive Constraints) ) , ( min r K y = ( , ) 0 y g x f x y s.t. ( , ) x y ? 0 1 ( y ) Even if g(x,y) is linear, this term can be nonlinear and nonconvex 0 where ( , ) g x Kr 0 n 1 , is a vector of binary val ues r y + a is large constant K Computation time
