A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius Imperial College London Designing and implementing algorithms for mixed-integer nonlinear optimization , 22 February 2018, Schloss Dagstuhl, Germany
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Outline Introduction Bilevel Programming Applications Bilevel problem formulation Properties and challenges B&S Algorithm 1. Special tree management with auxiliary lists 2. Bounding 3. Node selection 4. Branching BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Classification of bilevel problems Numerical results BASBL performance solving test problems from BASBLib Comparison of different branching variable and node selection heuristics Preliminary results for MINLP Conclusions ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 2 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Bilevel Programming Applications The world is multilevel! Applications of Bilevel Programming are diverse and include: ◮ Parameter Estimation [Mitsos et al., 2008] ◮ Management of Multi-Divisional Firms [Ryu et al., 2004] ◮ Environmental Policies: Biofuel Production [Bard et al., 2000] ◮ Traffic Planning [Migdalas, 1995] ◮ Chemical Equilibria [Clark, 1990] ◮ Design of Transportation Networks [LeBlanc and Boyce, 1985] ◮ Agricultural Planning [Fortuny-Amat and McCarl, 1981] ◮ Optimisation of Strategic Defence [Bracken and McGill, 1974] ◮ Resource Allocation [Cassidy et al., 1971] ◮ Stackelberg Games: Market Economy [Stackelberg, 1934] ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 3 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Bilevel Programming Problem (BPP) formulation Generalization of Mathematical Programming A mathematical program that contains an optimization problem in the constraints. ◮ For each fixed x = ˆ x , y is the optimal solution of inner optimization problem: ◮ Two decision makers (DM) are present: upper level DM ( leader ) and lower level DM ( follower ), each one with their own decision variables : x ∈ X and ( y ∈ Y ), objectives F ( x , y ) and ( f ( x , y ) ) and constraints G ( x , y ) and ( g ( x , y ) ). ◮ The constraints of the upper (lower)-level involve the variables of lower (upper)-level and objectives are possibly conflicting. (Mixed-Integer) Bilevel Programming Problem min x , y F ( x , y ) s.t. G ( x , y ) ≤ 0 min y f ( x , y ) Y ( x ) ← − s.t. g ( x , y ) ≤ 0 Global optimal solution set y = ( y 1 , y 2 ) ∈ Y ⊂ Z m 1 × R m − m 1 x = ( x 1 , x 2 ) ∈ X ⊂ Z n 1 × R n − n 1 , y ∈ Y ( x ) ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 4 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and challenges of (MI)BPP 1. Bilevel problems are usually non-convex & often have disconnected feasible regions. 2. Multiple global solutions at the lower level can induce additional challenges ( Non- unique bilevel solution ). 3. General nonconvex form is very challenging ( 2 nonconvex problems in one ) ◮ Solving a nonconvex inner problem locally or partitioning the inner space and treating each subdomain independently may lead to invalid solutions/bounds . ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 5 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and challenges of (MI)BPP 1. Bilevel problems are usually non-convex & often have disconnected feasible regions. 2. Multiple global solutions at the lower level can induce additional challenges ( Non- unique bilevel solution ). 3. General nonconvex form is very challenging ( 2 nonconvex problems in one ) ◮ Solving a nonconvex inner problem locally or partitioning the inner space and treating each subdomain independently may lead to invalid solutions/bounds . Example 3.5 [Mitsos and Barton, 2007] 9 8 min y y 7 16 y 4 + 2 y 3 − 8 y 2 − 1 . 5 y + 0 . 5 s.t. min 6 y 5 y ∈ [ − 1 , 1] Unique solution: F ∗ = 0 . 5 4 f 3 2 Mitsos, A., Barton, P.I (2007). 1 A Test Set for Bilevel Programs, http://yoric.mit.edu/ 0 sites/default/files/bileveltestset.pdf f ∗ − 1 − 1 − 0 . 5 0 0 . 5 1 y ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 5 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and challenges of (MI)BPP Partitioning the Inner Region y ∈ [ − 1 , 0] y s.t. y ∈ arg min min f ( y ) y ∈ [ − 1 , 0] 8 7 6 5 F (1) = − 0 . 5 4 f 3 2 1 f (1) 0 − 1 − 0 . 5 0 0 . 5 1 y ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 6 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and challenges of (MI)BPP Partitioning the Inner Region y ∈ [ − 1 , 0] y s.t. y ∈ arg min min f ( y ) y ∈ [0 , 1] y s.t. y ∈ arg min min f ( y ) y ∈ [ − 1 , 0] y ∈ [0 , 1] 9 8 7 6 5 F (1) = − 0 . 5 F (2) = 0 . 5 4 f 3 2 1 f (1) 0 f (2) − 1 − 1 − 0 . 5 0 0 . 5 1 y ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 6 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and challenges of (MI)BPP Partitioning the Inner Region y ∈ [ − 1 , 0] y s.t. y ∈ arg min min f ( y ) y ∈ [0 , 1] y s.t. y ∈ arg min min f ( y ) y ∈ [ − 1 , 0] y ∈ [0 , 1] 9 8 7 6 5 F (1) = − 0 . 5 F (2) = 0 . 5 4 f 3 2 1 f (1) 0 f (2) − 1 − 1 − 0 . 5 0 0 . 5 1 y F (1) looks more promising and one could be drawn to the false conclusion that y (1) = − 0 . 5 is the solution when it is not feasible in the original problem. The inner problem should always be solved globally and over the whole Y . ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 6 / 23
Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions Properties and Challenges of (MI)BPP Constructing a Convergent Lower Bounding Problem ◮ Solution of MIBPP: ( x ∗ i , y ∗ i ) = (2 , 2) , F ( x ∗ i , y ∗ i ) = − 22 , f ( x ∗ i , y ∗ i ) = 2 ◮ When integrality requirements are relaxed the optimal solution is: ( x ∗ c , y ∗ c ) = (8 , 1) , F ( x ∗ c , y ∗ c ) = − 18 → does not provide a valid bound of the MIBPP! Example 1 [Moore and Bard, 1990] -42 Feasible set 4 0 3 x ∈ Z + F ( x, y ) = − x − 10 y min ≤ x y + -32 -33 -34 0 2 3 2 y s.t. min y ∈ Z + f ( x, y ) = y ≤ + 1 x 0 5 -21 -22 -23 -24 -25 -26 s.t. − 25 x + 20 y ≤ 30 2 y 2 − x + 2 y ≤ 10 5 1 2 x − y ≤ 15 -13 -14 -15 -16 -17 -18 1 ≤ − 2 x − 10 y ≤ − 15 − 2 x − 10 y ≤ − 15 y − x 0 2 Moore, J., and Bard, J.F. (1990). The Mixed 0 1 2 3 4 5 6 7 8 Integer Linear Bilevel Programming Problem , x Operations Research 38(5), pp. 911-921 ✠ : Claire S. Adjiman , Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 7 / 23
Recommend
More recommend
