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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 ),

• bjectives 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.

min

x,y F(x, y)

s.t. G(x, y) ≤ 0 min

y f(x, y)

s.t. g(x, y) ≤ 0 y = (y1, y2) ∈ Y ⊂ Zm1 × Rm−m1 x = (x1, x2) ∈ X ⊂ Zn1 × Rn−n1, y ∈ Y(x) (Mixed-Integer) Bilevel Programming Problem Y(x) ← −

Global optimal solution set ✠ : 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. min

y

y s.t. min

y

16y4 + 2y3 − 8y2 − 1.5y + 0.5 y ∈ [−1, 1] Example 3.5 [Mitsos and Barton, 2007]

Mitsos, A., Barton, P.I (2007). A Test Set for Bilevel Programs, http://yoric.mit.edu/ sites/default/files/bileveltestset.pdf −1 −0.5 0.5 1 −1 1 2 3 4 5 6 7 8 9

Unique solution: F ∗ = 0.5

f∗

y f

✠ : 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

min

y∈[−1,0] y s.t. y ∈ arg min y∈[−1,0]

f(y)

−1 −0.5 0.5 1 1 2 3 4 5 6 7 8

F (1) = −0.5

f(1)

y f

✠ : 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

min

y∈[−1,0] y s.t. y ∈ arg min y∈[−1,0]

f(y) min

y∈[0,1] y s.t. y ∈ arg min y∈[0,1]

f(y)

−1 −0.5 0.5 1 −1 1 2 3 4 5 6 7 8 9

F (1) = −0.5 F (2) = 0.5

f(1) f(2)

y f

✠ : 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

min

y∈[−1,0] y s.t. y ∈ arg min y∈[−1,0]

f(y) min

y∈[0,1] y s.t. y ∈ arg min y∈[0,1]

f(y)

−1 −0.5 0.5 1 −1 1 2 3 4 5 6 7 8 9

F (1) = −0.5 F (2) = 0.5

f(1) f(2)

y f 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!

min

x∈Z+ F(x, y) = −x − 10y

s.t. min

y∈Z+ f(x, y) = y

s.t. − 25x + 20y ≤ 30 x + 2y ≤ 10 2x − y ≤ 15 − 2x − 10y ≤ −15

Example 1 [Moore and Bard, 1990]

Moore, J., and Bard, J.F. (1990). The Mixed Integer Linear Bilevel Programming Problem, Operations Research 38(5), pp. 911-921 1 2 3 4 5 6 7 8 1 2 3 4

− 2 5 x + 2 y ≤ 3 x + 2 y ≤ 1 2 x − y ≤ 1 5 −2x − 10y ≤ −15

• 13
• 14
• 15
• 16
• 17
• 18
• 21
• 22
• 23
• 24
• 25
• 26
• 32
• 33
• 34
• 42

x y

Feasible set

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 7 / 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!

min

x∈Z+ F(x, y) = −x − 10y

s.t. min

y∈Z+ f(x, y) = y

s.t. − 25x + 20y ≤ 30 x + 2y ≤ 10 2x − y ≤ 15 − 2x − 10y ≤ −15

Example 1 [Moore and Bard, 1990]

Moore, J., and Bard, J.F. (1990). The Mixed Integer Linear Bilevel Programming Problem, Operations Research 38(5), pp. 911-921 1 2 3 4 5 6 7 8 1 2 3 4

− 2 5 x + 2 y ≤ 3 x + 2 y ≤ 1 2 x − y ≤ 1 5 −2x − 10y ≤ −15

• 13
• 14
• 15
• 16
• 17
• 18
• 21
• 22
• 23
• 24
• 25
• 26
• 32
• 33
• 34
• 42
• 13
• 14
• 15
• 16
• 17
• 18
• 21
• 22

x y

Feasible set

Y(x)

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 7 / 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!

min

x∈Z+ F(x, y) = −x − 10y

s.t. min

y∈Z+ f(x, y) = y

s.t. − 25x + 20y ≤ 30 x + 2y ≤ 10 2x − y ≤ 15 − 2x − 10y ≤ −15

Example 1 [Moore and Bard, 1990]

Moore, J., and Bard, J.F. (1990). The Mixed Integer Linear Bilevel Programming Problem, Operations Research 38(5), pp. 911-921 1 2 3 4 5 6 7 8 1 2 3 4

− 2 5 x + 2 y ≤ 3 x + 2 y ≤ 1 2 x − y ≤ 1 5 −2x − 10y ≤ −15

• 13
• 14
• 15
• 16
• 17
• 18
• 21
• 22
• 23
• 24
• 25
• 26
• 32
• 33
• 34
• 42
• 13
• 14
• 15
• 16
• 17
• 18
• 21
• 22
• 22

x y

Feasible set

Y(x) (x∗, y∗)

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 7 / 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!

min

x∈Z+ F(x, y) = −x − 10y

s.t. min

y∈Z+ f(x, y) = y

s.t. − 25x + 20y ≤ 30 x + 2y ≤ 10 2x − y ≤ 15 − 2x − 10y ≤ −15

Example 1 [Moore and Bard, 1990]

Moore, J., and Bard, J.F. (1990). The Mixed Integer Linear Bilevel Programming Problem, Operations Research 38(5), pp. 911-921 1 2 3 4 5 6 7 8 1 2 3 4

− 2 5 x + 2 y ≤ 3 2 x − y ≤ 1 5 −2x − 10y ≤ −15

• 22
• 18

x y

Relaxed set

Y(x)

(x∗

i , y∗ i )

(x∗

c, y∗ c ) ✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 7 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Proposed approaches for Bilevel Programming

◮ Influential works by Bard and co-authors (1982–2001), Dempe and co-authors (1987–2015) and helpful reviews by [Vicente and Calamai, 1994] and [Colson et al., 2007]

Continuous Case

◮ Most of the work has been focused on the KKT reformulation, e.g. [Bard, 1988], [Visweswaran et al. 1996] and [G¨ um¨ u¸ s and Floudas, 2001]. ◮ Parametric programming theory in [Fa´ ısca et al., 2007] ◮ Nonconvex bilevel programs are tackled in [Mitsos et al., 2008], [Kleniati and Adjiman, JOGO, 2014a,b], [Paulaviˇ cius and Adjiman] ◮ Associated well-known optimization problems in [Bhattacharjee et al., 2005a;b], [Floudas and Stein, 2007], [Mitsos et al., 2008a] and [Tsoukalas et al., 2009].

Discrete & Mixed-Integer Case

◮ Discrete linear bilevel problems in [Moore & Bard, 1990], [Savard & Judice, 1996], [Dempe, Kalashnikov & R´ ıos-Mercado, 2005], [Fangh¨ anel & Dempe, 2009] ◮ Limited nonlinearity in [Edmunds & Bard, 1992] ◮ Global optimization of nonlinear mixed-integer bilevel problems in [G¨ um¨ u¸ s and Floudas, 2005], [Mitsos, 2010], [Kleniati and Adjiman, CACE, 2014c], [Paulaviˇ cius and Adjiman]

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 8 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Main components of Branch-and-Sandwich algorithm The main components of B&S are:

• 1. Tree management (with auxiliary lists of nodes)
• 2. Bounding scheme with full and outer fathoming rules
• 2a. Bounds on the inner objective function f(x, y):

◮ Inner lower bound f (k) ◮ Inner upper bound ¯ f (k) ◮ Best inner upper bound f UB

Xp

◮ Optimal value at ¯ x w(¯ x)

• 2b. Bounds on the outer objective function F(x, y):

◮ Outer lower bound F (k) ◮ Outer upper bound ¯ F (k)

• 3. Node Selection
• 4. Branching

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs 9 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 1. Tree management with auxiliary lists

−1 1 −1 1

4 5 6 7 X1 X2 X Y

Node [(n + m)-rectangle]

A node k represents subdomain: X(k) × Y (k) ⊆ X × Y ◮ B&S allows branching on x and y variables through special tree management. ◮ B&S uses more than one lists to manage nodes:

L classical list of open nodes corresponding to the outer problem; LIn list of exclusively inner open nodes ; LXp: independent lists of outer & inner nodes that cover the whole Y for a subdomain Xp of X, 1 ≤ p ≤ np (number of partition sets Xp ⊆ X).

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs10 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 1. Tree management with auxiliary lists

−1 1 −1 1

4 5 6 7 X1 X2 X Y

Node [(n + m)-rectangle]

A node k represents subdomain: X(k) × Y (k) ⊆ X × Y ◮ B&S allows branching on x and y variables through special tree management. ◮ B&S uses more than one lists to manage nodes:

L classical list of open nodes corresponding to the outer problem; LIn list of exclusively inner open nodes ; LXp: independent lists of outer & inner nodes that cover the whole Y for a subdomain Xp of X, 1 ≤ p ≤ np (number of partition sets Xp ⊆ X).

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs10 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 1. Tree management with auxiliary lists

−1 1 −1 1

4 5 6 7 X1 X2 X Y L LIn LXp p Xp {4, 6, 7} {5} {4, 6} 1 [−1.0, 0.0] {5, 7} 2 [0.0, 1.0]

Node [(n + m)-rectangle]

A node k represents subdomain: X(k) × Y (k) ⊆ X × Y ◮ B&S allows branching on x and y variables through special tree management. ◮ B&S uses more than one lists to manage nodes:

L classical list of open nodes corresponding to the outer problem; LIn list of exclusively inner open nodes ; LXp: independent lists of outer & inner nodes that cover the whole Y for a subdomain Xp of X, 1 ≤ p ≤ np (number of partition sets Xp ⊆ X).

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs10 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2. Bounding

Single level BPP reformulation

Optimal value function (OVF) for the inner problem:

w(x) = min

y∈Y {f(x, y) s.t. g(x, y) ≤ 0}.

(OVF)

Dempe, S., (2002) Foundations of bilevel programming. Kluwer, Dordrecht

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs11 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2. Bounding

Single level BPP reformulation

Optimal value function (OVF) for the inner problem:

w(x) = min

y∈Y {f(x, y) s.t. g(x, y) ≤ 0}.

(OVF) min

x,y F(x, y)

s.t. G(x, y) ≤ 0 g(x, y) ≤ 0 f(x, y) ≤ w(x) (x, y) ∈ X × Y (slBPP) Single-level BPP [Dempe, 2002]

Useful Property of (slBPP) [Mitsos and Barton, 2006]

A restriction of the inner problem yields a relaxation of the overall problem and vice versa

Dempe, S., (2002) Foundations of bilevel programming. Kluwer, Dordrecht Mitsos, A., and Barton, P.I. (2006). Issues in the development of global optimization algorithms for bilevel programs with a nonconvex inner program. Technical Report, Massachusetts Institute of Technology.

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs11 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2a. Inner Bounding Scheme

Inner Lower Bounding (ILB(k)) problem

f(k) = min

x,y {f(x, y) s.t. g(x, y) ≤ 0}

Inner Upper Bounding (IUB(k)) problem

¯ f(k) = min

x0,y

x0 s.t. f(X, y) ≤ x0 g(X, y) ≤ 0 x0 ∈ R, y ∈ Y (k)

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs12 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2a. Inner Bounding Scheme

Inner Lower Bounding (ILB(k)) problem

f(k) = min

x,y {f(x, y) s.t. g(x, y) ≤ 0}

Inner Upper Bounding (IUB(k)) problem

¯ f(k) = min

x0,y

x0 s.t. f(X, y) ≤ x0 g(X, y) ≤ 0 x0 ∈ R, y ∈ Y (k)

Best Inner Upper Bound (BIUB(p))

fBIUB

Xp

= max

• min

j∈L1

Xp

{ ¯ f(j)}, . . . , min

j∈Ls

Xp

{ ¯ f(j)}

• −1

1 −1 1

6 5 7 8 10 9 LX1 LX2 Y

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs12 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2a. Inner Bounding Scheme

Inner Lower Bounding (ILB(k)) problem

f(k) = min

x,y {f(x, y) s.t. g(x, y) ≤ 0}

Inner Upper Bounding (IUB(k)) problem

¯ f(k) = min

x0,y

x0 s.t. f(X, y) ≤ x0 g(X, y) ≤ 0 x0 ∈ R, y ∈ Y (k)

Fully fathom node k when:

• 1. f(k) = ∞,
• 2. f(k) > fBIUB

Xp

, then delete k from L (or LIn) and LXp.

Best Inner Upper Bound (BIUB(p))

fBIUB

Xp

= max

• min

j∈L1

Xp

{ ¯ f(j)}, . . . , min

j∈Ls

Xp

{ ¯ f(j)}

• −1

1 −1 1

6 5 7 8 10 9 LX1 LX2 Y

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs12 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2b. Outer Bounding Scheme

Outer Lower Bounding (LB(k)) problem

F (k) = min

x,y,µ

F(x, y), s.t. G(x, y) ≤ 0 g(x, y) ≤ 0 f(x, y) ≤ fBIUB

Xp

, (x, y) ∈ X(k) × Y (k),

Outer Upper Bounding (UB(k)) problem

¯ F (k’) = min

y

F(¯ x, y), s.t. G(¯ x, y) ≤ 0 g(¯ x, y) ≤ 0 f(¯ x, y) ≤ w(k’)(¯ x) + εf, where k′ = arg min {w(k)(¯ x) : k ∈ LXp}

−1 1 −1 1

6 5 7 8 10 9 LX1 LX2 ¯ x Y ¯ x - solution from (LB(k)) problem.

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs13 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 2b. Outer Bounding Scheme

Outer Lower Bounding (LB(k)) problem

F (k) = min

x,y,µ

F(x, y), s.t. G(x, y) ≤ 0 g(x, y) ≤ 0 f(x, y) ≤ fBIUB

Xp

, (x, y) ∈ X(k) × Y (k),

Outer Upper Bounding (UB(k)) problem

¯ F (k’) = min

y

F(¯ x, y), s.t. G(¯ x, y) ≤ 0 g(¯ x, y) ≤ 0 f(¯ x, y) ≤ w(k’)(¯ x) + εf, where k′ = arg min {w(k)(¯ x) : k ∈ LXp}

Outer fathom node k if:

• 1. F (k) = ∞,
• 2. F (k) ≥ F UB − εF ,

then move k from L to LIn.

−1 1 −1 1

6 5 7 8 10 9 LX1 LX2 ¯ x Y ¯ x - solution from (LB(k)) problem.

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs13 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 3. Node selection

Step 1 Find an independent list LXp, with a node k ∈ LXp ∩ L having the lowest outer LB (F). ◮ If several nodes with the same F exist use a node with the smallest level l(k). Step 2 Select a node k ∈ L ∩ LXp and a node kIn ∈ LIn ∩ LXp with the smallest level l(k). ◮ If several nodes with same l exist the node is selected by using one of two strategies:

−1 1 −1 1

6 5 7 8 9 LX1 LX2 Y

1 2 6 7 3 4 8 9 5

¯ F(k) F(k) 0.0 0.0 −2 −2 0.0 ¯ f(k) 0.17 0.0 0.0 0.0 −0.33 f(k) −0.33 −0.17 0.0 −0.17 −0.83

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs14 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 3. Node selection

Step 1 Find an independent list LXp, with a node k ∈ LXp ∩ L having the lowest outer LB (F). ◮ If several nodes with the same F exist use a node with the smallest level l(k). Step 2 Select a node k ∈ L ∩ LXp and a node kIn ∈ LIn ∩ LXp with the smallest level l(k). ◮ If several nodes with same l exist the node is selected by using one of two strategies:

−1 1 −1 1

6 5 7 8 9 LX1 LX2 Y

1 2 6 7 3 4 8 9 5

¯ F(k) F(k) 0.0 0.0 −2 −2 0.0 ¯ f(k) 0.17 0.0 0.0 0.0 −0.33 f(k) −0.33 −0.17 0.0 −0.17 −0.83

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs14 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 3. Node selection

Step 1 Find an independent list LXp, with a node k ∈ LXp ∩ L having the lowest outer LB (F). ◮ If several nodes with the same F exist use a node with the smallest level l(k). Step 2 Select a node k ∈ L ∩ LXp and a node kIn ∈ LIn ∩ LXp with the smallest level l(k). ◮ If several nodes with same l exist the node is selected by using one of two strategies:

−1 1 −1 1

6 5 7 8 9 LX1 LX2 Y

1 2 6 7 3 4 8 9 5

¯ F(k) F(k) 0.0 0.0 −2 −2 0.0 ¯ f(k) 0.17 0.0 0.0 0.0 −0.33 f(k) −0.33 −0.17 0.0 −0.17 −0.83

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs14 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 3. Node selection

Step 1 Find an independent list LXp, with a node k ∈ LXp ∩ L having the lowest outer LB (F). ◮ If several nodes with the same F exist use a node with the smallest level l(k). Step 2 Select a node k ∈ L ∩ LXp and a node kIn ∈ LIn ∩ LXp with the smallest level l(k). ◮ If several nodes with same l exist the node is selected by using one of two strategies:

−1 1 −1 1

6 5 7 8 9 LX1 LX2 Y

1 2 6 7 3 4 8 9 5

¯ F(k) F(k) 0.0 0.0 −2 −2 0.0 ¯ f(k) 0.17 0.0 0.0 0.0 −0.33 f(k) −0.33 −0.17 0.0 −0.17 −0.83

◮ “Best-lf” selects with the lowest f ◮ “Best-l ¯ f” selects with the lowest ¯ f

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs14 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 3. Node selection

Step 1 Find an independent list LXp, with a node k ∈ LXp ∩ L having the lowest outer LB (F). ◮ If several nodes with the same F exist use a node with the smallest level l(k). Step 2 Select a node k ∈ L ∩ LXp and a node kIn ∈ LIn ∩ LXp with the smallest level l(k). ◮ If several nodes with same l exist the node is selected by using one of two strategies:

−1 1 −1 1

6 5 7 8 9 LX1 LX2 Y

1 2 6 7 3 4 8 9 5

¯ F(k) F(k) 0.0 0.0 −2 −2 0.0 ¯ f(k) 0.17 0.0 0.0 0.0 −0.33 f(k) −0.33 −0.17 0.0 −0.17 −0.83

◮ “Best-lf” selects with the lowest f ◮ “Best-l ¯ f” selects with the lowest ¯ f

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs14 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

• 4. Branching

◮ Branching point selected using exact bisection. ◮ Two different branching variable selection strategies:

−1 1 −1 1

1 2 3

X1 X2 X Y

Figure: Branch on the lowest variable index (XY)

L LXp p Xp {2,3} {2} 1 [−1.0,0.0] {3} 2 [0.0,1.0] −1 1 −1 1

2 3

X1 X Y

Figure: Branch on the highest variable index (YX)

L LXp p Xp {2,3} {2,3} 1 [−1.0,1.0]

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs15 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBL: Branch-And-Sandwich BiLever solver

Implemented in C++ within the MINOTAUR (http://wiki.mcs.anl.gov/minotaur)

Notes on BASBL implementation

◮ We have extended MINOTAUR search tree and node management facilities ◮ B&S algorithm uses the list of inner-open nodes and also independent lists and sublists in addition to traditional list of open nodes. ◮ Moreover, every node in branch-and-sandwich tree must hold lower and upper bounds for both the inner and outer problems.

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs16 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBL: Branch-And-Sandwich BiLever solver

Implemented in C++ within the MINOTAUR (http://wiki.mcs.anl.gov/minotaur)

Notes on BASBL implementation

◮ We have extended MINOTAUR search tree and node management facilities ◮ B&S algorithm uses the list of inner-open nodes and also independent lists and sublists in addition to traditional list of open nodes. ◮ Moreover, every node in branch-and-sandwich tree must hold lower and upper bounds for both the inner and outer problems.

Solvers

◮ We have implemented an interface with GAMS, since natively available MINOTAUR solvers only ensure local optimality for NLP problems. ◮ Globally subproblems can be solved either with BARON or ANTIGONE solvers.

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs16 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBL: Branch-And-Sandwich BiLever solver

Implemented in C++ within the MINOTAUR (http://wiki.mcs.anl.gov/minotaur)

Notes on BASBL implementation

◮ We have extended MINOTAUR search tree and node management facilities ◮ B&S algorithm uses the list of inner-open nodes and also independent lists and sublists in addition to traditional list of open nodes. ◮ Moreover, every node in branch-and-sandwich tree must hold lower and upper bounds for both the inner and outer problems.

Solvers

◮ We have implemented an interface with GAMS, since natively available MINOTAUR solvers only ensure local optimality for NLP problems. ◮ Globally subproblems can be solved either with BARON or ANTIGONE solvers.

Test problems and test facilities

◮ All bilevel instances are described and read through extended AMPL interface ◮ A Library of Bilevel Test Problems (description in a format compatible with BASBL): http://basblsolver.github.io/BASBLib/

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs16 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

AMPL interface

Description of bilevel problem sib 1997 01 in AMPL language

var x >= 0, <= 12.5; # Outer variable bounds var y >= 0, <= 50; # Inner variable bounds var l{1..3} >= 0, <= 200; # KKT multipliers bounds # Outer objective minimize outer_obj: 16*x^2 + 9*y^2; subject to # Outer constraints:

• uter_con_1:
• 4*x + y <= 0;

# Inner objective: inner_obj: (x + y -20)^4 = 0; # Inner constraints: inner_con_1: 4*x + y -50 <= 0; # KKT conditions: stationarity_1: 3*(x + y -20)^3 + l[1] - l[2] + l[3] = 0; complementarity_1: l[1]*(4*x + y -50) = 0; complementarity_2: l[2]*y = 0; complementarity_3: l[3]*(y - 50) = 0;

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs17 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBLib - A Library of Bilevel Test Problems [Paulavicius & Adjiman, 2017]

◮ An actively growing online collection BASBLib of general bilevel test problems, gathered from the various sources and devoted to bilevel programming. ◮ The library is designed as an open resource to which other researchers in the bilevel programming community can easily contribute. ◮ Bilevel programming involves two optimization problems (outer–inner), the proposed classification is based on the nature of these problems. ◮ Currently, we distinguish the following classes (types) of BP problems:

Paulavicius, R. & Adjiman, C. S. (2017). BASBLib - a library of bilevel test problems DOI:10.5281/zenodo.595940

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs18 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBLib - A Library of Bilevel Test Problems [Paulavicius & Adjiman, 2017]

◮ An actively growing online collection BASBLib of general bilevel test problems, gathered from the various sources and devoted to bilevel programming. ◮ The library is designed as an open resource to which other researchers in the bilevel programming community can easily contribute. ◮ Bilevel programming involves two optimization problems (outer–inner), the proposed classification is based on the nature of these problems. ◮ Currently, we distinguish the following classes (types) of BP problems:

Paulavicius, R. & Adjiman, C. S. (2017). BASBLib - a library of bilevel test problems DOI:10.5281/zenodo.595940

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs18 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBLib - A Library of Bilevel Test Problems [Paulavicius & Adjiman, 2017]

◮ An actively growing online collection BASBLib of general bilevel test problems, gathered from the various sources and devoted to bilevel programming. ◮ The library is designed as an open resource to which other researchers in the bilevel programming community can easily contribute. ◮ Bilevel programming involves two optimization problems (outer–inner), the proposed classification is based on the nature of these problems. ◮ Currently, we distinguish the following classes (types) of BP problems:

Paulavicius, R. & Adjiman, C. S. (2017). BASBLib - a library of bilevel test problems DOI:10.5281/zenodo.595940

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs18 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBLib - A Library of Bilevel Test Problems [Paulavicius & Adjiman, 2017]

◮ An actively growing online collection BASBLib of general bilevel test problems, gathered from the various sources and devoted to bilevel programming. ◮ The library is designed as an open resource to which other researchers in the bilevel programming community can easily contribute. ◮ Bilevel programming involves two optimization problems (outer–inner), the proposed classification is based on the nature of these problems. ◮ Currently, we distinguish the following classes (types) of BP problems:

◮ outer linear-inner linear bilevel problems (LP-LP) ◮ outer linear-inner quadratic bilevel problems (LP-QP) ◮ outer quadratic-inner quadratic bilevel problems (QP-QP) ◮ outer linear-inner nonlinear bilevel problems (LP-NLP) ◮ outer quadratic-inner nonlinear bilevel problems (QP-NLP) ◮ outer nonlinear-inner nonlinear bilevel problems (NLP-NLP)

Paulavicius, R. & Adjiman, C. S. (2017). BASBLib - a library of bilevel test problems DOI:10.5281/zenodo.595940

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs18 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Online BASBLib resources

http://basblsolver.github.io/BASBLib/

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs19 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Online BASBLib resources

DOI:10.5281/zenodo.595940 https://github.com/basblsolver/BASBLib

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs19 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBL performance solving test problems from BASBLib using default options

Type Iter. Nopt tB(s) tG(s) #ILB #IUB #LB #ISP #UB LP-LP 1.00 0.94 0.23 0.13 1.00 1.00 1.00 0.94 0.94 LP-QP 1.20 1.00 0.33 0.20 1.40 1.40 1.20 1.00 1.00 QP-QP 1.00 1.00 0.37 0.26 1.00 1.00 1.00 1.00 1.00 LP-NLP 23.00 6.54 12.96 8.54 81.69 78.46 27.46 8.46 8.46 QP-NLP 5.76 1.82 2.42 1.43 15.24 13.41 9.06 2.59 2.59 NLP-NLP 1.00 1.00 26.62 26.40 1.00 1.00 1.00 1.00 1.00 Table: Average results

Iter.

• total number of BASBL iterations

Nopt

• the number of UB problems solved

before the optimal solution was computed for the first time tB(s)

• the total time (in seconds of

wall-clock time) spent in BASBL execution tG(s)

• the total time (in seconds of

wall-clock time) spent in GAMS #

• the total number of

subproblems solved ✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs20 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

BASBL performance solving test problems from BASBLib using default options

Type Iter. Nopt tB(s) tG(s) #ILB #IUB #LB #ISP #UB LP-LP 1.00 0.94 0.23 0.13 1.00 1.00 1.00 0.94 0.94 LP-QP 1.20 1.00 0.33 0.20 1.40 1.40 1.20 1.00 1.00 QP-QP 1.00 1.00 0.37 0.26 1.00 1.00 1.00 1.00 1.00 LP-NLP 23.00 6.54 12.96 8.54 81.69 78.46 27.46 8.46 8.46 QP-NLP 5.76 1.82 2.42 1.43 15.24 13.41 9.06 2.59 2.59 NLP-NLP 1.00 1.00 26.62 26.40 1.00 1.00 1.00 1.00 1.00 Table: Average results

Iter.

• total number of BASBL iterations

Nopt

• the number of UB problems solved

before the optimal solution was computed for the first time tB(s)

• the total time (in seconds of

wall-clock time) spent in BASBL execution tG(s)

• the total time (in seconds of

wall-clock time) spent in GAMS #

• the total number of

subproblems solved

Type Iter. Nopt tB(s) tG(s) #ILB #IUB #LB #ISP #UB LP-LP 1.00 1.00 0.23 0.13 1.00 1.00 1.00 1.00 1.00 LP-QP 1.00 1.00 0.35 0.21 1.00 1.00 1.00 1.00 1.00 QP-QP 1.00 1.00 0.34 0.22 1.00 1.00 1.00 1.00 1.00 LP-NLP 1.00 1.00 0.45 0.28 1.00 1.00 1.00 1.00 1.00 QP-NLP 1.00 1.00 0.66 0.55 1.00 1.00 1.00 1.00 1.00 NLP-NLP 1.00 1.00 0.82 0.66 1.00 1.00 1.00 1.00 1.00 Table: Median results

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs20 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Comparison of different branching and node selection heuristics

m b

• 1
• 1
• 5

m b

• 1
• 1
• 6

m b

• 1
• 1
• 6

v m b

• 1
• 1
• 7

m b

• 1
• 1
• 8

m b

• 1
• 1
• 9

m b

• 1
• 1
• 1

m b

• 1
• 1
• 1

1 m b

• 1
• 1
• 1

3 20 40 60 80 100 120 140

5 130 7 7 4 7 5 36 7 5 131 7 8 4 7 5 34 7 8 88 4 11 6 4 8 26 11

Number of BASBL iterations Branch-YX-Select-lf Branch-YX-Select-l ¯ f Branch-XY-Select-lf

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs21 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Mixed Integer Case: Preliminary results

With εf = 10−5 and εF = 10−3

Number of variables No. Source Problem xi xc yi yc F UB Nodes 1 Moore & Bard (1990) Example 1 1 1 −22 7 2 Moore & Bard (1990) Example 2 1 1 5 13 3 Edmunds & Bard (1992) Equation 3 1 1 4 9 1 4 Sahin & Ciric (1998) Example 4 2 2 −400 1 5 Dempe (2002) Equation 8.11 2 2 −10.4 3 6 Mitsos (2010) am 1 0 0 1 01 1 1 −1 1 7 Mitsos (2010) am 1 1 1 0 01 1 1 1 0.5 11 8 Mitsos (2010) am 1 1 1 1 01 1 1 1 1 −1 13 9 Mitsos (2010) am 1 1 1 1 02 1 1 1 1 0.209 1 10 Mitsos (2010) am 3 3 3 3 01 3 3 3 3 −2.5 1

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs22 / 23

Introduction B&S Algorithm BASBL solver MINOTAUR toolkit & BASBL solver BASBLib - A Library of Bilevel Test Problems Numerical results Conclusions

Conclusions

Conclusions

◮ Introduced BASBL - a new solver for the global optimization of (mixed-integer) nonconvex bilevel problems is implemented within MINOTAUR ◮ Introduced BASBLib - an online test library, designed to facilitate contributions from the bilevel optimization community. ◮ Plenty of scope to improve bounding problems.

Thank You For Your Attention! Questions ?

BASBL sources

BASBL: http://basblsolver.github.io/home/ BASBLib: https://basblsolver.github.io/BASBLib

c.adjiman@imperial.ac.uk remigijus.paulavicius@imperial.ac.uk

✠ : Claire S. Adjiman, Polyxeni-M. Kleniati, Remigijus Paulaviˇ cius A deterministic global optimisation algorithm for mixed-integer nonlinear bilevel programs23 / 23