BranchandSandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results
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Abstract
In the first part of this work, we presented a global optimization algorithm, BranchandSandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem (Kleniati and Adjiman in J Glob Optim, 2014). The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branchandbound tree, where two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. In the present paper, the theoretical properties of the proposed algorithm are investigated and finite \(\varepsilon \)convergence to a global solution of the bilevel problem is proved. Thirtyfour problems from the literature are tackled successfully.
Keywords
Bilevel programming Nonconvex inner problem Branch and bound1 Introduction
Notice that due to the nonconvexity of the inner problem in (1), we adopt the socalled optimistic formulation of the bilevel problem that implies some cooperation between the leader and the follower. In particular, for different globally optimal solutions of the inner problem to which the follower is indifferent, we optimize in favor of the leader’s (outer) objective and constraints; hence, the outer minimization in (1) is with respect to the whole set of variables. For more details about the optimistic formulation and its alternative pessimistic formulation, the interested reader is referred to [5, 9] and [17, 28], respectively.
Problem (1) has been tackled extensively for specific classes of the participating functions. For instance, the linear and the nonlinear but convex bilevel programming problems have been addressed in [4, 5, 9, 12, 13, 26, 27]. A thorough bibliographic review can be found in [10]. On the other hand, general bilevel programming problems with a nonconvex inner problem have received much less attention due to their intrinsic difficulties and currently are only addressed in [21, 23]. The proposed methods therein are for very general nonlinear bilevel problems, restricted solely by the absence of inner equality constraints.
Remark 1
Part of the BranchandSandwich algorithm is based on the property of Remark 1. For instance, we compute a constant upper bound on \(w(x)\) for all the \(x\) values over the domain under consideration and then using this upper bound on \(w(x)\) in formulation (2), we derive a relaxation of the overall problem. This relaxation plays the role of our proposed lower bounding problem. In order to compute a valid upper bound on \(w(x)\) for all the \(x\) values, we employ a semiinfinite formulation for the proposed inner upper bounding problem which we tackle via its tractable KKT relaxation. For this reason, a regularity condition is imposed for all the \(x\) values. Then, it is possible to employ the inner KKT conditions where necessary, e.g., in the inner upper bounding problem and in the overall lower bounding problem.
In the present article, we focus on analyzing the theoretical properties of the algorithm and testing its application on suitable numerical test problems. In particular, we prove its finite \(\varepsilon \)convergence to a global solution of the bilevel problem and we present the numerical results from its application to thirtyfour test problems from the literature. We also illustrate in detail the stepbystep application of the proposed algorithm on two test problems.
The paper is organized as follows. In Sect. 2, we recapitulate all the necessary notation introduced in [16]. In Sect. 3, we present a complete analysis of the convergence properties of the BranchandSandwich algorithm along with our main convergence result. An additional theoretical result regarding the exhaustiveness of our partitioning scheme follows in Sect. 4. Illustrative examples and numerical results are presented in Sect. 5. Concluding remarks are discussed in Sect. 6. Finally, the Appendix complements Sect. 5.
2 Notation
The following assumptions, definitions and nomenclature are used throughout the paper. At the end of this section, we also provide a brief statement of the proposed algorithm.
2.1 Assumptions
In addition to common assumptions, such as continuity, twice differentiability of the participating functions and compactness of the host sets, we also make the assumption below.
Assumption 1
A constraint qualification holds for the inner problem (3) for all values of \(x\).
2.2 Definitions and nomenclature
In this work, our aim is to compute \(\varepsilon \)optimal solutions as defined below.
Definition 1
Definition 1 implies that we apply \(\varepsilon _f\)optimality in the inner problem throughout the paper. Next, we remind the reader of all essential definitions relevant to our branching and bounding schemes, while a full exposition can be found in Part I [16].
Definition 2
Summary of bounds
Description  Symbol  Problem name [16] 

(Nonconvex) inner lower bound  \(f^{(k),\mathrm{L}}\)  (ILB) 
(Convex) relaxed inner lower bound  \({{\underline{f}}}^{(k)}\)  (RILB) 
(Nonconvex) inner upper bound  \(f^{(k),\mathrm U}\)  (IUB) 
(Nonconvex) relaxed inner upper bound  \(\bar{f}^{(k)}\)  (RIUB) 
(Nonconvex) inner subproblem at given \(x\)  \(w^{(k)}(x)\)  (ISP) 
(Convex) relaxed inner subproblem at given \(x\)  \({{\underline{w}}}^{(k)}(x)\)  (RISP) 
(Nonconvex) outer lower bound  \({{\underline{F}}}^{(k)}\)  (LB) 
(Nonconvex) outer upper bound  \(\bar{F}^{(k)}\)  (UB) 
Summary of core lists of nodes. \(\mathcal{L} \cap \mathcal L_\mathrm{In}=\emptyset \)
Description  Symbol  Comment 

List of open nodes w.r.t. the overall problem  \(\mathcal{L}\)  
List of open nodes w.r.t. the inner problem only  \(\mathcal L_\mathrm{In}\)  Also called list of outerfathomed nodes 
Summary of auxiliary (pairwise disjoint) lists of nodes. \(\{\mathcal{X}_p\subseteq X:p \in P\}\) is a partition of \(X\)
Description  Symbol  Comment 

Independent list for partition set \(\mathcal{X}_1\)  \(\mathcal{L}^{1}\)  Contains nodes in \(\mathcal{L} \cup \mathcal L_\mathrm{In}\) corresponding to \(\mathcal{X}_1\times Y\) 
\(\vdots \)  \(\vdots \)  
Independent list for partition set \(\mathcal{X}_p\)  \(\mathcal{L}^{p}\)  Contains nodes in \(\mathcal{L} \cup \mathcal L_\mathrm{In}\) corresponding to \(\mathcal{X}_p\times Y\) 
Remark 2
Finally, in order to branch at the most promising nodes among all the ones that have not yet fully fathomed, i.e., the nodes in \(\mathcal{L}\cup \mathcal L_\mathrm{In}\), we apply the following selection rule.
Definition 3
 (i)
find a node in \(\mathcal{L}\) with lowest overall lower bound: \(k^\mathrm{LB} = \mathop {\text {arg min}}\limits \nolimits _{j \in \mathcal{L}} \{{{\underline{F}}}^{(j)}\}\);
 (ii)
find the corresponding \(\mathcal{X}_p\) subdomain, \(p\in P\), such that \(k^\mathrm{LB} \in \mathcal{L}^{p}\);
 (iii)select a node \(k \in \mathcal{L} \cap \mathcal{L}^{p}\) and a node \(k_\mathrm{In} \in \mathcal{L_\mathrm{In}} \cap \mathcal{L}^{p}\), if non empty, based on$$\begin{aligned} k:= \mathop {\text {arg min}}\limits _i \{{{\underline{f}}}^{(i)} \mid i := \mathop {\text {arg min}}\limits _{j \in \mathcal{L}^{p}}\{ l^{(j)}\}\}. \end{aligned}$$(ISR)
2.3 Algorithm
To end this section a brief statement of the BranchandSandwich algorithm is given below.
Algorithm 1
 Step 0:

Initialize.
 Step 1:

Compute inner and outer bounds at the root node.
 Step 2:

If \(\mathcal{L}=\emptyset \), stop; otherwise, select a list \(\mathcal{L}^{p}\), a node \(k \in \mathcal{L}^{p} \cap \mathcal{L}\) and, if relevant, a node \(k_\mathrm{In} \in \mathcal{L}^{p} \cap \mathcal L_\mathrm{In}\).
 Step 3:

Branch on selected node(s) to create child nodes & amend the lists of nodes (list management).
 Steps 4–5:

Compute inner bounds at child nodes in \(\mathcal{L}\cup \mathcal L_\mathrm{In}\). Apply full fathoming, if needed.
 Steps 6–7:

Compute outer bounds at child nodes in \(\mathcal{L}\). Apply outer fathoming, if needed. Goto Step 2.
3 Proof of convergence
A general theory concerning the convergence of branchandbounds methods can be found in [14, 15]. In this section, we apply this theory to both branchandbound schemes, inner and outer, of the BranchandSandwich algorithm introduced in [16] in order to set the foundations for its overall convergence, proved in Theorem 6, to an \(\varepsilon \)optimal solution of problem (1). It is assumed throughout that an exhaustive partitioning scheme is used. In Sect. 4, we show that this is true for the partitioning scheme used in the algorithm. We first recapitulate some relevant definitions used in our theoretical results.
Definition 4
Definition 5
Definition 6
[15, Def. IV.8.] The “fathombyinfeasibility” rule is called certain in the limit if every infinite decreasing nested sequence \(\{k_{q}\}\) of successively refined nodes converges to a feasible singleton.
Definition 7
Definition 8
[15, Def. IV.6.] A selection operation is said to be bound improving if, at least each time after a finite number of iterations, at least one node where the actual lower bound is attained is selected for further partition.
Remark 3
3.1 Inner convergence properties
Theorem 1
Proof
Corollary 1
The lowest inner lower bound in \(\mathcal{L}^{p_q}\) is convergent to \(w(\bar{x})\) as \(\mathcal{X}_{p_{q}}\) converges to singleton \(\bar{x}\).
Theorem 2
 (i)
a consistent bounding scheme;
 (ii)
a boundimproving selection operation;
 (iii)
it is finite \(\varepsilon _f\)convergent.^{1}
Proof
(i) By Corollary 1; (ii) the inner selection rule (ISR) is boundimproving; (iii) by (i) and (ii) based on [15, Th. IV.3.]. \(\square \)
3.2 Outer convergence properties
Theorem 3
The fathoming rule by outer infeasibility of the BranchandSandwich algorithm is certain in the limit.
Proof
Remark 4
Finally, we can show that the BranchandSandwich algorithm is convergent based on [15, Th. IV.3.]. In particular, we show that the bounding scheme is consistent (cf. Definition 7) and the selection operation is bound improving (cf. Definition 8).
Theorem 4
The bounding scheme of the BranchandSandwich algorithm is consistent.
Proof
Theorem 5
The selection operation of the BranchandSandwich algorithm (cf. Definition 3) is bound improving.
Proof
From Definition 3, recall that (i) \(k^\mathrm{LB} \in \mathcal{L}\) is such that \(k^\mathrm{LB} = \mathop {\text {arg min}}\limits \nolimits _{j \in \mathcal{L}} \{{{\underline{F}}}^{(j)}\}\), (ii) \(\mathcal{X}_p\), \(p\in P\), is such that \(k^\mathrm{LB} \in \mathcal{L}^{p}\), and (iii) \(k \in \mathcal{L}\) is such that \(k \in \mathcal{L}^{p}\) satisfying (ISR). If \(k = k^\mathrm{LB}\) then the node with the lowest overall bound is selected and our proof is complete. Let us now examine the case where \(k\ne k^\mathrm{LB}\). Observe that the pair \((l^{(k)},{{\underline{f}}}^{(k)})\) dominates the pair \((l^{(k^\mathrm{LB})},{{\underline{f}}}^{(k^\mathrm{LB})})\), since otherwise node \(k^\mathrm{LB}\) would have been selected. In other words, we have that either \(l^{(k)} < l^{(k^\mathrm{LB})}\) or \({{\underline{f}}}^{(k)} < {{\underline{f}}}^{(k^\mathrm{LB})} \le f^{\mathrm{UB},p}\). The branching strategy (cf. [16, Def. 6]) and the inner bounding scheme imply that the values \(l^{(k)}\), \({{\underline{f}}}^{(k)}\) are nondecreasing, while Lemma 1 in [16] implies that \(f^{\mathrm{UB},p}\) is nonincreasing, over refined host sets. Hence, node \(k^\mathrm{LB}\) will eventually be either fathomed by inner value dominance, i.e., \({{\underline{f}}}^{(k^\mathrm{LB})} > f^{\mathrm{UB},p}\), in which case another node will hold the lowest overall lower bound and then the same arguments apply, or selected for exploration based on \(l^{(k^\mathrm{LB})} \le l^{(k)}\) and/or \({{\underline{f}}}^{(k^\mathrm{LB})} \le {{\underline{f}}}^{(k)}\) being satisfied. The node with the lowest overall lower bound is guaranteed to be selected after finitely many iterations due to the finite \(\varepsilon _f\)convergence of the inner branchandbound scheme shown in Theorem 2. \(\square \)
Our main result regarding the convergence of BranchandSandwich is now stated.
Theorem 6
The BranchandSandwich algorithm is \(\varepsilon \)convergent, such that at termination we have \(F^\mathrm{UB}  F^\mathrm{LB} \le \varepsilon _F\), where \(F^\mathrm{LB} = \min _{k \in \mathcal{L}} {{\underline{F}}}^{(k)}\), and the incumbent solution \((x^\mathrm{UB},y^\mathrm{UB})\) is an \(\varepsilon \)optimal solution of problem (1).
Proof
The finite \(\varepsilon \)convergence property of BranchandSandwich follows from Theorem IV.3. in [15] and Theorems 4–5. Furthermore, termination of the algorithm implies that \(F^\mathrm{LB} \ge F^\mathrm{UB} \varepsilon _F\), i.e., condition (7) of outer \(\varepsilon _F\)optimality is true. Using Theorem 3 in [16], this implies that point \((x^\mathrm{UB},y^\mathrm{UB})\) is \(\varepsilon \)optimal in (1) (cf. Definition 1). \(\square \)
4 Proof of exhaustiveness
This section concerns the exhaustiveness of our subdivision process (c.f. BranchandSandwich subdivision process [16, Sect. 4.2, Def. 6]) and complements Sect. 3 above. Prior to our proof of exhaustiveness, we state below a wellknown preliminary lemma.
Lemma 1
 (i)
for infinitely many \(q\) the subdivision of \(k_{q}\) is a bisection;
 (ii)there exists a constant \(\rho \in (0,1)\) such that for every \(q\):$$\begin{aligned} \delta (k_{q+1}) \le \rho \delta (k_{q}). \end{aligned}$$
Theorem 7
(Theorem 1 in [16]) The subdivision process of the BranchandSandwich algorithm is exhaustive.
Proof
Remark 5
5 Numerical results and examples
In this section, we report the application of the BranchandSandwich algorithm to 34 literature problems. In the first two columns of Table 4 the problem number, the original name and the source of all the instances on which BranchandSandwich was tested are shown. These include a few literature problems from [5, 6, 8, 18, 19, 25] and all the test problems from [20] for which Assumption 1 is satisfied.^{2} The two problem instances that appear to have no source in Table 4, i.e., problem No. 15 and problem No. 34, are variants of Example 3.13 and Example 3.28, respectively, from [20], and are stated in the Appendix. Both satisfy Assumption 1; the former thanks to the Abadie constraint qualification and the latter thanks to the linear/concave constraint qualification. While all problems are solved in this paper, in the third column of Table 4, we state the example numbers for the few problems selected for a more detailed discussion.
Next, from left to right, starting with the fourth column of Table 4, we report whether or not the inner problem is nonconvex (NC), the dimension of the outer variable vector, the dimension of the inner variable vector, the number of constraints in the outer problem, and the number of constraints in the inner problem.
In order to tackle problem No. 30, we first reformulated it by replacing variable \(y_2 \in [0.1,10]\) with a new variable \(y_2'=1/y_2\) with the consistent bounds. This transformation was proposed in [20, Example 4.5] to ensure the satisfiability of the Abadie constraint qualification since it yields linear constraints for the inner problem. Therefore, Assumption 1 is satisfied and our algorithm applies.
Preliminary numerical results with \(\varepsilon _f=10^{5}\) and \(\varepsilon _F=10^{3}\) for all problems, except No. 19–20 where \(\varepsilon _F=10^{1}\). \(N_\mathrm{opt}\) is the number of outer upper bounding problems solved before the optimal solution is computed for the first time
No.  BranchandSandwich method  Mitsos et al. method [21]  

\(F^\mathrm{*}\)  \(F^\mathrm{UB}\)  \(N_\mathrm{opt}\)  #UBD  #LBD  #Nodes  \(N_\mathrm{opt}\)  #UBD  #LBD  
1  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
2  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(3\) 
3  \(\infty \)  \(\infty \)  \(0\)  \(0\)  \(1\)  \(1\)  \(0\)  \(1\)  \(3\) 
4  \(1\)  \(1\)  \(2\)  \(2\)  \(3\)  \(3\)  \(1\)  \(1\)  \(3\) 
5  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(3\) 
6  \(0.5\)  \(0.5\)  \(6\)  \(6\)  \(11\)  \(11\)  \(1\)  \(1\)  \(3\) 
7  \(1\)  \(1\)  \(2\)  \(2\)  \(2\)  \(3\)  \(1\)  \(1\)  \(1\) 
8  \(\infty \)  \(\infty \)  \(0\)  \(0\)  \(1\)  \(1\)  \(0\)  \(1\)  \(3\) 
9  \(0\)  \(0\)  \(1\)  \(1\)  \(1\)  \(1\)  \(2\)  \(2\)  \(2\) 
10  \(1\)  \(1\)  \(2\)  \(2\)  \(2\)  \(3\)  \(1\)  \(1\)  \(1\) 
11  \(0.5\)  \(0.5\)  \(5\)  \(5\)  \(7\)  \(11\)  \(1\)  \(1\)  \(3\) 
12  \(0.8\)  \(0.8\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\)  \(1\) 
13  \(0\)  \(0\)  \(3\)  \(4\)  \(11\)  \(11\)  \(14\)  \(10\)  \(17\) 
14  \(1\)  \(1\)  \(4\)  \(7\)  \(15\)  \(27\)  \(2\)  \(2\)  \(3\) 
15  \(1\)  \(1\)  \(4\)  \(7\)  \(15\)  \(23\)  –  –  – 
16  \(0.25\)  \(0.25\)  \(5\)  \(5\)  \(10\)  \(15\)  \(1\)  \(3\)  \(7\) 
17  \(0\)  \(0\)  \(2\)  \(5\)  \(8\)  \(13\)  \(1\)  \(1\)  \(3\) 
18  \(2\)  \(2\)  \(7\)  \(7\)  \(13\)  \(19\)  \(1\)  \(1\)  \(3\) 
19  \(0.1875\)  \(0.1875\)  \(4\)  \(7\)  \(35\)  \(55\)  \(2\)  \(18\)  \(37\) 
20  \(0.25\)  \(0.25\)  \(13\)  \(13\)  \(25\)  \(49\)  \(3\)  \(2\)  \(3\) 
21  \(0.258\)  \(0.259\)  \(4\)  \(4\)  \(7\)  \(11\)  \(18\)  \(14\)  \(27\) 
22  \(0.3125\)  \(0.3125\)  \(5\)  \(11\)  \(26\)  \(39\)  \(3\)  \(3\)  \(5\) 
23  \(0.2095\)  \(0.2094\)  \(2\)  \(2\)  \(3\)  \(3\)  \(1\)  \(2\)  \(5\) 
24  \(0.2095\)  \(0.2094\)  \(2\)  \(2\)  \(3\)  \(3\)  \(8\)  \(5\)  \(11\) 
25  \(1.755\)  \(1.755\)  \(5\)  \(5\)  \(10\)  \(11\)  \(19\)  \(14\)  \(27\) 
26  \(0\)  \(0\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
27  \(0\)  \(0\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
28  \(17\)  \(17\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
29  \(22.5\)  \(22.5\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
30  \(0.193616\)  \(0.193616\)  \(2\)  \(2\)  \(2\)  \(3\)  \(1\)  \(1\)  \(1\) 
31  \(1.75\)  \(1.75\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
32  \(29.2\)  \(29.2\)  \(1\)  \(1\)  \(1\)  \(1\)  –  –  – 
33  \(2.35\)  \(2.35\)  \(1\)  \(1\)  \(1\)  \(1\)  \(2\)  \(2\)  \(5\) 
34\(^\mathrm{a}\)  \(10\)  \(10\)  \(1\)  \(2\)  \(3\)  \(3\)  2\(^\mathrm{a}\)  2\(^\mathrm{a}\)  2\(^\mathrm{a}\) 
In problems No. 1–10, 12 and 18, the BranchandSandwich algorithm achieved higher accuracy in the outer objective value than required to meet the convergence tolerance, e.g., \(\varepsilon _F=10^{5}\), for the same number of nodes. There were also many instances for which convergence was achieved at the root node, such as problems No. 1–3, 5, 8–9, 12, 18, 25–29, 31–34. For instance, consider problem 12, which despite being a variation of No. 11, requires one node only for termination. For this example, the lower bounding problem computes the optimal solution of the bilevel problem at the root node; then, the convex relaxation of the inner problem for the obtained \(x\) yields the actual optimal objective value of the inner problem for this \(x\) value and, as a result, a feasible outer upper bounding problem. On the other hand, No. 19–20 were the slowest to converge because their convergence was dependent on eliminating KKT inner suboptimal points that were yielding an outer objective value lower than the actual optimal value. For instance, No. 20 terminated with \(F^\mathrm{UB}=F^*=0.25\), but with a lower bound of \(F^\mathrm{LB} = 0.19\).
The number of nodes reported in Table 5 is based on using the proposed subdivision process of BranchandSandwich (cf. [16, Def. 6]) and on solving the inner upper bounding and outer lower bounding problems, respectively (RIUB) and (LB), to global optimality, as required, using the \(\alpha \)BB solver [1, 2, 3]. The outer upper bounding problem (UB), although nonconvex, does not need to be solved to global optimality. Hence, for all the examples tested, all (UB) subproblems were solved with the local solver MINOS [22]. Finally, the inner lower bounding problems (ILB) and the inner subproblems (ISP) were solved using the proposed convex relaxed problems (RILB) and (RISP), respectively. These problems were derived by employing the \(\alpha \)BB convexifications techniques [2,3] and then solved with MINOS.
Before comparing the performance of our approach and the Mitsos et al. approach, we note that a direct comparison of performance is difficult because the formulation and size of the subproblems solved in both approaches are different. This is compounded by the fact that only some subproblems are solved to global optimality in BranchandSandwich whereas all subproblems are solved to global optimality in the Mitsos et al. approach. Thus, one must be cautious in drawing conclusions from the early analysis presented here. It can be seen from Table 5 that in many cases, the numbers of subproblems in both algorithms are comparable. For example, fewer subproblems are solved with the Mistos et al. approach in problems 16, 17, 18, and fewer subproblems are solved with the BranchandSandwich algorithm for problems 13, 21 and 25. These results indicate that the partitioning of the \(Y\) space can prove very beneficial. This preliminary comparison is very encouraging and motivates the further development of the BranchandSandwich algorithm.
In the remainder of this section, the BranchandSandwich algorithm is demonstrated in detail for two examples, problem No. 11 and problem No. 17. The computed values in the worked examples are displayed with up to two significant digits.
Example 1
 Step 0:
 \(\mathrm{Iter}=0\) and we set \(\mathcal{L}=\mathcal L_\mathrm{In}=\emptyset \), \(F^\mathrm{UB}=\infty \), \((x^\mathrm{UB}, y^\mathrm{UB})=\emptyset \).
 Step 1:
 We set \(n_\mathrm{node}=1\) and compute \({{\underline{f}}}^{(1)}=2.50\) and \(\bar{f}^{(1)}=0.17\). The latter value sets \(f_X^\mathrm{UB}=0.17\). Next, we compute \({{\underline{F}}}^{(1)}=2\) at \((x^{(1)},y^{(1)})=(1,1)\). Since the outer lower bounding is feasible, we add node \(1\) to the list \(\mathcal{L}\) with all the computed information:where the last field is the level of node \(1\): \(l^{(1)}=0\). Also, we set \(p=1\) and$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L} =&\,&\{1:&2.50&0.17&2&1&0&\}, \end{array} \end{aligned}$$We then compute \({{\underline{w}}}(1)=1.50\), which yields \(\bar{F}^{(1)}=\infty \).$$\begin{aligned} \mathcal{X}_1=\left[ 1,1\right] , \mathcal{L}^{1}=\{1\}, f^{\mathrm{UB},1}=f_X^\mathrm{UB}=0.17. \end{aligned}$$
 Step 2:

\(\mathbf{Iter=1}\) (since \(\mathcal{L}\ne \emptyset \)). We select node \(1 \in \mathcal{L} \cap \mathcal{L}^{1}\) and remove it from \(\mathcal{L}\). As a result, at this point \(\mathcal{L}=\emptyset \) and \(\mathcal{L}^{1} =\{1\}\).
 Step 3:
 We branch on \(y=0\), i.e., the midpoint of variable \(y\), and create nodes:This results in \(\mathcal{L}^{1} =\{2,3\}\) corresponding to \(\mathcal{X}_1=[1,1]\).$$\begin{aligned} \begin{array}{llllr} 2 &{}:=&{} \{(x,y) \in \mathrm{I\!R}^2 \mid &{}1\le x\le 1,&{}1\le y\le 0\}, \\ 3 &{}:=&{} \{(x,y) \in \mathrm{I\!R}^2 \mid &{} 1\le x\le 1, &{}0\le y\le 1\}. \end{array} \end{aligned}$$
 Step 4:
 We compute \({{\underline{f}}}^{(2)}= 0.60\) and \({{\underline{f}}}^{(3)}= 0.96\) and add both nodes to \(\mathcal{L}\):The first and last properties, i.e., \({{\underline{f}}}^{(i)}\) and \(l^{(i)}\), \(i=1,2\), where \(l^{(2)}=l^{(3)}=1\), are set based on the new nodes \(2\) and \(3\); the other values are inherited from node \(1\).$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L}= &{} &{} \{2: &{} 0.60 &{} 0.17 &{} 2 &{} 1 &{} 1 \\ &{} &{} 3: &{} 0.96 &{} 0.17 &{} 2 &{} 1 &{} 1\}.&{}\\ \end{array} \end{aligned}$$
 Step 5:

We compute \(\bar{f}^{(2)}= 0\) and \(\bar{f}^{(3)}=0.17\). The former value updates \(\bar{f}^{(2)}\) in \(\mathcal{L}\), as well as the best inner upper bound for list \(\mathcal{L}^{1}\): \(f^{\mathrm{UB},1}=0\).
 Step 6:

We compute \({{\underline{F}}}^{(2)}= 2\) at \((1,1)\) and \({{\underline{F}}}^{(3)}= 1\) at \((1,0)\).
 Step 7:

For \(i=2\), we set \(\bar{x}=x^{(2)}=1\) and compute \({{\underline{w}}}^{(2)}(\bar{x})=0.24\) and \({{\underline{w}}}^{(3)}(\bar{x})=0.83\). The lowest value is given at node \(3\), where we compute \(\bar{F}^{(3)} =0\) at \((1,1)\) and, as a result, we update the incumbent value to \(F^\mathrm{UB}=0\). For \(i=3\), we set \(\bar{x}=x^{(3)}=1\), but we already have computed \({{\underline{w}}}^{(2)}(\bar{x})\), \({{\underline{w}}}^{(3)}(\bar{x})\) and \(\bar{F}^{(3)}\) for \(\bar{x}=1\); hence, no further computation is made. Set \(n_\mathrm{node}=3\).
 Step 2:

\(\mathbf{Iter=2}\). The selection rule of Definition 3 initially chooses node \(2 \in \mathcal{L}\), which then points to the domain \(\mathcal{X}_1\) via list \(\mathcal{L}^{1}\), since \(2 \in \mathcal{L} \cap \mathcal{L}^{1}\). Node \(3 \in \mathcal{L} \cap \mathcal{L}^{1}\) is then selected due to (ISR) and removed from \(\mathcal{L}\). As a result, at this point \(\mathcal{L}=\{2\}\) and \(\mathcal{L}^{1} =\{2,3\}\).
 Step 3:

We branch on \(x=0\), resulting in \(\mathcal{L}^{1} =\{\{2,4\},\{2,5\}\}\) corresponding to \(\mathcal{X}_1=[1,1]\).
 Step 4:
 We compute \({{\underline{f}}}^{(4)}= 0.83\) and \({{\underline{f}}}^{(5)}= 0.49\). Both are added to \(\mathcal{L}\):$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L}= &{} &{} \{2: &{} 0.60 &{} 0 &{} 2 &{} 1 &{} 1 \\ &{} &{} 4: &{} 0.83 &{} 0.17 &{} 1 &{} 1 &{} 2 \\ &{} &{} 5: &{} 0.49 &{} 0.17 &{} 1 &{} 1 &{} 2\}.&{}\\ \end{array} \end{aligned}$$
 Step 5:

We compute \(\bar{f}^{(4)}= 0\) and \(\bar{f}^{(5)}= 0.17\).
 Step 6:
 We compute \({{\underline{F}}}^{(4)}= 1\) and \({{\underline{F}}}^{(5)} = 0\), with the latter value leading to the outer fathoming of node \(5\), namely:Also, recall \(\mathcal{L}^{1} =\{\{2,4\},\{2,5\}\}\), with \(f^{\mathrm{UB},1}=0\) and \(\mathcal{X}_1=[1,1]\).$$\begin{aligned} \begin{array}{ll@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \mathcal{L}&{}= &{} \{2: 0.60 &{} 0 &{} 2 &{} 1 &{} 1 \\ &{} &{} 4: 0.83 &{} 0 &{} 1 &{} 1 &{} 2\};&{} \\ \mathcal{L_\mathrm{In}}&{}=&{} \{5: 0.49 &{} 0.17 &{}  &{}  &{} 2\}.&{} \end{array} \end{aligned}$$
 Step 7:

For \(i=4\), we compute \({{\underline{w}}}^{(4)}(\bar{x})=0.83\) (we already know \({{\underline{w}}}^{(2)}(\bar{x})=0.24\) for \(\bar{x}=1\)). The lowest value of \({{\underline{w}}}^{(j)}(\bar{x})\), \(j \in \mathcal{L}^{1}\), being at node \(4\) leads to computing \(\bar{F}^{(4)} =0\) and no update of the incumbent is needed. For \(i=5\), the upper bounding procedure does not apply because \(5\) is no longer in \(\mathcal{L}\). Set \(n_\mathrm{node}=5\).
 Step 2:

\(\mathbf{Iter=3}\). Node \(2\) is selected based on Definition 3 and removed from \(\mathcal{L}\).
 Step 3:

At node \(2\), we branch on \(x=0\), resulting in \(\mathcal{L}^{1} = \{4,6\}\) and \(\mathcal{L}^{2} = \{5,7\}\) with \(\mathcal{X}_1 = \left[ 1,0\right] \) and \(\mathcal{X}_2 = \left[ 0,1\right] \), respectively. The corresponding best inner upper bounds are set to \(f^{\mathrm{UB},1}=f^{\mathrm{UB},2}=0\).
 Step 4:

We compute \({{\underline{f}}}^{(6)}= 0.41\) and \({{\underline{f}}}^{(7)}= 0.20\). Both nodes are added to \(\mathcal{L}\).
 Step 5:

We compute \(\bar{f}^{(6)} = \bar{f}^{(7)}= 0\).
 Step 6:
 We compute \({{\underline{F}}}^{(6)} =2\) and \({{\underline{F}}}^{(7)}= 0\), with \(7\) being outer fathomed:Recall \(\mathcal{L}^{2} =\{5,7\}\), with \(f^{\mathrm{UB},2}=0\) and \(\mathcal{X}_2=[0,1]\). This independent list can now be discarded as it no longer holds any node from \(\mathcal{L}\) (cf. Listdeletion fathoming rules [16, Definition 10]). Thus:$$\begin{aligned} \begin{array}{ll@{\quad }ll@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \mathcal{L}&{}=&{} \{4: &{} 0.83 &{} 0 &{} 1 &{} 1 &{} 2 \\ &{}&{} 6: &{} 0.41 &{} 0 &{} 2 &{} 1 &{} 2\}; \\ \mathcal{L_\mathrm{In}}&{}=&{} \{5: &{} 0.49 &{} 0.17 &{}  &{}  &{} 2 \\ &{}&{} 7: &{} 0.20 &{} 0 &{}  &{}  &{} 2\}. \end{array} \end{aligned}$$One independent list remains: \(\mathcal{L}^{1} =\{4,6\}\), with \(f^{\mathrm{UB},1}=0\) and \(\mathcal{X}_1=[1,0]\).$$\begin{aligned} \begin{array}{llll@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} \mathcal{L} &{}=&{} \{4: &{} 0.83 &{} 0 &{} 1 &{} 1 &{} 2 \\ &{}&{} 6: &{} 0.41 &{} 0 &{} 2 &{} 1 &{} 2\};&{}\\ \mathcal{L_\mathrm{In}} &{}=&{} \emptyset . \end{array} \end{aligned}$$
 Step 7:

For \(i=6\), we compute \({{\underline{w}}}^{(6)}(\bar{x})=0.24\) for \(\bar{x}=1\) and compare it with \({{\underline{w}}}^{(4)}(\bar{x})=0.83\); no extra computation needs to be made since the lowest value is at node \(4\), where \(\bar{F}^{(4)}\) for \(\bar{x}=1\) has been computed. Set \(n_\mathrm{node}=7\).
 Step 2:

\(\mathbf{Iter=4}\) and node \(4\in \mathcal{L} \cap \mathcal{L}^{1}\) is selected.
 Step 3:
 At node \(4\), we branch on \(y=0.5\) and create nodes \(8\) and \(9\):$$\begin{aligned} \mathcal{L}^{1} = \{6,8,9\} \ \text {with} \ f^{\mathrm{UB},1}=0 \ \text {and} \ \mathcal{X}_1=[1,0]. \end{aligned}$$
 Step 4:

We compute \({{\underline{f}}}^{(8)}= 0.17\), \({{\underline{f}}}^{(9)}= 0.83\) and add both nodes to \(\mathcal{L}\).
 Step 5:

We compute \(\bar{f}^{(8)}= 0\), \(\bar{f}^{(9)}= 0.33\). The latter value updates the best inner upper bound of \(\mathcal{L}^{1}\), i.e., \(f^{\mathrm{UB},1}=0.33\). This update causes the full fathoming of node \(8\) since \({{\underline{f}}}^{(8)} > f^{\mathrm{UB},1}\) (i.e., node \(8\) is removed from all the lists).
 Step 6:
 We compute \({{\underline{F}}}^{(9)}= 0\) that leads to the outer fathoming of node \(9\). Then the lists are:and \(\mathcal{L}^{1} =\{6,9\}\), with \(f^{\mathrm{UB},1}=0.33\) and \(\mathcal{X}_1=\left[ 1,0\right] \).$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L} &{}= &{} \{6: &{} 0.41 &{} 0 &{} 2 &{} 1 &{} 2&{}\};\\ \mathcal{L_\mathrm{In}} &{} =&{} \{9: &{} 0.83 &{} 0.33 &{}  &{}  &{} 3&{}\}, \end{array} \end{aligned}$$
 Step 7:

The upper bounding procedure does not apply since none of the new nodes are in \(\mathcal{L}\) anymore. Set \(n_\mathrm{node}=9\) and go back to Step 2.
 Step 2:

\(\mathbf{Iter=5}\) and nodes \(6\in \mathcal{L} \cap \mathcal{L}^{1}\) and \(9\in \mathcal{L_\mathrm{In}} \cap \mathcal{L}^{1}\) are selected.
 Step 3:
 At node \(6\), we branch on \(y=0.5\) and create nodes \(10\) and \(11\). This gives \(\mathcal{L}^{1} =\{9,10,11\}\). At node \(9\), we branch on \(x=0.5\) and create nodes \(12\) and \(13\). This gives the final form of the remaining independent list:$$\begin{aligned} \mathcal{L}^{1} = \{\{10,11,12\},\{10,11,13\}\} \ \text {with} \ f^{\mathrm{UB},1}=0 \ \text {and} \ \mathcal{X}_1=[1,0]. \end{aligned}$$
 Step 4:
 We compute \({{\underline{f}}}^{(10)}= 0.20\), \({{\underline{f}}}^{(11)}= 0.11\), \({{\underline{f}}}^{(12)}= 0.83\) and \({{\underline{f}}}^{(13)}= 0.58\). The first two nodes are not added to \(\mathcal{L}\) since \({{\underline{f}}}^{(10)}, {{\underline{f}}}^{(11)}> f^{\mathrm{UB},1}\); the other two nodes are added to \(\mathcal L_\mathrm{In}\):Full fathoming of nodes \(10\) and \(11\) invokes the list deletion procedure, which deletes list \(\mathcal{L}^{1} =\{\{12\},\{13\}\}\) and results in \(\mathcal{L_\mathrm{In}}=\emptyset \).$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L} &{}= &{} \emptyset ;\\ \mathcal{L_\mathrm{In}} &{}=&{} \{12: &{} 0.83 &{} 0.33 &{}  &{}  &{} 4\\ &{} &{} \{13: &{} 0.58 &{} 0.33 &{}  &{}  &{} 4\}. \end{array} \end{aligned}$$
 Step 5:

There is no new node in \(\mathcal{L}\) or \(\mathcal L_\mathrm{In}\); thus, we return to Step 2.
 Step 2:

\(\mathcal{L}=\emptyset \); terminate with \(F^\mathrm{UB}=0\) at \((1,1)\).
Remark 6
In the example above, branching on outerfathomed nodes (here, node \(9\)) is not strictly necessary. However, in the majority of the examples considered it is essential to ensure convergence of the bounds on the inner problem objective function.
Example 2
 Step 0:
 \(\mathrm{Iter}=0\) and we set \(\mathcal{L}=\mathcal L_\mathrm{In}=\emptyset \), \(F^\mathrm{UB}=\infty \).
 Step 1:
 \(n_\mathrm{node}=1\) and we compute \({{\underline{f}}}^{(1)}=1\), \(\bar{f}^{(1)}=0.57\) and \({{\underline{F}}}^{(1)}=0.50\) at \((x^{(1)},y^{(1)})=(0.71,0.50)\). We add node \(1\) to the list \(\mathcal{L}\) with all the computed information:and set:$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L} =&\,&\{1:&1&0.57&0.50&0.71&0\}, \end{array} \end{aligned}$$We then compute \(w(0.71)=0.71\), which yields \(\bar{F}^{(1)}=0.50\) at \((0.71,0.50)\). This updates the incumbent value: \(F^\mathrm{UB}=0.50\).$$\begin{aligned} p=1, \mathcal{X}_1=\left[ 0.1,1\right] , \mathcal{L}^{1}=\{1\}, f^{\mathrm{UB},1}=0.57. \end{aligned}$$
 Step 2:

\( \mathrm{\mathbf{Iter}}=1\) (no termination was achieved). We select node \(1 \in \mathcal{L} \cap \mathcal{L}^{1}\) and remove it from \(\mathcal{L}\).
 Step 3:

We branch on \(y=0\), resulting in \(\mathcal{L}^{1} =\{2,3\}\), where \(\mathcal{X}_1=[0.1,1]\).
 Step 4:
 We compute \({{\underline{f}}}^{(2)}= 0\) and \({{\underline{f}}}^{(3)}= 1\) and add both nodes to \(\mathcal{L}\):$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L}&{}=&{} \{2: &{} 0 &{} 0.57 &{} 0.50 &{} 0.71 &{} 1 \\ &{}&{} 3:&{}\!\! 1 &{} 0.57 &{} 0.50 &{} 0.71 &{} 1\}.&{}\\ \end{array} \end{aligned}$$
 Step 5:

We compute \(\bar{f}^{(2)}= 0.57\) and \(\bar{f}^{(3)}= 0.10\) and update the best inner upper bound for list \(\mathcal{L}^{1}\): \(f^{\mathrm{UB},1}=0.10\). This results in node \(2\) being fully fathomed.
 Step 6:
 We compute \({{\underline{F}}}^{(3)}= 0.50\) at \((0.65,0.50)\). At this point list \(\mathcal{L}\) is:But node \(3\) is outer fathomed due to outer value dominance (cf. Outer fathoming rules [16, Definition 9]) yielding:$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L}=&\,&\{3:&1&0.10&0.50&0.65&1\}.&\end{array} \end{aligned}$$and \(\mathcal{L}^{1}=\{3\}\) with \(f^{\mathrm{UB},1}=0.10\) and \({\mathcal{X}_1} = \left[ 0.1,1\right] \). It is clear that list \(\mathcal{L}^{1}\) no longer holds nodes from \(\mathcal{L}\) and can be discarded (cf. Listdeletion fathoming rules [16, Definition 10]).$$\begin{aligned} \begin{array}{lllllllll} \mathcal{L} &{}= &{} \emptyset ;\\ \mathcal{L_\mathrm{In}} &{}=&{} \{3: &{} 1 &{} 0.10 &{}  &{}  &{} 1&{}\}, \end{array} \end{aligned}$$
 Step 7:

\(n_\mathrm{node}=3\). There is no open node left, leading back to Step 2.
 Step 2:

\(\mathcal{L}=\emptyset \); terminate with \(F^\mathrm{UB}=0.50\) at \((0.65,0.50)\).
However, node \(2\) can be outerfathomed, i.e., moved to the list \(\mathcal L_\mathrm{In}\), because \(\bar{F}^{(2)}=\infty \). Branching on the outerfathomed node \(2\) for another two levels leads to the creation of \(4\) descendants in total from node \(2\). Recall that fathoming based on the bounding information of the overall problem implies that the optimal solution of the bilevel problem cannot be found in the fathomed region but does not imply anything for the optimal solution of the inner problem (in the \(X\) subdomain under consideration). We have introduced the concept of outer fathoming to recognize and make use of such situations. The fathomingbyinfeasibility rule, for example, may be satisfied due to infeasible outer constraints, if present. In the case of Problem No. 11, where no outer constraints are present, the infeasibility of the lower bounding problem can only be due to inner information, which means that in this case full fathoming of node \(2\) could have been detected. In general, if we were able to detect/check where the infeasibility of the outer lower bounding comes from, then the BranchandSandwich algorithm in its most general form, i.e., using the convex relaxed problems (RILB) and (RISP), would require \(7\) nodes for problem No. 11.
6 Conclusions
The present paper constitutes the second part of our work on developing a branchandbound scheme, the BranchandSandwich algorithm, for the solution of optimistic bilevel programming problems that satisfy an appropriate regularity condition in the inner problem. In this article, we explored the convergence properties of the BranchandSandwich algorithm and we proved finite convergence to an \(\varepsilon \)optimal global solution.
We demonstrated our algorithm in detail for two test cases. With this detailed exposition, we wanted to (i) illustrate the proposed branching scheme and the use of auxiliary lists; (ii) highlight the flexibility that our method offers in how the proposed bounding problems are tackled; and also, (iii) point out that with the inner bounding scheme and the corresponding fathoming rules, it is possible to eliminate large portions of the inner space.
The BranchandSandwich algorithm was tested on 34 small problems with promising numerical results. The full implementation of the algorithm and computational performance are the focus of current work. Alternative choices in the way each step of the proposed algorithm is tackled, as well as different branching strategies, need to be explored. The implementation of the proposed method will also allow us to evaluate its performance on larger problems.
Footnotes
 1.
Given a convergence tolerance \(\varepsilon > 0\), a procedure is said to be finite \(\varepsilon \)convergent if it converges after a finite number of steps to an \(\varepsilon \)optimal solution of the problem being solved [7, p. 291].
 2.BranchandSandwich was applicable to all problem instances in [20] except for Examples 3.23, 3.25, 3.27 and 3.28.Table 4
Problem instances and their statistics
No.
Example in [source]
Example in this work (Part I or Part II)
NC Inner Problem?
#Outer var. (\(n\))
#Inner var. (\(m\))
#Outer con. (\(o\))
#Inner con. (\(r\))
1
4.2 [19]
Yes
\(0\)
\(1\)
\(0\)
\(0\)
2
3.1 [20]
No
\(0\)
\(1\)
\(0\)
\(0\)
3
3.2 [20]
No
\(0\)
\(1\)
\(1\)
\(0\)
4
3.3 [20]
Yes
\(0\)
\(1\)
\(0\)
\(1\)
5
3.4 [20]
Yes
\(0\)
\(1\)
\(0\)
\(0\)
6
3.5 [20]
1 [16]
Yes
\(0\)
\(1\)
\(0\)
\(0\)
7
3.6 [20]
Yes
\(0\)
\(1\)
\(0\)
\(0\)
8
3.7 [20]
No
\(0\)
\(1\)
\(1\)
\(0\)
9
3.8 [20]
No
\(1\)
\(1\)
\(2\)
\(0\)
10
3.9 [20]
Yes
\(1\)
\(1\)
\(1\)
\(0\)
11
3.10 [20]
2
Yes
\(1\)
\(1\)
\(0\)
\(0\)
12
3.11 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
13
3.12 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
14
3.13 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
15
3
Yes
\(1\)
\(1\)
\(0\)
\(0\)
16
3.14 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
17
3.15 [20]
1
Yes
\(1\)
\(1\)
\(0\)
\(0\)
18
3.16 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
19
3.17 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
20
3.18 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
21
3.19 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
22
3.20 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
23
3.21 [20]
2 [16]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
24
3.22 [20]
Yes
\(1\)
\(1\)
\(0\)
\(1\)
25
3.24 [20]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
26
4.1 [18]
Yes
\(1\)
\(1\)
\(0\)
\(0\)
27
5.6 [8]
Yes
\(1\)
\(1\)
\(0\)
\(2\)
28
7.1.1 [5]
Yes
\(1\)
\(1\)
\(0\)
\(3\)
29
1 [25]
No
\(1\)
\(1\)
\(0\)
\(3\)
30
4.5 [20]
Yes
\(1\)
\(2\)
\(0\)
\(2\)
31
2 [6]
No
\(2\)
\(2\)
\(0\)
\(3\)
32
1 [6]
No
\(2\)
\(3\)
\(0\)
\(3\)
33
3.26 [20]
Yes
\(2\)
\(3\)
\(3\)
\(0\)
34
4
Yes
\(5\)
\(5\)
\(3\)
\(1\)
Notes
Acknowledgments
We gratefully acknowledge funding by the Leverhulme Trust through the Philip Leverhulme Prize and by the EPSRC through a Leadership Fellowship [EP/J003840/1].
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