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Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints

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Abstract

Discretization-based algorithms are proposed for the global solution of mixed-integer nonlinear generalized semi-infinite (GSIP) and bilevel (BLP) programs with lower-level equality constraints coupling the lower and upper level. The algorithms are extensions, respectively, of the algorithm proposed by Mitsos and Tsoukalas (J Glob Optim 61(1):1–17, 2015. https://doi.org/10.1007/s10898-014-0146-6) and by Mitsos (J Glob Optim 47(4):557–582, 2010. https://doi.org/10.1007/s10898-009-9479-y). As their predecessors, the algorithms are based on bounding procedures, which achieve convergence through a successive discretization of the lower-level variable space. In order to cope with convergence issues introduced by coupling equality constraints, a subset of the lower-level variables is treated as dependent variables fixed by the equality constraints while the remaining lower-level variables are discretized. Proofs of finite termination with \(\varepsilon \)-optimality are provided under appropriate assumptions, the preeminent of which are the existence, uniqueness, and continuity of the solution to the equality constraints. The performance of the proposed algorithms is assessed based on numerical experiments.

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Acknowledgements

We gratefully acknowledge the financial support provided by Réseau de transport d’électricité (RTE, France) through the project “Bilevel Optimization for Worst-case Analysis of Power Grids” and by the Excellence Initiative of the German federal state governments through the Cluster of Excellence “Tailor Made Fuels from Biomass” (EXC 236). Moll Glass is grateful for her scholarship from the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes).

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  1. Aachener Verfarhenstechnik, RWTH Aachen University, Aachen, Germany

    Hatim Djelassi, Moll Glass & Alexander Mitsos

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  1. Hatim Djelassi
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  2. Moll Glass
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  3. Alexander Mitsos
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Correspondence to Alexander Mitsos.

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Appendices

Proofs

1.1 Proof of Lemma 6

Proof

Consider any fixed \((\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) = \bar{\varvec{w}}^u \in \mathcal {W}^{u,s}(f^{u,*}+{\hat{\varepsilon }}^u,\varepsilon ^{ EQ})\) with \({\hat{\varepsilon }}^{ EQ}> \varepsilon ^{ EQ} > 0\). By Assumption 7, for any \(\varepsilon ^l_1 > 0\) there exists \((\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) \in \mathcal {W}^l \times \mathcal {Z}^l\) such that

$$\begin{aligned} \begin{array}{llll} &{}\varvec{g}^{l,1}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) &{}<&{} \varvec{0} \\ &{} \varvec{g}^{l,2}(\bar{\varvec{w}}^l) &{}\le &{} \varvec{0} \\ &{} ||\varvec{h}^l(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)||_\infty &{}<&{} \varepsilon ^{ EQ} \\ &{} \varvec{g}^{l,3}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) &{}<&{} \varvec{0} \\ \end{array} \end{aligned}$$

and

$$\begin{aligned} f^{l}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) \le f^{l,*}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) + \varepsilon ^l_1. \end{aligned}$$
(9)

By continuity of \(f^l(\cdot ,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)\) on \(\mathcal {X}^u\) (Assumption 2), for all \(\varepsilon ^l_2 > 0\), there exist \(\delta _1 > 0\) such that

$$\begin{aligned} f^l(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)< f^l(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) + \varepsilon ^l_2, \; \forall \varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \delta _1. \end{aligned}$$
(10)

Combining (9) and (10) yields that for all \(\varepsilon ^l_1,\varepsilon ^l_2 > 0\) there exists \(\delta _1 > 0\) such that

$$\begin{aligned} f^l(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)< f^{l,*}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) + \varepsilon ^l_1 + \varepsilon ^l_2, \; \forall \varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \delta _1. \end{aligned}$$
(11)

Similarly, by continuity of \(\varvec{g}^{l,1}(\cdot ,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)\), \(||\varvec{h}^l(\cdot ,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)||_\infty \), and \(\varvec{g}^{l,3}(\cdot ,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)\) on \(\mathcal {X}^u\) (Assumption 2), there exists \(\delta _2 > 0\) such that

$$\begin{aligned} \left. \begin{array}{llll} &{} \varvec{g}^{l,1}(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) &{}\le \varvec{0} \\ &{} ||\varvec{h}^l(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)||_\infty &{}\le \varepsilon ^{ EQ} \\ &{} \varvec{g}^{l,3}(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l) &{}\le \varvec{0} \\ \end{array}\right\} , \; \forall \varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \delta _2. \end{aligned}$$

Together with \(\varvec{g}^{l,2}(\bar{\varvec{w}}^l) \le \varvec{0}\), it follows that for all \(\varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \delta _2\), \((\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l)\) is feasible in (BLP-LLP). Therefore and by definition of \(f^{l,*}\), it follows that there exists \(\delta _2 > 0\) such that

$$\begin{aligned} f^{l,*}(\varvec{x}^u,\bar{\varvec{y}}^u) \le f(\varvec{x}^u,\bar{\varvec{y}}^u,\bar{\varvec{w}}^l,\bar{\varvec{z}}^l), \; \forall \varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \delta _2. \end{aligned}$$
(12)

Combining (11) and (12) yields that for all \(\varepsilon ^l_1,\varepsilon ^l_2 > 0\) there exists \(\delta _1,\delta _2 > 0\) such that

$$\begin{aligned} f^{l,*}(\varvec{x}^u,\bar{\varvec{y}}^u)< f^{l,*}(\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) + \varepsilon ^l_1 + \varepsilon ^l_2, \; \forall \varvec{x}^u \in \mathcal {X}^u : ||\varvec{x}^u-\bar{\varvec{x}}^u|| < \min \{\delta _1,\delta _2\}, \end{aligned}$$

which proves that \(f^{l,*}(\cdot ,\bar{\varvec{y}}^u)\) is upper semi-continuous at \(\bar{\varvec{x}}^u\) for all \(\bar{\varvec{w}}^u = (\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) \in \mathcal {W}^{u,s}(f^{u,*}+{\hat{\varepsilon }}^u,\varepsilon ^{ EQ})\) with \({\hat{\varepsilon }}^{ EQ}> \varepsilon ^{ EQ} > 0\).

By Lemma 2.2.1 and Theorem 4.2.1(1) in Bank et al. [1] and Assumptions 1 and 2, \(f^{l,*}(\cdot ,\bar{\varvec{y}}^u)\) is lower semi-continuous at \(\bar{\varvec{x}}^u\) for all \((\bar{\varvec{x}}^u,\bar{\varvec{y}}^u) \in \mathcal {W}^{u,s}(\infty ,\varepsilon ^{ EQ}) \supseteq \mathcal {W}^{u,s}(f^{u,*}+{\hat{\varepsilon }}^u,\varepsilon ^{ EQ})\) with \({\hat{\varepsilon }}^{ EQ}> \varepsilon ^{ EQ} > 0\). \(\square \)

1.2 Proof of Lemma 7

Proof

Let for now \(f^{u,*}\) denote the infimum of (BLP) without asserting that the minimum is attained. By Definition 6 of the level sets, the infimum of (BLP) is equivalent to

$$\begin{aligned}&\begin{array}{lcll} &{}f^{u,*} = \underset{\varvec{w}^u,\varvec{w}^l,\varvec{z}^l}{\text {inf}}\, f^u(\varvec{w}^u,\varvec{w}^l,\varvec{z}^l)&{} \\ &{}\text {s.t.}\,f^{l}(\varvec{w}^u,\varvec{w}^l,\varvec{z}^l) &{}\le &{} f^{l,*}(\varvec{w}^u) \\ &{}(\varvec{w}^u,\varvec{w}^l,\varvec{z}^l) &{}\in &{} \mathcal {V}^s(f^{u,*},\varepsilon ^{ EQ}) \\ \end{array} \\&\qquad \varvec{w}^u \in \mathcal {W}^u, \quad \varvec{w}^l \in \mathcal {W}^l, \quad \varvec{z}^l \in \mathcal {Z}^l.\nonumber \end{aligned}$$
(13)

Only points \(\varvec{w}^u \in \mathcal {W}^{u,s}(f^{u,*},\varepsilon ^{ EQ}) \subseteq \mathcal {W}^{u,s}(f^{u,*} + {\hat{\varepsilon }}^u,{\hat{\varepsilon }}^{ EQ})\) are contained in the feasible set of this augmented problem. Therefore, \(f^{l,*}(\cdot ,\varvec{y}^u)\) is continuous on the feasible set of (13) (Lemma 6), which in turn is closed (and possibly empty) by the closedness of the level sets (Assumptions 1 and 2). In the case of an empty feasible set, (13) is infeasible; otherwise it attains its minimum by Weierstrass’ theorem. As a consequence, either (BLP) is infeasible or its minimum exists. \(\square \)

1.3 Proof of Theorem 2

Proof

(BLP-LBD) is a valid relaxation of (BLP) and is solved according to Definition 3 with \(\varepsilon ^{ MINLP}\) and \(\varepsilon ^{ EQ}\). By assumption it holds that \(\varepsilon ^{ MINLP} < {\hat{\varepsilon }}^u\) and \(\varepsilon ^{ EQ} < {\hat{\varepsilon }}^{ EQ}\). Therefore, it holds that \((\bar{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}}) \in \mathcal {W}^{u,s}(f^{u,*} + {\hat{\varepsilon }}^u, \varepsilon ^{ EQ})\) for all points furnished by (BLP-LBD). If (BLP-LBD) is infeasible in any iteration of Algorithm 3, (BLP) is infeasible by virtue of (BLP-LBD) being a valid relaxation and the algorithm terminates.

Otherwise, we will show that for any \(\delta > 0\), after finitely many iterations of Algorithm 3, a point \((\hat{\varvec{x}}^{u,{ LBD}},\hat{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) is furnished by (BLP-LBD) that satisfies \(\hat{\varvec{y}}^{u,{ LBD}} = \bar{\varvec{y}}^{u,{ LBD}}\) and \(\hat{\varvec{x}}^{u,{ LBD}} \in \mathcal {B}_\delta (\bar{\varvec{x}}^{u,{ LBD}})\), where \((\bar{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\bar{\varvec{w}}^{l,{ LBD}},\bar{\varvec{z}}^{l,{ LBD}})\) is a solution previously furnished by (BLP-LBD). Furthermore, we will show that there exists \(\delta > 0\) such that such a solution \((\hat{\varvec{x}}^{u,{ LBD}},\hat{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) is \(\tfrac{1}{2}\varepsilon ^u\)-optimal. Finally, we will show that Algorithm 3 terminates once such a solution is found.

By compactness of \(\mathcal {X}^u\) each open cover of \(\mathcal {X}^u\) has a finite subcover. In particular, this is true for an open cover composed of open neighborhoods \(\mathcal {B}_\delta (\varvec{x}^{u})\) around points \(\varvec{x}^{u} \in \mathcal {X}^u\) with \(\delta > 0\). Together with integrality of \(\mathcal {Y}^u\) it follows that for any \(\delta > 0\), (BLP-LBD) furnishes solutions \((\bar{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\bar{\varvec{w}}^{l,{ LBD}},\bar{\varvec{z}}^{l,{ LBD}})\) and \((\hat{\varvec{x}}^{u,{ LBD}},\hat{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) with \(\hat{\varvec{y}}^{u,{ LBD}} = \bar{\varvec{y}}^{u,{ LBD}}\) and \(\hat{\varvec{x}}^{u,{ LBD}} \in \mathcal {B}_\delta (\bar{\varvec{x}}^{u,{ LBD}})\) within finitely many iterations.

With \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) being furnished by (BLP-LBD), it follows by construction that \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}}) \in \mathcal {W}^{u,s}(f^{u,*} + {\hat{\varepsilon }}^u, \varepsilon ^{ EQ})\) and \(\varvec{z}^{l,k} \in \mathcal {Z}^{l,*}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,k},\varepsilon ^{ EQ})\) for all \(k \in \mathcal {K}\). In particular, it holds that \(\varvec{z}^{l,K} \in \mathcal {Z}^{l,*}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,K},\varepsilon ^{ EQ})\) where \(K \in \mathcal {K}\) and \(\varvec{w}^{l,K}\) was generated for \((\bar{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}})\). It follows by Lemma 9, \(\varepsilon ^{ EQ}\) sufficiently small, and \(\varepsilon ^{l,{ UBD}} > \varepsilon ^{l,{ AUX}}\) that there exists \(\delta > 0\) such that for \(\hat{\varvec{x}}^{u,{ LBD}} \in \mathcal {B}_\delta (\bar{\varvec{x}}^{u,{ LBD}})\) and \(\varvec{z}^{l,K} \in \mathcal {Z}^{l,*}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,K},\varepsilon ^{ EQ})\) it holds that

$$\begin{aligned} \begin{array}{llll} &{} f^l(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,K},\varvec{z}^{l,K}) &{}\le &{} f^{l,*}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}}) + \varepsilon ^{l,{ UBD}} \\ &{} \varvec{g}^{l,2}(\varvec{w}^{l,K}) &{}\le &{} \varvec{0} \\ &{} \varvec{g}^{l,1}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,K},\varvec{z}^{l,K}) &{}\le &{} \varvec{0}. \\ \end{array} \end{aligned}$$

Consequently, the Kth logical constraint enforces

$$\begin{aligned} f^l(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}}) \le f^l(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\varvec{w}^{l,K},\varvec{z}^{l,K}), \end{aligned}$$

which in turn yields

$$\begin{aligned} f^l(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}}) \le f^{l,*}(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}}) + \varepsilon ^{l,{ UBD}}. \end{aligned}$$

It follows that \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) satisfies all lower and upper-level constraints and that \((\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) is \(\varepsilon ^{l,{ UBD}}\)-optimal in (BLP-LLP) for \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}})\). Furthermore, the lower bound generated from the solution \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) satisfies

$$\begin{aligned} { LBD} = f^{u,{ LBD},-} \ge f^u(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}}) - \varepsilon ^{ MINLP}. \end{aligned}$$

Finally, \((\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}})\) is feasible in (BLP-UBD) and the upper bound generated from its solution \((\hat{\varvec{w}}^{l,{ UBD}},\hat{\varvec{z}}^{l,{ UBD}})\) satisfies

$$\begin{aligned} { UBD}= & {} f^u(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ UBD}},\hat{\varvec{z}}^{l,{ UBD}})\\\le & {} f^u(\hat{\varvec{x}}^{u,{ LBD}},\bar{\varvec{y}}^{u,{ LBD}},\hat{\varvec{w}}^{l,{ LBD}},\hat{\varvec{z}}^{l,{ LBD}}) + \varepsilon ^{ MINLP}. \end{aligned}$$

With \(\varepsilon ^{ MINLP} < \tfrac{1}{2}\varepsilon ^u\), it holds that \({ UBD} - { LBD}< \varepsilon ^u\) and Algorithm 3 terminates. \(\square \)

GSIP test set

This section comprises the newly proposed equality-constrained GSIP test problems. The new test problems are obtained by replacing certain expressions in their original counterparts by new (dependent) variables and adding equality and selection constraints. The equivalence of the equality-constrained instances with their original counterparts can be verified by solving the equality constraints analytically for the dependent variables and in the case of multiple solutions applying the selection constraints. Substituting the so-obtained unique solution for the dependent variables in the test problems yields their original counterparts.

Test casejongen-4-2-eq The original version of the test case is proposed in [20, Example 4.2]. The equality-constrained version is given by

figure i

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2\).

Test casejongen-4-1-eq The original version of the test case is proposed in [20, Example 4.1]. The equality-constrained version is given by

figure j

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_2 - (x^l_1)^3\).

Test caseruckmann-5-2-eq The original version of the test case is proposed in [37, Example 5.2]. The equality-constrained version is given by

figure k

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2 + (x^l_2)^2\).

Test casestill-303-eq The original version of the test case is proposed in [49, p. 303]. The equality-constrained version is given by

figure l

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1+1)^2 + (x^u_1)^2\).

Test case ruckmann-3-1-eq The original version of the test case is proposed in [36, Example 3.1]. The equality-constrained version is given by

figure m

The solution of the equality constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2\) and \(z^{l,*}_2(\varvec{w}^u,\varvec{w}^l) = x^u_2 x^l_1\).

Test caseruckmann-5-1-eq The original version of the test case is proposed in [37, Example 5.1]. The equality-constrained version is given by

figure n

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1+x^l_2)^2\).

Test casevazquez-3-3-eq The original version of the test case is proposed in [15, Example 3.3]. The equality-constrained version is given by

figure o

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^5\).

Test casevazquez-2-2-eq The original version of the test case is proposed in [15, Example 2.2]. In [24, 30], the sign of the lower-level inequality constraint is inverted when compared to [15]. The same is done here for the sake of comparability of results. The equality-constrained version is given by

figure p

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_1 - (x^l_1)^2\).

Test caselemonidis-9-eq The original version of the test case is proposed in [24, p. 155]. The equality-constrained version is given by

figure q

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^u_1)^2 x^l_1\).

Test caseruckmann-4-5-eq The original version of the test case is proposed in [38, Example 4.5]. The equality-constrained version is given by

figure r

The solution of the equality constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_1 - x^l_1\) and \(z^{l,*}_2(\varvec{w}^u,\varvec{w}^l) = x^u_2 - x^l_1\).

Test caselemonidis-12-eq The original version of the test case is proposed in [24, p. 159]. The equality-constrained version is given by

figure s

The solution of the equality and selection constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2\).

Test caselemonidis-13-eq The original version of the test case is proposed in [24, p. 160]. The equality-constrained version is given by

figure t

The solution of the equality constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_2 x^l_1\) and \(z^{l,*}_2(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2\).

Test caselemonidis-14-eq The original version of the test case is proposed in [24, p. 162]. The equality-constrained version is given by

figure u

The solution of the equality and selection constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_2)^2\).

BLP test set

This section comprises the newly proposed equality-constrained BLP test problems. The new test problems are obtained by replacing certain expressions in their original counterparts by new (dependent) variables and adding equality and selection constraints. The equivalence of the equality-constrained instances with their original counterparts can be verified by solving the equality constraints analytically for the dependent variables and in the case of multiple solutions applying the selection constraints. Substituting the so-obtained unique solution for the dependent variables in the test problems yields their original counterparts.

Test caseam-1-0-0-1-01-eq The original version of the test case is proposed in [28, p. 578]. The equality-constrained version is given by

figure v

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_1 y^l_1\).

Test caseam-1-1-1-0-01-eq The original version of the test case is proposed in [28, p. 578]. The equality-constrained version is given by

figure w

The solution of the equality constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_1 + x^l_1\) and \(z^{l,*}_2(\varvec{w}^u,\varvec{w}^l) = x^u_1 - x^l_1\).

Test caseam-1-1-1-1-01-eq The original version of the test case is proposed in [28, p. 578]. The equality-constrained version is given by

figure x

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = x^u_1 (x^l_1)^2\).

Test caseam-1-1-1-1-02-eq The original version of the test case is proposed in [28, p. 579]. The equality-constrained version is given by

figure y

The solution of the equality and selection constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^2\).

Test caseam-3-3-3-3-01-eq The original version of the test case is proposed in [28, p. 579]. The equality-constrained version is given by

figure z

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (x^l_1)^3 x^l_2 y^u_1\).

Test caseedmunds-bard-eq The original version of the test case is proposed in [12, p. 159]. The equality-constrained version is given by

figure aa

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = (y^l_1)^2\).

Test casejan-chern-eq The original version of the test case is proposed in [19, p. 583]. The equality-constrained version is given by

figure ab

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = y^l_1\).

Test casemoore-bard-eq The original version of the test case is proposed in [33, Example 2]. The equality-constrained version is given by

figure ac

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = y^u_1 + 2 y^l_1\).

Test casesahin-ciric-eq The original version of the test case is proposed in [41, Example 4]. The equality-constrained version is given by

figure ad

The solution of the equality constraints yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = y^l_1 y^l_2\), \(z^{l,*}_2(\varvec{w}^u,\varvec{w}^l) = (1-y^l_1) y^l_2\), and \(z^{l,*}_3(\varvec{w}^u,\varvec{w}^l) = y^l_1 (1-y^l_2)\).

Test casethirwani-arora-eq The original version of the test case is proposed in [53, Example 1]. The equality-constrained version is given by

figure ae

The solution of the equality constraint yields \(z^{l,*}_1(\varvec{w}^u,\varvec{w}^l) = 6 y^l_1 + 4 y^u_1\).

Numerical results

See Tables 1, 2, and 3.

Table 1 Numerical results for Algorithm 1
Full size table
Table 2 Numerical results for Algorithm 2
Full size table
Table 3 Numerical results for Algorithm 3
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Djelassi, H., Glass, M. & Mitsos, A. Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints. J Glob Optim 75, 341–392 (2019). https://doi.org/10.1007/s10898-019-00764-3

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