Abstract
Nearoptimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of nearoptimal robust multilevel problems, where nearoptimal robustness is modelled through additional adversarial decisionmakers. Nearoptimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without nearoptimality robustness under general conditions.
Introduction
Multilevel optimization is a class of mathematical optimization problems where other problems are embedded in the constraints. They are well suited to model sequential decisionmaking processes, where a first decisionmaker, the leader intrinsically integrates the reaction of another decisionmaker, the follower, into their decisionmaking problem. In recent years, most of the research focuses on the study and design of efficient solution methods for the case of two levels, namely bilevel problems [1], which fostered a growing range of applications.
Nearoptimal robustness, defined in [2], is an extension of bilevel optimization. In this setting, the upper level anticipates limited deviations of the lower level from an optimal solution and aims at a solution that remains feasible for any feasible and nearoptimal solution of the lower level. This protection of the upper level against uncertain deviations of the lowerlevel has led to the characterization of nearoptimality robustness as a robust optimization approach for bilevel optimization. The models where the upper level is protected against all optimal lower level responses (that is, without deviations) are referred to as pessimistic bilevel optimization models. In nearoptimal robustness, the lowerlevel response corresponds to the uncertain parameter and the maximum deviation of the objective value from an optimal solution to the uncertainty budget. Because the set of nearoptimal lowerlevel solutions potentially has infinite cardinality and depends on the upperlevel decision itself, nearoptimality robustness adds generalized semiinfinite constraints to the bilevel problem. The additional constraint can also be viewed as a form of robustness under decisiondependent uncertainty.
In this paper, we prove complexity results on multilevel problems to which nearoptimality robustness constraints are added under various forms. We show that under fairly general conditions, the nearoptimal robust version of a multilevel problem remains on the same level of the polynomial hierarchy as the canonical problem. These results are nontrivial assuming that the polynomial hierarchy does not collapse and open the possibility of solution algorithms for nearoptimal robust multilevel problems as efficient as for their canonical counterpart. Even though we focus on nearoptimal robust multilevel problems, the complexity results we establish hold for all multilevel problems that present the same hierarchical structure, i.e. the same anticipation and parameterization between levels as the nearoptimal formulation with the adversarial problems, as defined in Sect. 3. In particular, the results extend to pessimistic multilevel problems, which can be viewed as a special case of the equivalent nearoptimal robust multilevel problem.
The rest of this paper is organized as follows. Section 2 introduces the notation and the background on nearoptimality robustness and existing complexity results in multilevel optimization. Section 3 presents complexity results for the nearoptimal robust version of bilevel problems, where the lower level belongs to \({\mathcal {P}}\) and \({{\mathcal {N}}}{{\mathcal {P}}}\). These results are extended in Section 4 to multilevel optimization problems, focusing on integer multilevel linear problems with nearoptimal deviations of the topmost intermediate level. Section 5 provides complexity results for a generalized form of nearoptimal robustness in integer multilevel problems, where multiple decisionmakers anticipate nearoptimal reactions of a lower level. Finally, we draw some conclusions in Sect. 6.
Multilevel optimization and nearoptimality robustness
In this section, we introduce the notation and terminology for bilevel optimization and nearoptimality robustness, and highlight prior complexity results in multilevel optimization. Let us define a bilevel problem as:
We denote by \({\mathcal {X}}\) and \({\mathcal {Y}}\) the domain of upper and lowerlevel variables respectively. We use the convenience notation \(\left[ \![n\right] \!] = \{1, \dots ,n\}\) for a natural n.
Problem (1) is illposed, since multiple solutions to the lower level may exist [3, Ch. 1]. Models often rely on additional assumptions to alleviate this ambiguity, the two most common being the optimistic and pessimistic approaches. In the optimistic case (BiP), the lower level selects an optimal decision that most favours the upper level. In this setting, the lowerlevel decision can be taken by the upper level, as long as it is optimal for the lowerlevel problem. The upper level can thus optimize over both x and v, leading to:
Constraint (2d) implies that v is feasible for the lower level and that f(x, v) is the optimal value of the lowerlevel problem, parameterized by x.
The pessimistic approach assumes that the lower level chooses an optimal solution that is the worst for the upperlevel objective as in [1] or with respect to the upperlevel constraints as in [4].
The nearoptimal robust version of (BiP) considers that the lowerlevel solution may not be optimal but nearoptimal with respect to the lowerlevel objective function. The tolerance for nearoptimality, denoted by \(\delta \) is expressed as a maximum deviation of the objective value from optimality. The problem solved at the upper level must integrate this deviation and protects the feasibility of its constraints for any nearoptimal lowerlevel decision. The problem is formulated as:
\({\mathcal {Z}}(x;\delta )\) denotes the nearoptimal set, i.e. the set of nearoptimal lowerlevel solutions, depending on both the upperlevel decision x and \(\delta \). (NORBiP) is a generalization of the pessimistic bilevel problem since the latter is both a special case and a relaxation of (NORBiP) [2]. We refer to (BiP) as the canonical problem for (NORBiP) [or equivalently Problem (4)] and (NORBiP) as the nearoptimal robust version of (BiP). In the formulation of (NORBiP), the upperlevel objective depends on decision variables of both levels, but is not protected against nearoptimal deviations. A more conservative formulation also protecting the objective by moving it to the constraints in an epigraph formulation [2] is given by:
The optimal values of the three problems are ordered as:
We next provide a review of complexity results for bilevel and multilevel optimization problems. Bilevel problems are \({{\mathcal {N}}}{{\mathcal {P}}}\)hard in general, even when the objective functions and constraints at both levels are linear [5]. When the lowerlevel problem is convex, a common solution approach consists in replacing it with its KKT conditions [6, 7], which are necessary and sufficient if the problem satisfies certain constraint qualifications. This approach results in a single optimization problem with complementarity constraints, of which the decision problem is \({{\mathcal {N}}}{{\mathcal {P}}}\)complete [8]. A specific form of the threelevel problem is investigated in [9], where only the objective value of the bottomlevel problem appears in the objective functions of the first and second levels. If these conditions hold and all objectives and constraints are linear, the problem can be reduced to a single level one with complementarity constraints of polynomial size.
Pessimistic bilevel problems for which no upperlevel constraint depends on lowerlevel variables are studied in [10]. The problem of finding an optimal solution to the pessimistic case is shown to be \({{\mathcal {N}}}{{\mathcal {P}}}\)hard, even if a solution to the optimistic counterpart of the same problem is provided. A variant is also defined, where the lower level may pick a suboptimal response only impacting the upperlevel objective. This variant is comparable to the ObjectiveRobust NearOptimal Bilevel Problem defined in [2]. In [11], the lowerlevel is assumed to respond to the upper level with a decision derived from a heuristic algorithm from a predefined set. An uncertain bilevel setting with a pure binary lower level is considered in [12]. Lowerlevel response uncertainty is encoded as a maximum Hamming distance of the nearoptimal decision to the optimal one. In [4], the independent case of the pessimistic bilevel problem is studied, corresponding to a special case of (NORBiP) with \(\delta =0\) and all lowerlevel constraints independent of the upperlevel variables. It is shown that the linear independent pessimistic bilevel problem, and consequently the linear nearoptimal robust bilevel problem, can be solved in polynomial time while it is strongly \({{\mathcal {N}}}{{\mathcal {P}}}\)hard in the nonlinear case.
When the lowerlevel problem cannot be solved in polynomial time, the bilevel problem is in general \(\varSigma _2^P\)hard. The notion of \(\varSigma _2^P\)hardness and classes of the polynomial hierarchy are recalled in Sect. 3. Despite this complexity result, new algorithms and corresponding implementations have been developed to solve these problems and in particular, mixedinteger linear bilevel problems [13,14,15]. Variants of the bilevel knapsack were investigated in [16], and proven to be \(\varSigma _2^P\)hard as the generic mixedinteger bilevel problem.
Multilevel optimization was initially investigated in [17] in the case of linear constraints and objectives at all levels. In this setting, the problem is shown to be in \(\varSigma _s^P\), with \(s+1\) being the number of levels. The linear bilevel problem corresponds to \(s=1\) and is in \(\varSigma _1^P\equiv {{\mathcal {N}}}{{\mathcal {P}}}\). If, on the contrary, at least the bottomlevel problem involves integrality constraints (or more generally belongs to \({{\mathcal {N}}}{{\mathcal {P}}}\) but not \({\mathcal {P}}\)), the multilevel problem with s levels belongs to \(\varSigma _s^P\). A model unifying multistage stochastic and multilevel problems is defined in [18], based on a risk function capturing the component of the objective function which is unknown to a decisionmaker at their stage. Complexity and completeness results in the polynomial hierarchy above the first level are compiled in [19]. We also refer the interested reader to Kleinert et al. [20] for a recent review on complexity results and computational approaches in bilevel optimization.
As highlighted in [18], most results in the literature on complexity of multilevel optimization use \({{\mathcal {N}}}{{\mathcal {P}}}\)hardness as the sole characterization. This only indicates that a given problem is at least as hard as all problems in \({{\mathcal {N}}}{{\mathcal {P}}}\) and that no polynomialtime solution method should be expected unless \({{\mathcal {N}}}{{\mathcal {P}}} = {\mathcal {P}}\).
We characterize nearoptimal robust multilevel problems not only on the hardness or “lower bound” on complexity, i.e. being at least as hard as all problems in a given class but through their complexity “upper bound”, i.e. the class of the polynomial hierarchy they belong to. The linear optimistic bilevel problem is for instance strongly \({{\mathcal {N}}}{{\mathcal {P}}}\)hard, but belongs to \({{\mathcal {N}}}{{\mathcal {P}}}\) and is therefore not \(\varSigma _2^P\)hard. The results are established for (NORBiP) and directly apply to (NORBiPAlt), to the constraintbased pessimistic bilevel problem from [4], and to the more classical objectivebased pessimistic bilevel formulation which can be reformulated as constraintbased.
Nearoptimal robust bilevel problems
We establish in this section complexity results for nearoptimal robust bilevel problems for which the lower level \({\mathcal {L}}\) is a singlelevel problem parameterized by the upperlevel decision. (NORBiP) can be reformulated as in [2] by replacing each kth semiinfinite Constraint (3c) with the lowerlevel solution \(z_k\) in \({\mathcal {Z}}(x;\delta )\) that yields the highest value of \(G_k(x, z_k)\):
From a gametheoretical perspective, the nearoptimal robust version of a bilevel problem can be seen as a threeplayer hierarchical game. The upper level \({\mathcal {U}}\) and lower level \({\mathcal {L}}\) are identical to the canonical bilevel problem. The third level is the adversarial problem \({\mathcal {A}}\) and selects the worst nearoptimal lowerlevel solution with respect to upperlevel constraints, as represented by the embedded maximization in Constraint (4d). If the upperlevel problem has multiple constraints, the adversarial problem can be decomposed into problems \({\mathcal {A}}_k, k \in \left[ \![m_u\right] \!]\), where \(m_u\) is the number of upperlevel constraints. The interaction among the three players is depicted in Fig. 1a. The adversarial problem can be split into \(m_u\) adversarial problems as done in [2], each finding the worstcase with respect to one of the upperlevel constraints. The canonical problem refers to the optimistic bilevel problem without nearoptimal robustness constraints. We refer to the variable v as the canonical lowerlevel decision.
The complexity classes of the polynomial hierarchy are only defined for decision problems. We consider that an optimization problem belongs to a given class if that class contains the decision problem of determining if there exists a feasible solution for which the objective value at least as good as a given bound.
Definition 1
The decision problem associated with an optimization problem is in \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \), with \({\mathcal {H}}\) a set of realvalued functions on a vector space \({\mathcal {Y}}\), iff:

i.
it belongs to \({\mathcal {P}}\);

ii.
for any \(h \in {\mathcal {H}}\), the problem with an additional linear constraint and an objective function set as \(h(\cdot )\) is also in \({\mathcal {P}}\).
A broad range of problems in \({\mathcal {P}}\) are also in \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \) for certain sets of functions \({\mathcal {H}}\) (see Example 1 for linear problems and linear functions and Example 2 for some combinatorial problems in \({\mathcal {P}}\)). The classes \({{\mathcal {N}}}{{\mathcal {P}}}^*\left[ {\mathcal {H}}\right] \) and \(\varSigma _s^{P*}\left[ {\mathcal {H}}\right] \) are defined in a similar way. We next consider two examples illustrating these definitions.
Example 1
Denoting by \({\mathcal {H}}_L\) the set of linear functions from the space of lowerlevel variables to \({\mathbb {R}}\), linear optimization problems are in \({\mathcal {P}}^{*}\left[ {\mathcal {H}}_L\right] \), since any given problem with an additional linear constraint and a different linear objective function is also a linear optimization problem.
Example 2
Denoting by \({\mathcal {H}}_L\) the set of linear functions from the space of lowerlevel variables to \({\mathbb {R}}\), combinatorial optimization problems in \({\mathcal {P}}\) which can be formulated as linear optimization problems with totally unimodular matrices are not in \({\mathcal {P}}^{*}\left[ {\mathcal {H}}_L\right] \) in general. Such problems include network flow or bipartite matching problems. Indeed, adding a linear constraint may break the integrality of solutions of the linear relaxation of the lowerlevel problem.
The polynomial hierarchy is first defined in [21] and a link to multilevel games is established in [17]. The complexity class at the sth level of the polynomial hierarchy is denoted by \(\varSigma _s^{P}\), defined recursively as \(\varSigma _0^{P} = {\mathcal {P}}\), \(\varSigma _1^{P} = {{\mathcal {N}}}{{\mathcal {P}}}\), and problems of the class \(\varSigma _s^{P}\), \(s>1\) being solvable in nondeterministic polynomial time, provided an oracle for problems of class \(\varSigma _{s1}^{P}\). In particular, a positive answer to a decision problem in \({{\mathcal {N}}}{{\mathcal {P}}}\) can be verified, given a certificate, in polynomial time. If the decision problem associated with an optimization problem is in \({{\mathcal {N}}}{{\mathcal {P}}}\), and given a potential solution, the objective value of the solution can be compared to a given bound and the feasibility can be verified in polynomial time. We reformulate these statements in the following proposition:
Proposition 1
[17] An optimization problem is in \(\varSigma _{s+1}^P\) if verifying that a given solution is feasible and attains a given bound can be done in polynomial time, when equipped with an oracle solving problems in \(\varSigma _{s}^P\) in a single step.
Proposition 1 is the main property of the classes of the polynomial hierarchy used to determine the complexity of nearoptimal robust bilevel problems in various settings throughout this paper.
Lemma 1
Given a bilevel problem in the form of Problem (2), if the lowerlevel problem is in \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \), and
then the adversarial problem (4d) is in \({\mathcal {P}}\).
Proof
The lowerlevel problem can equivalently be written in an epigraph form:
Given a solution of the lowerlevel problem (v, w) and an upperlevel constraint \(G_k(x,y) \le 0\), the adversarial problem is defined by:
Compared to the lowerlevel problem, the adversarial problem contains an additional linear constraint \(u \le w\) and an objective function updated to \(G(x,\cdot )\). \(\square \)
Lemma 1 highlights that the restriction imposed by the class \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \) on the lower level ensures the complexity class for the adversarial problem. This result is now leveraged in Theorem 1.
Theorem 1
Given a bilevel problem (P), if there exists \({\mathcal {H}}\) such that the lowerlevel problem is in \({{\mathcal {N}}}{{\mathcal {P}}}^*\left[ {\mathcal {H}}\right] \) and
then the nearoptimal robust version of the bilevel problem is in \(\varSigma _{2}^P\) similarly to the canonical problem.
Proof
The proof relies on the ability to verify that a given solution (x, v) results in an objective value at least as low as a bound \(\varGamma \) according to Proposition 1. This verification can be carried out with the following steps:

i.
compute the upperlevel objective value F(x, v) and verify that \(F(x, v) \le \varGamma \);

ii.
verify that upperlevel constraints are satisfied;

iii.
verify that lowerlevel constraints are satisfied;

iv.
compute the optimum value \({\mathcal {L}}(x)\) of the lowerlevel problem parameterized by x and check if:
$$\begin{aligned} f(x,v) \le \min _{y} {\mathcal {L}}(x); \end{aligned}$$ 
v.
Compute the worst case: Find
$$\begin{aligned} z_k \in \mathop {\mathrm {arg \, max}}\limits _{y\in {\mathcal {Y}}}\, {\mathcal {A}}_k(x, v)\,\,\, \forall k \in \left[ \![m_u\right] \!], \end{aligned}$$where \({\mathcal {A}}_k(x, v)\) is the kth adversarial problem parameterized by (x, v);

vi.
Verify nearoptimal robustness: \(\forall k \in \left[ \![m_u\right] \!]\), verify that the kth upperlevel constraint is feasible for the worstcase \(z_k\).
Steps i and ii can be carried out in polynomial time by assumption. Step iii requires to check the feasibility of a solution to a problem in \({{\mathcal {N}}}{{\mathcal {P}}}\). This can be done in polynomial time. Step iv consists in solving the lowerlevel problem, while Step v corresponds to solving \(m_u\) adversarial problems belonging to \({{\mathcal {N}}}{{\mathcal {P}}}\) based on Lemma 1. \(\square \)
Theorem 2
Given a bilevel problem (P), if the lowerlevel problem is convex and in \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \) with \({\mathcal {H}}\) a set of convex functions, and if the upperlevel constraints are such that \(G_k(x,\cdot ) \in {\mathcal {H}}\), then the nearoptimal robust version of the bilevel problem is in \({{\mathcal {N}}}{{\mathcal {P}}}\). If the upperlevel constraints are convex nonaffine with respect to the lowerlevel constraints, the nearoptimal robust version is in general not in \({{\mathcal {N}}}{{\mathcal {P}}}\).
Proof
If the upperlevel constraints are concave with respect to the lowerlevel variables, the adversarial problem defined as:
is convex. Furthermore, by definition of \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \), the adversarial problem is in \({\mathcal {P}}\).
Applying the same reasoning as in the proof of Theorem 1, Steps 13 are identical and can be carried out in polynomial time. Step 4 can be performed in polynomial time since \({\mathcal {L}}\) is in \({\mathcal {P}}\). Step 5 is also performed in polynomial time since \(\forall k \in \left[ \![m_u\right] \!]\), each kth adversarial problem (5) is a convex problem that can be solved in polynomial time since \({\mathcal {L}}\) is in \({\mathcal {P}}^*\left[ {\mathcal {H}}\right] \). Step 6 simply is a simple comparison of two quantities.
If the upperlevel constraints are convex nonaffine with respect to the lowerlevel variables, Problem (5) maximizes a convex nonaffine function over a convex set. Such a problem is \({{\mathcal {N}}}{{\mathcal {P}}}\)hard in general. Therefore, the verification that a given solution is feasible and satisfies a predefined bound on the objective value requires solving the \(m_u\) \({{\mathcal {N}}}{{\mathcal {P}}}\)hard adversarial problems. If \({\mathcal {L}}\) is in \({{\mathcal {N}}}{{\mathcal {P}}}^*\left[ {\mathcal {H}}\right] \), then these adversarial problems are in \({{\mathcal {N}}}{{\mathcal {P}}}\) by Eq. (3), and the nearoptimal robust problem is in \(\varSigma _{2}^P\) according to Proposition 1. \(\square \)
Nearoptimal robust mixedinteger multilevel problems
In this section, we study the complexity of a nearoptimal robust version of mixedinteger multilevel linear problems (MIMLP), where the lower level itself is a slevel problem and is \(\varSigma _s^P\)hard. The canonical multilevel problem is, therefore, \(\varSigma _{s+1}^P\)hard [17]. For some instances of mixedinteger bilevel, the optimal value can be approached arbitrarily but not reached [22]. To avoid such pathological cases, we restrict our attention to multilevel problems satisfying the criterion for mixedinteger bilevel problems from Fischetti et al. [13]:
Property 1
The continuous variables at any level s do not appear in the problems at levels that are lower than s (the levels deciding after s).
More specifically, we will focus on mixedinteger multilevel linear problems where the uppermost lower level \({\mathcal {L}}_1\) may pick a solution deviating from the optimal value, while we ignore deviations of the levels \({\mathcal {L}}_{i>1}\). This problem is noted (\(\text {NOMIMLP}_s\)) and depicted in Fig. 1b.
The adversarial problem corresponds to a decision of the level \({\mathcal {L}}_1\) different from the canonical decision. This decision induces a different reaction from the subsequent levels \({\mathcal {L}}_2\), \({\mathcal {L}}_3\). Since the toplevel constraints depend on the joint reaction of all following levels, we will note \(z_{ki} = (z_{k1}, z_{k2}, z_{k3})\) the worstcase joint nearoptimal solution of all lower levels with respect to the toplevel constraint k.
Theorem 3
If \({\mathcal {L}}_1\) is in \(\varSigma _{s}^{P*}\left[ {\mathcal {H}}_L\right] \), the decision problem associated with (\(\text {NOMIMLP}_s\)) is in \(\varSigma _{s+1}^P\) as the canonical multilevel problem.
Proof
Given a solution to all levels \((x_U,v_{1}, v_{2},\dots v_s)\) and a bound \(\varGamma \), verifying that this solution is (i) feasible, (ii) nearoptimal robust of parameter \(\delta \), and (iii) has an objective value at least as good as the bound \(\varGamma \) can be done through the following steps:

i.
Compute the objective value and verify that it is lower than \(\varGamma \);

ii.
verify variable integrality;

iii.
solve the problem \({\mathcal {L}}_1\), parameterized by \(x_U\), and verify that the solution \((v_{1}, v_{2},\dots )\) is optimal;

iv.
\(\forall k \in \left[ \![m_u\right] \!]\), solve the kth adversarial problem. Let \(z_k=(z_{k1}, z_{k2}\dots z_{ks})\) be the solution;

v.
\(\forall k \in \left[ \![m_u\right] \!]\), verify that the kth upperlevel constraint is feasible for the adversarial solution \(z_k\).
Steps i, ii, and v can be performed in polynomial time. Step iii requires solving a problem in \(\varSigma _{s}^P\), while Step iv consists in solving \(m_u\) problems in \(\varSigma _{s}^P\), since \({\mathcal {L}}_1\) is in \(\varSigma _{s}^{P*}\left[ {\mathcal {H}}_L\right] \). Checking the validity of a solution thus requires solving problems in \(\varSigma _{s}^P\) and is itself in \(\varSigma _{s+1}^P\), like the original problem. \(\square \)
Generalized nearoptimal robust multilevel problem
In this section, we study the complexity of a variant of the problem presented in Sect. 4 with \(s+1\) decisionmakers at multiple top levels \({\mathcal {U}}_1, {\mathcal {U}}_2, \dots , {\mathcal {U}}_s\) and a single bottom level \({\mathcal {L}}\). We denote by \({\mathcal {U}}_1\) the topmost level. We assume that the bottomlevel entity may choose a solution deviating from optimality. This requires that the entities at all \({\mathcal {U}}_i\, \forall i \in \{1\ldots s\}\) levels anticipate this deviation, thus solving a nearoptimal robust problem to protect their feasibility from it. The variant, coined \(\text {GNORMP}_s\), is illustrated in Fig. 1c, d. We assume throughout this section that Property 1 holds in order to avoid the unreachability problem previously mentioned. The decision variables of all upper levels are denoted by \(x_{(i)}\), and the objective functions by \(F_{(i)}(x_{(1)}, x_{(2)},\ldots , x_{(s)})\). The lowerlevel canonical decision is denoted v as in previous sections.
If the lowest level \({\mathcal {L}}\) belongs to \({{\mathcal {N}}}{{\mathcal {P}}}\), \({\mathcal {U}}_s\) belongs to \(\varSigma _{2}^P\) and the original problem is in \(\varSigma _{s+1}^P\). In a more general multilevel case, if the lowest level \({\mathcal {L}}\) solves a problem in \(\varSigma _{r}^P\), \({\mathcal {U}}_s\) solves a problem in \(\varSigma _{r+1}^P\) and \({\mathcal {U}}_1\) in \(\varSigma _{r+s}^P\).
We note that for all fixed decisions \(x_{(i)} \forall i \in \{1\ldots s1\}\), \({\mathcal {U}}_s\) is a nearoptimal robust bilevel problem. This differs from the model presented in Sect. 4 where, for a fixed upperlevel decision, the topmost lower level \({\mathcal {L}}_1\) is the same parameterized problem as in the canonical setting. Furthermore, as all levels \({\mathcal {U}}_i\) anticipate deviations of the lowerlevel decision in the nearoptimal set, the worst case can be formulated with respect to the constraints of each of these levels. In conclusion, distinct adversarial problems \({\mathcal {A}}_{i}\, \forall i \in \{1\ldots s\}\) can be formulated. Each upper level \({\mathcal {U}}_i\) integrates the reaction of the corresponding adversarial problem in its nearoptimality robustness constraint. This formulation of \((\text {GNORMP}_s)\) is depicted Fig. 1d.
Theorem 4
Given a \(s+1\)level problem \((\mathrm{GNORMP}_s)\), if the bottomlevel problem parameterized by all upperlevel decisions \({\mathcal {L}}(x_{(1)}, x_{(2)}\dots , x_{(s)})\) is in \(\varSigma _{r}^{P*}\left[ {\mathcal {H}}_L\right] \), then \((\mathrm{GNORMP}_s)\) is in \(\varSigma _{r+s}^P\) like the corresponding canonical bilevel problem.
Proof
We denote by \(x_U = (x_{(1)}, x_{(2)},\dots , x_{(s)})\) and \(m_{U_i}\) the number of constraints of problem \({\mathcal {U}}_i\). As for Theorem 1, this proof is based on the complexity of verifying that a given solution \((x_U, v)\) is feasible and results in an objective value below a given bound. The verification requires the following steps:

i.
compute the toplevel objective value and assert that it is below the bound;

ii.
verify feasibility of \((x_U, v)\) with respect to the constraints at all levels;

iii.
verify optimality of v for \({\mathcal {L}}\) parameterized by \(x_U\);

iv.
verify optimality of \(x_{(i)}\) for the nearoptimal robust problem solved by the ith level \({\mathcal {U}}_i(x_{(1)}, x_{(2)}\dots x_{(i1)}; \delta )\) parameterized by all the decisions at levels above and the nearoptimality tolerance \(\delta \);

v.
compute the nearoptimal lowerlevel solution \(z_{k}\) which is the worstcase with respect to the kth constraint of the topmost level \(\forall k \in [\![m_{U_1}]\!]\);

vi.
verify that each \(k \in [\![m_{U_1}]\!]\) toplevel constraint is satisfied with respect to the corresponding worstcase solution \(z_k\).
Steps iii are performed in polynomial time. Step iii requires solving Problem \({\mathcal {L}}(x_U)\), belonging to \(\varSigma _r^P\). Step iv consists in solving a generalized nearoptimal robust multilevel problem \((\text {GNORMP}_{s1})\) with one level less than the current problem. Step v requires the solution of \(m_{U_1}\) adversarial problems belonging to \(\varSigma _r^P\) since \({\mathcal {L}}\) is in \(\varSigma _{r}^{P*}\left[ {\mathcal {H}}_L\right] \). Step vi is an elementary comparison of two quantities for each \(k\in \left[ \![m_u\right] \!]\). The step of highest complexity is Step 4. If it requires to solve a problem in \(\varSigma _{r+s1}^P\), then \((\text {GNORMP}_{s})\) is in \(\varSigma _{r+s}^P\) similarly to its canonical problem.
Let us assume that Step iv requires to solve a problem outside \(\varSigma _{r+s1}^P\). Then \((\text {GNORMP}_{s1})\) is not in \(\varSigma _{r+s1}^P\) as the associated canonical problem, and that Step iv requires to solve a problem not in \(\varSigma _{r+s2}^P\). By recurrence, \((\text {GNORMP}_{1})\) is not in \(\varSigma _{r+1}^P\). However, \((\text {GNORMP}_{1})\) is a nearoptimal robust bilevel problem where the lower level itself in \(\varSigma _{r}^P\); this corresponds to the setting of Section 4. This contradicts Theorem 3, \((\text {GNORMP}_{s1})\) is therefore in \(\varSigma _{r+s1}^P\). Verifying the feasibility of a given solution to \((\text {GNORMP}_{s})\) requires solving a problem at most in \(\varSigma _{r+s1}^P\). Based on Proposition 1, \((\text {GNORMP}_{s})\) is in \(\varSigma _{r+s}^P\) as its canonical multilevel problem. \(\square \)
In conclusion, Theorem 4 shows that adding nearoptimality robustness at an arbitrary level of the multilevel problem does not increase its complexity in the polynomial hierarchy. By combining this property with the possibility to add nearoptimal deviation at an intermediate level as in Theorem 3, nearoptimality robustness can be added at multiple levels of a multilevel model without changing its complexity class in the polynomial hierarchy.
Conclusion
In this paper, we have shown that for many configurations of bilevel and multilevel optimization problems, adding nearoptimality robustness to the canonical problem does not increase its complexity in the polynomial hierarchy. This result is obtained even though nearoptimality robustness constraints add another level to the multilevel problem, which in general would change the complexity class.
We defined the class \(\varSigma _s^{P*}\left[ {\mathcal {H}}\right] \) as a slight restriction on problems from \(\varSigma _s^{P}\), ensuring that the adversarial problem derived from a lowerlevel problem from \(\varSigma _s^{P}\) lies in the same complexity class. While the definition of \(\varSigma _s^{P*}\left[ {\mathcal {H}}\right] \) is general enough to capture many nonlinear multilevel problems, it avoids specific cases where the modified objective or additional linear constraint changes the complexity class for the adversarial problem.
Future work will consider specialized solution algorithms for some classes of nearoptimal robust bilevel and multilevel problems.
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Acknowledgements
The work of Mathieu Besançon was supported by the Mermoz scholarship and the GdR RO.
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Besançon, M., Anjos, M.F. & Brotcorne, L. Complexity of nearoptimal robust versions of multilevel optimization problems. Optim Lett 15, 2597–2610 (2021). https://doi.org/10.1007/s11590021017549
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Keywords
 Nearoptimal robustness
 Multilevel optimization
 Complexity theory
Mathematics Subject Classification
 91A65
 90C26
 90C10
 90C10
 90C11
 90C60