Robust Max-Min Benchmark Overview

• Robust max–min benchmark is a paradigm for evaluating optimization methods using worst-case criteria, uncertainty sets, and instance hardness as demonstrated in grouped bandits and discrete optimization.
• Benchmark constructions employ analytical testbeds, hard-instance generators, and gold-standard exact methods to provide actionable insights into algorithm performance under adverse conditions.
• Evaluation metrics such as sample complexity, optimality gap, and computational hardness help quantify robustness and scalability in diverse optimization applications.

Searching arXiv for recent and foundational papers related to robust max–min benchmarks and min–max optimization. Robust max–min benchmark denotes a family of benchmark constructions for optimization and learning problems in which performance is evaluated through a worst-case criterion, a minimum-over-components criterion, or both. In the cited literature, the expression appears in several technically distinct roles: a theoretical benchmark for identifying the group whose worst arm has the highest mean in max–min grouped bandits (Wang et al., 2021); a gold-standard robust max–min fairness benchmark for secure MISO cognitive radio with SWIPT under bounded CSI uncertainty (Zhou et al., 2016); benchmark libraries and hard-instance generators for robust discrete optimization (Goerigk et al., 2022, Goerigk et al., 2018); and analytical or certifiably exact testbeds for nonconcave min–max optimization and max–min bilinear completely positive programs (Maheshwari et al., 17 Aug 2025, Gao et al., 16 Feb 2026). This suggests that the topic is best understood as a benchmark paradigm organized around uncertainty sets, worst-case objectives, hardness profiles, and certifiable performance yardsticks rather than as a single standardized dataset.

1. Conceptual core and terminology

The common structural feature is an optimization objective that treats adverse realizations as decisive. In max–min grouped bandits, the objective is to identify the group whose worst arm has the largest mean,

$G^*=\arg\max_{G\in{\cal G}}\min_{i\in G}\mu_i,$

with fixed-confidence identification and sample complexity as the central benchmark quantities (Wang et al., 2021). In one-stage robust discrete optimization, the canonical form is

$\min_{x\in X}\max_{c\in U} c^T x,$

while min–max regret replaces $c^T x$ by $c^T x-\mathrm{opt}(c)$ (Goerigk et al., 2022). In covering settings, the corresponding max–min question asks which feasible uncertainty set $A$ is most expensive to cover, namely $\max_{A\in\Omega} f(A)$ (Gupta et al., 2010), or under a cardinality model, which $k$ demands are hardest to cover (0912.1045).

Terminology is not entirely uniform. The ordered weighted averaging formulation explicitly states that classic robust min–max optimization is a special case and labels it “robust min–max (robust max–min)” when $b=0$ and $w=(1,0,\dots,0)$, yielding

$\min_{x\in\mathcal{X}} \max_{i=1,\dots,K} c_i^T x$

(Baak et al., 2023). In another direction, Wald’s maximin model is presented as a min–max Stackelberg game with dependent strategy sets,

$\min_{x\in X}\max_{c\in U} c^T x,$0

linking robust optimization to sequential game-theoretic formulations (Goktas et al., 2021).

A robust max–min benchmark therefore refers not merely to a problem class, but to a means of evaluating methods against worst-case structure. Depending on the domain, the benchmark may be an information-theoretic lower bound, a library of hard instances, an analytical family with known global optima, or an exact algorithm whose solution quality is certifiable.

2. Canonical benchmark formulations

Several benchmark formulations recur across the literature.

Domain	Canonical objective	Benchmark role
Grouped bandits	$\min_{x\in X}\max_{c\in U} c^T x,$1	Sample-complexity benchmark
Robust discrete optimization	$\min_{x\in X}\max_{c\in U} c^T x,$2	Instance library and hard instances
Secure MISO CR with SWIPT	$\min_{x\in X}\max_{c\in U} c^T x,$3 under worst-case constraints	Gold-standard robust fairness benchmark
Convex–non-concave min–max	$\min_{x\in X}\max_{c\in U} c^T x,$4	Analytical global-solution testbed
CP/COP max–min bilinear models	$\min_{x\in X}\max_{c\in U} c^T x,$5	Certifiably exact SDP benchmark

In grouped bandits, the structural novelty is that arms are arranged in possibly-overlapping groups, and every arm belongs to at least one group. The benchmark is instance-dependent through the gaps $\min_{x\in X}\max_{c\in U} c^T x,$6, with the operational interpretation that once the confidence radius on arm $\min_{x\in X}\max_{c\in U} c^T x,$7 falls below $\min_{x\in X}\max_{c\in U} c^T x,$8, arm $\min_{x\in X}\max_{c\in U} c^T x,$9 can be eliminated or the optimal group can be certified (Wang et al., 2021).

In secure MISO cognitive radio with SWIPT, the robust max–min benchmark is a fairness objective over energy harvesting receivers. A slack variable $c^T x$0 is introduced so that the original max-min-EH problem becomes a maximization of $c^T x$1 subject to worst-case harvested-energy, secrecy-rate, secondary-user energy-harvesting, interference-cap, power-budget, and rank constraints under bounded CSI-error sets $c^T x$2 and $c^T x$3 (Zhou et al., 2016).

In robust discrete optimization, the benchmark formulation is parameterized by the nominal problem and the uncertainty set. The selection problem with $c^T x$4 and $c^T x$5 is used as the nominal ground problem, while interval, discrete, and budgeted uncertainty sets provide controlled robustness regimes (Goerigk et al., 2022). In the OWA framework, the same robust min–max problem is embedded into a broader spectrum that also includes Hurwicz and CVaR-type criteria (Baak et al., 2023).

In recent global min–max optimization work, benchmark problems are constructed to have analytical solutions despite non-concavity in the inner variable. A hand-crafted family over $c^T x$6, $c^T x$7 with

$c^T x$8

has closed-form global value $c^T x$9 and optimal aggregates $c^T x-\mathrm{opt}(c)$0, $c^T x-\mathrm{opt}(c)$1 (Maheshwari et al., 17 Aug 2025).

3. Benchmark construction strategies

One benchmark strategy is theoretical calibration by upper and lower bounds. In max–min grouped bandits, the upper bounds from successive elimination and the lower bounds from information-theoretic arguments are proposed as the benchmark itself: future algorithms should be measured against the instance-dependent complexity $c^T x-\mathrm{opt}(c)$2 and the minimax complexity $c^T x-\mathrm{opt}(c)$3, and in practice one may compare the total number of pulls to $c^T x-\mathrm{opt}(c)$4 and $c^T x-\mathrm{opt}(c)$5 (Wang et al., 2021).

A second strategy is explicit hard-instance generation. For robust min–max discrete optimization, uniform sampling is presented as a baseline, but structured sampling and optimization-based generation are designed to produce substantially harder instances. The “bimodal” and “symmetry” discrete generators already increase difficulty, and the HIRO approach modifies scenarios within an $c^T x-\mathrm{opt}(c)$6-neighborhood of radius $c^T x-\mathrm{opt}(c)$7 so as to maximize the robust objective value of the best solution through a bilevel max–min–max construction (Goerigk et al., 2022). The earlier “Generating Hard Instances for Robust Combinatorial Optimization” paper formulates an instance-generation model

$c^T x-\mathrm{opt}(c)$8

and reports that generated instances can be up to 500 times harder than standard random ones for Selection and TSP (Goerigk et al., 2018).

A third strategy is analytical benchmarking. EXOTIC introduces a class of convex–non-concave min–max problems with analytical global solutions, thereby providing a testbed where the global optimum is known exactly and gradient-based methods can be compared by true optimality gap rather than surrogate stationarity (Maheshwari et al., 17 Aug 2025). A related certifiable strategy appears in max–min bilinear completely positive programs, where a hierarchy of semidefinite relaxations is paired with flat-truncation conditions; when flatness holds, the relaxation is exact and optimal $c^T x-\mathrm{opt}(c)$9 of the original CP/COP program can be extracted (Gao et al., 16 Feb 2026).

A fourth strategy is the construction of a gold-standard exact method. In the secure MISO SWIPT setting, a one-dimensional search over $A$0 combined with S-Procedure-based LMI conversion and semidefinite relaxation yields the exact optimum under the stated assumptions, and because the resulting $A$1 and $A$2 are provably rank-one, the method is explicitly proposed as a gold-standard robust max–min fairness benchmark (Zhou et al., 2016).

4. Evaluation metrics and performance yardsticks

The benchmark role of a robust max–min formulation depends on the metric used to compare algorithms. In grouped bandits, the central metric is fixed-confidence sample complexity. The elimination algorithm stops after at most

$A$3

under the stated sub-Gaussian and uniqueness assumptions, while the StableOpt variant yields an instance-independent simple-regret bound and a fixed-confidence requirement $A$4 (Wang et al., 2021).

In analytical min–max benchmarks such as EXOTIC, the preferred measures are the optimality gap or regret, the number of inner convex-solver calls, the convergence rate of the gap versus total inner iterations $A$5, and the near-optimality dimension $A$6 of the outer objective (Maheshwari et al., 17 Aug 2025). This benchmark style is designed for algorithms that do not have access to exact global solutions except through the benchmark construction itself.

In robust Bayesian optimization, the benchmark metric is simple robust-regret at iteration $A$7,

$A$8

with $A$9 defined from the current GP posterior. The reported protocol uses 100 random restarts, 5 random initial evaluations, up to 30 further BO iterations, and mean $\max_{A\in\Omega} f(A)$0 standard deviation plots of $\max_{A\in\Omega} f(A)$1 on synthetic benchmark functions with boundary optima, non-differentiable argmax transitions, and frequently changing worst-case profiles (Weichert et al., 2021).

In large-scale robust optimization, ProM evaluates methods by objective gap, constraint violation, CPU time, and iteration count. The reported benchmark comparisons against reformulation, cutting-plane, OCO, and SGSP emphasize scaling behavior and oracle complexity, with $\max_{A\in\Omega} f(A)$2 or $\max_{A\in\Omega} f(A)$3 guarantees depending on smoothness (Tu et al., 2024). In robust discrete optimization, computational hardness is measured directly through CPLEX solve times and time-limit hits; for example, the benchmark suite records average solution time, variation with $\max_{A\in\Omega} f(A)$4, and the effect of HIRO budgets $\max_{A\in\Omega} f(A)$5 (Goerigk et al., 2022).

These metrics are not interchangeable. A sample-complexity benchmark, an optimality-gap benchmark, and a wall-clock hardness benchmark all evaluate worst-case reasoning, but they quantify different algorithmic bottlenecks: information acquisition, global nonconvex search, and discrete combinatorial difficulty.

5. Benchmark algorithms, baselines, and certificates

Benchmarks in this area are usually accompanied by explicit baseline algorithms. In grouped bandits, the tailored successive elimination algorithm maintains armwise upper and lower confidence bounds, computes the set of possible worst arms $\max_{A\in\Omega} f(A)$6, prunes the current set of potentially optimal groups $\max_{A\in\Omega} f(A)$7, and samples only arms that remain possible witnesses of optimality; the alternative StableOpt variant adapts the “stable-GP” method of Bogunović et al. to the finite-arm setting (Wang et al., 2021).

In secure MISO cognitive radio, the baseline is exact rather than heuristic. The bounded CSI-error constraints are converted into LMIs with the S-Procedure, auxiliary variables $\max_{A\in\Omega} f(A)$8 and $\max_{A\in\Omega} f(A)$9 are introduced, and a one-dimensional line search over

$k$0

is used to solve a convex SDP at each $k$1. The KKT analysis shows $k$2 and likewise rank-one $k$3, which makes the solution both computationally reproducible and structurally interpretable (Zhou et al., 2016).

In certifiable polynomial or semidefinite benchmark settings, the benchmark includes proof machinery. The CP/COP hierarchy uses moment matrices $k$4, localizing constraints, and SOS inner approximations of copositivity; flat truncation provides an explicit certificate that the SDP is exact and that equilibrium strategies can be extracted, as illustrated on the cyclic Colonel Blotto game (Gao et al., 16 Feb 2026). In two-stage stochastic min–max mixed-integer programming, the Lagrangian-integrated L-shaped ($k$5) method serves as the main benchmark solver and is compared against integer L-shaped and deterministic expanded methods; on bi-parameterized network interdiction, the reported result is that $k$6 solved all instances in 23 seconds on average, whereas the benchmark method failed to solve any instance within 3600 seconds (Kang et al., 2 Jan 2025).

For nonconvex-nonconcave and nonsmooth settings, scalable baselines replace exactness by stationarity or oracle complexity guarantees. The SSPG method for min–sum–max problems uses log–sum–exp smoothing, minibatching, and proximal-gradient updates to obtain almost-sure convergence to a Clarke/directional stationary point and $k$7 iteration complexity (Liu et al., 24 Feb 2025). The convex–concave min–max Stackelberg framework supplies first-order baselines with polynomial convergence, and Fisher-market equilibria are used as a practical benchmark across smoothness regimes (Goktas et al., 2021).

6. Tightness, hardness, and open issues

A central issue is whether a benchmark is uniformly tight across structurally different instances. The grouped-bandit study states that when groups are disjoint and each group has $k$8 arms, upper and lower bounds match up to logarithmic factors, but in heavily overlapping instances or when one suboptimal group has many arms and only one “bad” arm, the elimination algorithm may waste samples on good arms and the group-level lower bound can also be loose. The paper identifies this as a manifestation of the deeper difficulty of “good-arm identification” and states that designing uniformly tight strategies remains an open challenge (Wang et al., 2021).

Another issue is scalability of exact or certifiable benchmarks. The CP/COP semidefinite hierarchy is exact in tested Blotto instances, but scalability is limited by SDP size growing combinatorially in $k$9, and decomposition or chordal-sparsity techniques are proposed as possible remedies (Gao et al., 16 Feb 2026). In min–max–min robustness under discrete budgeted uncertainty, the exact row-and-column generation algorithm is described as exact but only viable for very small $b=0$0, with exact RCG impractical beyond $b=0$1 and heuristics required for larger instances (Goerigk et al., 2019).

A further controversy concerns the adequacy of local or gradient-based methods for global max–min benchmarking. The EXOTIC work states that beyond convex-concave min–max optimization, gradient-based methods may be arbitrarily far from global optima, which motivates benchmark families with analytical global solutions and tree-based global search (Maheshwari et al., 17 Aug 2025). Related complexity barriers also appear in OWA-based robust optimization: with $b=0$2 part of the input and non-decreasing weights, the OWA problem is stated to be strongly NP-hard and not approximable unless $b=0$3, while for non-increasing weights the approximation landscape depends delicately on $b=0$4 and on the relationship to min–max and $b=0$5-norm approximations (Baak et al., 2023).

Finally, benchmark meaning itself is heterogeneous. In some works it denotes a benchmark library of instances (Goerigk et al., 2022); in others, a worst-case complexity target (Wang et al., 2021); in others, a gold-standard exact solver (Zhou et al., 2016) or a certifiable relaxation hierarchy (Gao et al., 16 Feb 2026). A plausible implication is that robust max–min benchmarking remains methodologically plural: no single benchmark captures the full range of worst-case optimization behavior across bandits, combinatorial optimization, wireless communications, Bayesian optimization, and nonconvex min–max games.