Pith. sign in

REVIEW 3 major objections 5 minor 47 references

Non-linear Multi-objective Optimization with Probabilistic Branch and Bound

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that one noisy simulation per design point is enough for a branch-and-bound algorithm to eventually capture the entire Pareto optimal set, with estimators converging to the true objective values.

desk verdict A valuable empirical heuristic with a load-bearing asymptotic proof gap that needs major repair before the convergence claims are credible. read the letter →

arxiv 2506.04554 v1 pith:RC6NJAYG submitted 2025-06-05 math.OC cs.LG

Hao Huang , Zelda B. Zabinsky This is my paper · ORCID
classification math.OCcs.LG MSC 90C2990C1590C26
keywords multi-objectiveoptimizationsimulationprobabilisticbranchandboundParetooptimalsetsingleobservationstochasticefficientfrontierasymptoticconvergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

MOPBnB(so) is a branch-and-bound algorithm for stochastic multi-objective optimization that evaluates each sampled design once, with a single noisy simulation, and estimates each objective value by averaging over neighboring samples inside a shrinking ball. The paper's main claim is asymptotic: if samples accumulate in those balls fast enough, then with probability one the algorithm eventually captures the true Pareto optimal set, the solutions not dominated in every objective, and the estimated objective values converge to the true values. A finite-time bound, with probability at least $1-\alpha$, covers the deterministic noiseless case and says every unpruned subregion contains at least one high-quality solution. If this is right, cutting the number of simulations by orders of magnitude does not sacrifice eventual correctness, which matters for expensive simulation models where replication is the bottleneck.

What carries the argument

The load-bearing object is the single-observation neighborhood estimator: at iteration $k$, the $m$ objective values at sample $x_{i,j}$ are estimated as the average of all noisy observations $y_{i,r}$ inside the ball $B(x_{i,j}, r_k)$, with radius $r_k = r_0/B^{k/n}$ shrinking geometrically in dimension $n$ and branching factor $B$. The estimator is carried by Lemma 1, adapted from the single-observation adaptive-search literature, which guarantees $\hat{f}_\ell(x) \to f_\ell(x)$ almost surely under Assumptions 2-4, including bounded noise and a sample count in the shrinking balls that grows faster than the balls shrink. Around this estimator, the branch-and-bound loop samples $n_k$ points in active subregions and $k^c$ points in pruned subregions, marks a subregion pruned if it contains no non-dominated estimated sample, branches subregions with non-dominated samples into $B$ children, and permits later reclassification of pruned regions. Theorems 3 and 4 transfer the pointwise convergence of the estimator onto the global pruning and branching process.

What would settle it

Run MOPBnB(so) on a two-objective test problem, keep a subregion pruned at an early iteration, and record the number of samples that fall inside the ball of radius $r_k = r_0/B^{k/n}$ around a Pareto point in that subregion. If the expected count decays like $O(k^2/B^k)$ rather than growing faster than the ball shrinks, then Assumption 4 is violated for exactly the pruned-region case used in the proof of Theorem 3, and the claimed almost-sure non-pruning of Pareto-containing subregions would not follow.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that multiple replications are not needed for asymptotic Pareto-set capture. Theorem 3 states that any subregion intersecting the true Pareto optimal set $S_E$ is eventually never pruned: $P(\lim_{k\to\infty} \sigma \in \Sigma_k \mid \sigma \cap S_E \neq \varnothing)=1$, and Theorem 4 states that every non-pruned subregion containing Pareto points is eventually contained in the high-quality set $L(\delta,S)$. The proof combines a single-observation estimator, which averages all observed values in a ball of radius $r_k$ around a sample point, with the branch-and-bound rule that prunes only subregions whose sampled points are all dominated. Reclassification of pruned subregions is allowed, so a region pruned under noisy early estimates can re-enter the active partition. The conclusions are asymptotic as $k\to\infty$, not finite-time, in the stochastic case.

Load-bearing premise

The load-bearing premise is Assumption 4, that the number of samples inside each shrinking ball grows faster than the ball shrinks, together with its use on subregions pruned early in the run where the algorithm only adds $k^c$ samples per iteration; Theorem 4 also needs Remark 1's link between small objective-function differences and membership in the high-quality set $L(\delta,S)$.

Editorial extensions

If this is right

  • A single simulation per sampled design is enough for eventual capture of the Pareto optimal set and convergence of the efficient-frontier estimates; replication-based procedures are not required for asymptotic correctness.
  • Pruned subregions must continue to receive a small number of samples each iteration, $k^c$, so that reclassification remains possible and the single-observation estimator has points to average near boundaries.
  • In the deterministic case, the finite-time probability bound of at least $1-\alpha$ from the earlier MOPBnB analysis still applies, because the sampling and branching structure is unchanged and extra sampling in pruned regions only helps.
  • The practical effect is a large reduction in simulation calls: on the reported FF and ZDT1 experiments, the same iterations used millions of evaluations with replications versus thousands for MOPBnB(so).
  • If the convergence theorems hold, the algorithm supplies an approximation of the Pareto set whose non-dominated samples are close to and well spread along the true efficient frontier in the M1 and M2 metrics, although the extent metric M3 shows pruning can lose some edges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not stated in the paper, the same single-observation estimator could be combined with other partition rules, such as adaptive or surrogate-guided splits, and the asymptotic-capture argument would carry over as long as the sample-growth condition holds on every subregion, including long-pruned ones.
  • A natural testable extension is a finite-time version of Theorem 3: the proof gives only an almost-sure limit, so a reader could attempt to bound the probability that a Pareto-containing subregion is pruned after iteration $K$, which would require an explicit rate for the sample-growth condition in pruned regions.
  • The numerical evidence suggests the bottleneck for the extent metric M3 is the permanent loss of parts of the frontier from pruning; making reclassification thresholds more conservative might trade some speed on the closeness metric for broader coverage, a direction the paper does not explore.
  • For heavy-tailed or unbounded noise, Assumption 3 fails, and the averaging estimator would likely need a different noise-handling rule, such as trimmed means, before the same asymptotic guarantee could be expected.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This manuscript proposes MOPBnB(so), a partition-based algorithm for stochastic multi-objective optimization over mixed continuous/integer domains. The algorithm evaluates each sampled design point once and estimates objective values by averaging observations of neighboring points within a shrinking ball; subregions whose sampled points are dominated are pruned, with possible reclassification, and surviving subregions are branched. The authors provide a finite-time bound for deterministic problems (Theorem 1, cited from earlier work), a finite-time bound for the replication-based variant (Theorem 2, cited), and two asymptotic results: Theorem 3 claims that subregions containing true Pareto points are almost surely never pruned, and Theorem 4 claims that such unpruned subregions are eventually subsets of the high-quality set L(δ,S). Numerical comparisons on ZDT1, ZDT2, ZDT3, and FF test problems show MOPBnB(so) using far fewer function evaluations than MOPBnB(wr) and generally better M1 and M2 performance than NSGA-II, at the cost of smaller frontier extent (M3).

Significance. If the asymptotic theorems were valid, the paper would make a useful contribution: it would show that one simulation replication per design point suffices for eventual Pareto-set approximation in stochastic multiobjective optimization, a substantial computational saving over replication-based methods. The algorithmic idea is clearly presented, the pseudocode is explicit, and the numerical comparison is informative, with documented function-evaluation counts and multiple performance metrics. The paper also builds transparently on published finite-time and single-observation results. However, the central asymptotic claims are not supported as written: the proof of Theorem 3 applies a lemma whose key assumption is violated by the algorithm's own sampling schedule in pruned regions, and Theorem 4 relies on an incorrect inequality in Remark 1. These are not cosmetic issues, so the contribution is not yet established at the level claimed in the abstract.

major comments (3)
  1. [Section 3, Eq. (2); Appendix proof of Theorem 3] Lemma 1 is invoked for the closest sampled point x' in a pruned subregion σ, but the sampling rule for pruned regions cannot satisfy Assumption 4 under the manuscript's own radius schedule. Algorithm 1 adds only kc samples to each pruned region on iteration k, so after k iterations a region that remains pruned contains O(k^2) samples. With r_k = r_0/B^{k/n} (Section 4), the volume of B(x,r_k) is proportional to r_k^n = r_0^n/B^k, and the expected number of samples in that ball is O(k^2/B^k), which tends to zero. Consequently the estimator in Eq. (3) is eventually undefined on such regions with probability one, and the convergence statement from Lemma 1 cannot be applied to x' in Eqs. (16)-(18). The sentence before Lemma 1 claiming that the kc sample size 'satisfies Assumption 4' is therefore false for the stated geometric schedule, and Theorem 3's asymptotic capture claim is unsupported.
  2. [Section 3, Remark 1; Appendix proof of Theorem 4] The implication in Remark 1 is incorrect. From |f_l(x) - f_l(x*)| < ε for all l and some x* in S_E, the distance D(x) in Eq. (4) is at most sqrt(m) ε, not sqrt(m) ε^2. Since Remark 1 sets y(δ,S) = sqrt(m) ε^2, for the relevant small-tolerance regime ε < 1 one has sqrt(m) ε > y(δ,S), so x need not belong to L(δ,S). The proof of Theorem 4 relies on this implication when it moves from x not in L(δ,S) to the existence of l with |f_l(x) - f_l(x')| ≥ ε in Eq. (22); the contrapositive is not valid. Replacing ε^2 by ε in Remark 1 would fix the inequality, but as written Theorem 4 does not follow.
  3. [Appendix proof of Theorem 3, Eqs. (16)-(18)] Independently of the Assumption 4 problem, the proof has not established that the closest sampled point x' converges to the Pareto point x or that the bound ρ in Eq. (16) tends to zero. Equation (18) treats ρ = 0 as an almost-sure event, but ρ is introduced as a finite upper bound on the objective difference; showing ρ → 0 requires a lower bound on the density of samples near x, which is exactly the missing Assumption 4 condition. Thus the local-consistency step of the proof is incomplete even if the sampling schedule is modified.
minor comments (5)
  1. [Section 4] The numerical experiments use Gaussian noise ξ_x ~ N(0, 0.1), which is unbounded and hence violates Assumption 3; the authors should either use bounded noise or state explicitly that the experiments are heuristic and outside the assumptions of Theorems 3-4.
  2. [Theorem 3 statement] The event 'σ ∈ Σ_k as k → ∞' is ambiguous because a fixed subregion is removed from Σ_k when it is branched; the authors should state the result in terms of the nested subregions containing a fixed Pareto point.
  3. [Theorem 1 statement] Equation (8) and the surrounding text contain a formatting error in the union/intersection expression that makes the statement of Theorem 1 hard to read.
  4. [Assumption 3 and Algorithm 1] The symbol α is used both for the confidence parameter in Algorithm 1 and for the uniform noise bound in Assumption 3; renaming one would avoid confusion.
  5. [Theorem 2 statement] Theorem 2's probability bound (1-α)(1-mα) is only meaningful when mα < 1; the required condition on m and α should be stated.

Circularity Check

0 steps flagged · score 2.0 of 10

No construction-level circularity: the central asymptotic claims are not definitional or fitted, although they rely on prior self-authored convergence results and contain a correctness gap.

full rationale

The derivation chain from Algorithm 1 to Theorems 3 and 4 does not reduce to its inputs by construction. The key imported convergence statement, Lemma 1, is taken from Kiatsupaibul, Smith, and Zabinsky (2020), which shares an author with the present paper; however, it is a published, parameter-free result about single-observation estimators whose stated assumptions (Assumptions 2-4) do not include the target conclusion that Pareto-containing subregions are never pruned. It therefore functions as external evidence rather than a self-referential premise. Similarly, Theorems 1 and 2 are cited to Huang and Zabinsky (2014) without proof, but they support only the inherited finite-time deterministic and replication-based analysis and are not fitted to any data in this paper. No free parameter is calibrated to force Theorems 3 or 4, and no prediction is a renamed fitted input. The main weaknesses flagged by a skeptical reading, namely the assertion that the kc samples in pruned subregions satisfy Assumption 4 under the geometric radius schedule, and Remark 1's use of sqrt(m)epsilon^2 = y(delta,S), are potential correctness gaps in the proof rather than cases where the conclusion is equivalent to the premise by definition. Circularity is therefore not established; the low score reflects the presence of self-citations that are not constructionally circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central theorems rest on imported convergence lemmas and on Assumptions 2 to 4; the parameter sequences r_k and n_k are user-chosen and are not shown to satisfy Assumption 4 in pruned subregions. No physical entities are invented, and no data-derived constants are fitted.

free parameters (5)
  • ball radius schedule r_k = r_k = r0 / B^(k/n), r0 = 0.1 in experiments
    Chosen by hand; must satisfy Assumption 4 for convergence, but in pruned subregions the expected number of samples in the ball decays rather than grows.
  • sample count n_k in non-pruned subregions = ceil(ln(alpha_k)/ln(1-delta)) with alpha_k = alpha/B^k
    Chosen to make uniform-sampling miss probability compatible with alpha_k; in non-pruned regions density grows, while in pruned regions the analogous guarantee is not shown.
  • pruned-region per-iteration sample count kc = c = 50 in experiments
    Intended to support estimation in pruned subregions and reclassification; the proof does not establish that kc samples keep ball counts growing as r_k shrinks.
  • branching factor B = B = 2 in experiments
    User-defined partition parameter; the analysis assumes one region splits into B new subregions each iteration.
  • confidence alpha and closeness delta = alpha = 0.1, delta = 0.1 in experiments
    User-defined parameters defining the probability bound and the high-quality set L(delta,S); not fitted to data.
assumptions (6)
  • domain assumption Objective functions are continuous on the convex part of S (Assumption 2).
    Used in Theorem 3's proof to bound objective differences between a true Pareto point and the closest sampled point.
  • domain assumption Random errors are uniformly bounded over x in S (Assumption 3).
    Required by Lemma 1 and the asymptotic theorems; the Gaussian multiplicative noise in the experiments does not satisfy it.
  • ad hoc to paper Sample counts inside shrinking balls grow faster than radii shrink (Assumption 4).
    Core premise for estimator convergence; not verified for pruned subregions where only kc samples per iteration are drawn over a fixed region, and the chosen r_k schedule appears to violate it there.
  • standard math Single-observation estimator converges almost surely under Assumptions 2 to 4 (Lemma 1 from Kiatsupaibul et al. 2020).
    External lemma with overlapping authorship; the new proofs depend on applying it to all sampled points, including those in pruned subregions.
  • standard math Finite-time bounds of MOPBnB(wr) from Huang and Zabinsky (2014) remain valid (Theorems 1 and 2).
    Proofs are not reproduced; the deterministic finite-time result is cited to the authors' earlier paper.
  • ad hoc to paper Remark 1: there exists epsilon with sqrt(m)epsilon^2 = y(delta,S), and f-closeness implies membership in L(delta,S).
    The stated relation is algebraically wrong as written (the correct relation should be sqrt(m)epsilon = y(delta,S)), and Theorem 4 relies on this remark.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-linear Multi-objective Optimization with Probabilistic Branch and Bound." pith.science (2026). https://pith.science/paper/RC6NJAYG

@misc{pith2026250604554,
  author       = {Pith},
  title        = {Pith review of: Non-linear Multi-objective Optimization with Probabilistic Branch and Bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RC6NJAYG}},
  note         = {Machine review of arXiv:2506.04554}
}
read the original abstract

A multiple objective simulation optimization algorithm named Multiple Objective Probabilistic Branch and Bound with Single Observation (MOPBnB(so)) is presented for approximating the Pareto optimal set and the associated efficient frontier for stochastic multi-objective optimization problems. MOPBnB(so) evaluates a noisy function exactly once at any solution and uses neighboring solutions to estimate the objective functions, in contrast to a variant that uses multiple replications at a solution to estimate the objective functions. A finite-time performance analysis for deterministic multi-objective problems provides a bound on the probability that MOPBnB(so) captures the Pareto optimal set. Asymptotic convergence of MOPBnB(so) on stochastic problems is derived, in that the algorithm captures the Pareto optimal set and the estimations converge to the true objective function values. Numerical results reveal that the variant with multiple replications is extremely intensive in terms of computational resources compared to MOPBnB(so). In addition, numerical results show that MOPBnB(so) outperforms a genetic algorithm NSGA-II on test problems.

Figures

Figures reproduced from arXiv: 2506.04554 by the authors.

Figure 1
Figure 1. Approximated efficient frontier and Pareto optimal set of FF with [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Approximated efficient frontier and Pareto optimal set of ZDT1 with [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. M1 metric of FF, ZDT1, ZDT2, and ZDT3 with [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: M2 metric of FF, ZDT1, ZDT2, and ZDT3 with [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: M3 metric of FF, ZDT1, ZDT2, and ZDT3 with [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 44 canonical work pages

  1. [1]

    \ Suresh, S

    al2019revisiting APACrefauthors Al-Dujaili, A. \ Suresh, S. APACrefauthors \ 2019 . Revisiting norm optimization for multi-objective black-box problems: a finite-time analysis Revisiting norm optimization for multi-objective black-box problems: a finite-time analysis . Journal of Global Optimization 73 3 659--673

    work page 2019

  2. [2]

    \ Lee, J S

    andradottir2021pareto APACrefauthors Andrad \'o ttir, S. \ Lee, J S. APACrefauthors \ 2021 . Pareto set estimation with guaranteed probability of correct selection Pareto set estimation with guaranteed probability of correct selection . European Journal of Operational Research 292 1 286--298

    work page 2021

  3. [3]

    \ Hunter, S R

    applegate2019r APACrefauthors Applegate, E A. \ Hunter, S R. APACrefauthors \ 2019 . R-MGSPLINE: retrospective multi-gradient search for multi-objective simulation optimization on integer lattices R-mgspline: retrospective multi-gradient search for multi-objective simulation optimization on integer lattices . 2019 Winter Simulation Conference (WSC) 2019 w...

    work page 2019

  4. [4]

    , Dunnett, C W

    Beck54 APACrefauthors Bechhofer, R E. , Dunnett, C W. \ Sobel, M. APACrefauthors \ 1954 . A Two-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with a Common Unknown Variance A two-sample multiple decision procedure for ranking means of normal populations with a common unknown variance . Biometrika 41 170--176

    work page 1954

  5. [5]

    , Charkhgard, H

    boland2019preprocessing APACrefauthors Boland, N. , Charkhgard, H. \ Savelsbergh, M. APACrefauthors \ 2019 . Preprocessing and cut generation techniques for multi-objective binary programming Preprocessing and cut generation techniques for multi-objective binary programming . European Journal of Operational Research 274 3 858--875

    work page 2019

  6. [6]

    , Lee, L H

    chew2009differentiated APACrefauthors Chew, E P. , Lee, L H. , Teng, S. \ Koh, C H. APACrefauthors \ 2009 . Differentiated service inventory optimization using nested partitions and MOCBA Differentiated service inventory optimization using nested partitions and MOCBA . Computers & Operations Research 36 5 1703--1710

    work page 2009

  7. [7]

    \ Kim, T G

    choi2018pareto APACrefauthors Choi, S H. \ Kim, T G. APACrefauthors \ 2018 . Pareto set selection for multiobjective stochastic simulation model Pareto set selection for multiobjective stochastic simulation model . IEEE Transactions on Systems, Man, and Cybernetics: Systems 50 11 4256--4269

    work page 2018

  8. [8]

    APACrefauthors \ 2001

    Deb01 APACrefauthors Deb, K. APACrefauthors \ 2001 . Multi-objective Optimization Using Evolutionary Algorithms Multi-objective optimization using evolutionary algorithms \ ( 16)\ [Book]. New York John Wiley & Sons, Chichester

    work page 2001

Show all 47 references
  1. [9]

    , Pratap, A

    deb2002fast APACrefauthors Deb, K. , Pratap, A. , Agarwal, S. \ Meyarivan, T. APACrefauthors \ 2002 . A fast and elitist multiobjective genetic algorithm: NSGA-II A fast and elitist multiobjective genetic algorithm: NSGA-II . IEEE transactions on evolutionary computation 6 2 182--197

    2002

  2. [10]

    \ Gandibleux, X

    ehrgott2002multiple APACrefauthors Ehrgott, M. \ Gandibleux, X. APACrefauthors \ 2002 . Multiple criteria optimization: state of the art annotated bibliographic surveys Multiple criteria optimization: state of the art annotated bibliographic surveys . Kluwer Academic Publishers

    2002

  3. [11]

    u ger, C. \ Sch \

    eichfelder2017decision APACrefauthors Eichfelder, G. , Kr \"u ger, C. \ Sch \"o bel, A. APACrefauthors \ 2017 . Decision uncertainty in multiobjective optimization Decision uncertainty in multiobjective optimization . Journal of Global Optimization 69 2 485--510

    2017

  4. [12]

    , Niebling, J

    eichfelder2020algorithmic APACrefauthors Eichfelder, G. , Niebling, J. \ Rockt \"a schel, S. APACrefauthors \ 2020 . An algorithmic approach to multiobjective optimization with decision uncertainty An algorithmic approach to multiobjective optimization with decision uncertaint...

    2020

  5. [13]

    \ Hunter, S R

    feldman2018score APACrefauthors Feldman, G. \ Hunter, S R. APACrefauthors \ 2018 . SCORE allocations for bi-objective ranking and selection Score allocations for bi-objective ranking and selection . ACM Transactions on Modeling and Computer Simulation (TOMACS) 28 1 1--28

    2018

  6. [14]

    , Bect, J

    feliot2017bayesian APACrefauthors Feliot, P. , Bect, J. \ Vazquez, E. APACrefauthors \ 2017 . A B ayesian approach to constrained single-and multi-objective optimization A B ayesian approach to constrained single-and multi-objective optimization . Journal of Global Optimizatio...

    2017

  7. [15]

    , Fonseca, C M

    figueira2017easy APACrefauthors Figueira, J R. , Fonseca, C M. , Halffmann, P. , Klamroth, K. , Paquete, L. , Ruzika, S. Willems, D. APACrefauthors \ 2017 . Easy to say they are Hard, but Hard to see they are Easy—Towards a Categorization of Tractable Multiobjective Combinator...

    2017

  8. [16]

    \ Fleming, P J

    fonseca1993genetic APACrefauthors Fonseca, C M. \ Fleming, P J. APACrefauthors \ 1993 . Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. Genetic algorithms for multiobjective optimization: formulation, discussion and generalizatio...

    1993

  9. [17]

    APACrefauthors \ 2002

    Fu02 APACrefauthors Fu, M C. APACrefauthors \ 2002 . Optimization for simulation: Theory vs. Practice Optimization for simulation: Theory vs. practice \ [Journal Article]. INFORMS Journal on Computing 14 3 192-215 . APACrefURL http://joc.journal.informs.org/content/14/3/192 APACrefURL

    2002

  10. [18]

    APACrefauthors \ 2015

    fu2015handbook APACrefauthors Fu, M C. APACrefauthors \ 2015 . Handbook of simulation optimization Handbook of simulation optimization \ ( 216). Springer

    2015

  11. [19]

    , Jalali, H

    gonzalez2020multiobjective APACrefauthors Gonzalez, S R. , Jalali, H. \ Van Nieuwenhuyse, I. APACrefauthors \ 2020 . A multiobjective stochastic simulation optimization algorithm A multiobjective stochastic simulation optimization algorithm . European Journal of Operational Re...

    2020

  12. [20]

    u nster, Germany, March 19-22, 2017, Proceedings 9 Evolutionary multi-criterion optimization: 9th international conference, emo 2017, m \

    horn2017first APACrefauthors Horn, D. , Dagge, M. , Sun, X. \ Bischl, B. APACrefauthors \ 2017 . First investigations on noisy model-based multi-objective optimization First investigations on noisy model-based multi-objective optimization . Evolutionary Multi-Criterion Optimiz...

    2017

  13. [21]

    \ Cheng, C W

    huang2024multiple APACrefauthors Huang, H. \ Cheng, C W. APACrefauthors \ 2024 . Multiple Objective Simulation Optimization for Emergency Department Resource Allocation Multiple objective simulation optimization for emergency department resource allocation . Asia-Pacific Journ...

    2024

  14. [22]

    \ Zabinsky, Z B

    Hu13:WinSim APACrefauthors Huang, H. \ Zabinsky, Z B. APACrefauthors \ 2013 . Adaptive Probabilistic Branch and Bound with Confidence Intervals for Level Set Approximation Adaptive probabilistic branch and bound with confidence intervals for level set approximation \ [Conferen...

    2013

  15. [23]

    \ Zabinsky, Z B

    Huang14 APACrefauthors Huang, H. \ Zabinsky, Z B. APACrefauthors \ 2014 . Multiple Objective Probabilistic Branch and Bound for P areto Optimal Approximation Multiple objective probabilistic branch and bound for P areto optimal approximation \ [Conference Proceedings]. S. Jain...

    2014

  16. [24]

    , Applegate, E A

    hunter2019introduction APACrefauthors Hunter, S R. , Applegate, E A. , Arora, V. , Chong, B. , Cooper, K. , Rinc \'o n-Guevara, O. \ Vivas-Valencia, C. APACrefauthors \ 2019 . An introduction to multiobjective simulation optimization An introduction to multiobjective simulatio...

    2019

  17. [25]

    , Smith, R L

    kiatsupaibul2018single APACrefauthors Kiatsupaibul, S. , Smith, R L. \ Zabinsky, Z B. APACrefauthors \ 2018 . Single observation adaptive search for continuous simulation optimization Single observation adaptive search for continuous simulation optimization . Operations Research

    2018

  18. [26]

    , Smith, R L

    FiniteSOSA APACrefauthors Kiatsupaibul, S. , Smith, R L. \ Zabinsky, Z B. APACrefauthors \ 2020 . Single Observation Adaptive Search for Discrete and Continuous Stochastic Optimization Single observation adaptive search for discrete and continuous stochastic optimization . INF...

    2020

  19. [27]

    \ Ryu, J h

    kim2011sample APACrefauthors Kim, S. \ Ryu, J h. APACrefauthors \ 2011 . The sample average approximation method for multi-objective stochastic optimization The sample average approximation method for multi-objective stochastic optimization . Proceedings of the 2011 Winter Sim...

    2011

  20. [28]

    , Wagner, T

    koch2015efficient APACrefauthors Koch, P. , Wagner, T. , Emmerich, M T. , B \"a ck, T. \ Konen, W. APACrefauthors \ 2015 . Efficient multi-criteria optimization on noisy machine learning problems Efficient multi-criteria optimization on noisy machine learning problems . Applie...

    2015

  21. [29]

    , Luccio, F

    MOKung75onfinding APACrefauthors Kung, H T. , Luccio, F. \ Preparata, F P. APACrefauthors \ 1975 . On finding the maxima of a set of vectors On finding the maxima of a set of vectors . Journal of the ACM 22 469--476

    1975

  22. [30]

    , Chew, E P

    MOO04OCBA APACrefauthors Lee, L H. , Chew, E P. , Teng, S. \ Goldsman, D. APACrefauthors \ 2004 . Optimal computing budget allocation for multi-objective simulation models Optimal computing budget allocation for multi-objective simulation models \ [Conference Proceedings]. Pro...

    2004 arXiv

  23. [31]

    , Chew, E P

    MOO10OCBA APACrefauthors Lee, L H. , Chew, E P. , Teng, S. \ Goldsman, D. APACrefauthors \ 2010 . Finding the non-dominated P areto set for multi-objective simulation models Finding the non-dominated P areto set for multi-objective simulation models \ [Journal Article]. IIE Tr...

    2010 doi

  24. [32]

    , Lee, L H

    li2015mo APACrefauthors Li, H. , Lee, L H. , Chew, E P. \ Lendermann, P. APACrefauthors \ 2015 . MO-COMPASS : A fast convergent search algorithm for multi-objective discrete optimization via simulation MO-COMPASS : A fast convergent search algorithm for multi-objective discret...

    2015

  25. [33]

    , Liu, W

    li2017optimal APACrefauthors Li, J. , Liu, W. , Pedrielli, G. , Lee, L H. \ Chew, E P. APACrefauthors \ 2017 . Optimal computing budget allocation to select the nondominated systems—A large deviations perspective Optimal computing budget allocation to select the nondominated s...

    2017

  26. [34]

    liu2020multi APACrefauthors Liu, Q. , Li, X. , Liu, H. \ Guo, Z. APACrefauthors \ 2020 . Multi-objective metaheuristics for discrete optimization problems: A review of the state-of-the-art Multi-objective metaheuristics for discrete optimization problems: A review of the state...

    2020

  27. [35]

    \ Zabinsky, Z B

    MeteZabinsky2014 APACrefauthors Mete, H O. \ Zabinsky, Z B. APACrefauthors \ 2014 . Multi-objective Interacting Particle Algorithm for Global Optimization Multi-objective interacting particle algorithm for global optimization . INFORMS Journal on Computing 26 500--513

    2014

  28. [36]

    APACrefauthors \ 2012

    miettinen2012nonlinear APACrefauthors Miettinen, K. APACrefauthors \ 2012 . Nonlinear multiobjective optimization Nonlinear multiobjective optimization \ ( 12). Springer Science & Business Media

    2012

  29. [37]

    , Khandait, T

    pedrielli2023part APACrefauthors Pedrielli, G. , Khandait, T. , Cao, Y. , Thibeault, Q. , Huang, H. , Castillo-Effen, M. \ Fainekos, G. APACrefauthors \ 2023 . Part-x: A family of stochastic algorithms for search-based test generation with probabilistic guarantees Part-x: A fa...

    2023

  30. [38]

    \ Van Nieuwenhuyse, I

    rojas2020survey APACrefauthors Rojas-Gonzalez, S. \ Van Nieuwenhuyse, I. APACrefauthors \ 2020 . A survey on kriging-based infill algorithms for multiobjective simulation optimization A survey on kriging-based infill algorithms for multiobjective simulation optimization . Comp...

    2020

  31. [39]

    , Liang, K H

    sarker2002new APACrefauthors Sarker, R. , Liang, K H. \ Newton, C. APACrefauthors \ 2002 . A new multiobjective evolutionary algorithm A new multiobjective evolutionary algorithm . European Journal of Operational Research 140 1 12--23

    2002

  32. [40]

    , Pant, M

    verma2021comprehensive APACrefauthors Verma, S. , Pant, M. \ Snasel, V. APACrefauthors \ 2021 . A comprehensive review on NSGA-II for multi-objective combinatorial optimization problems A comprehensive review on NSGA-II for multi-objective combinatorial optimization problems ....

    2021

  33. [41]

    , Lee, L H

    wang2015multi APACrefauthors Wang, Y. , Lee, L H. , Chew, E P. , Lam, S S W. , Low, S K. , Ong, M E H. \ Li, H. APACrefauthors \ 2015 . Multi-objective optimization for a hospital inpatient flow process via discrete event simulation Multi-objective optimization for a hospital ...

    2015

  34. [42]

    , Wang, L

    yang2011multi APACrefauthors Yang, R. , Wang, L. \ Wang, Z. APACrefauthors \ 2011 . Multi-objective particle swarm optimization for decision-making in building automation Multi-objective particle swarm optimization for decision-making in building automation . Power and Energy ...

    2011

  35. [43]

    \ Bekker, J

    yoon2020multi APACrefauthors Yoon, M. \ Bekker, J. APACrefauthors \ 2020 . Multi-objective simulation optimisation on discrete sets: a literature review Multi-objective simulation optimisation on discrete sets: a literature review . International Journal of Operational Researc...

    2020

  36. [44]

    , Ngadiman, M S

    yusoff2011overview APACrefauthors Yusoff, Y. , Ngadiman, M S. \ Zain, A M. APACrefauthors \ 2011 . Overview of NSGA-II for optimizing machining process parameters Overview of NSGA-II for optimizing machining process parameters . Procedia Engineering 15 3978--3983

    2011

  37. [45]

    \ Huang, H

    zabinsky2020partition APACrefauthors Zabinsky, Z B. \ Huang, H. APACrefauthors \ 2020 . A partition-based optimization approach for level set approximation: Probabilistic branch and bound A partition-based optimization approach for level set approximation: Probabilistic branch...

    2020

  38. [46]

    , " * write output.state after.block = add.period write newline

    ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution journal key keywords month note number organization pages publisher school series title type url volume year archivePrefix primaryClass adsurl adsnote version label extra.labe...

  39. [47]

    write newline

    " write newline "" before.all 'output.state := FUNCTION add.period duplicate empty 'skip "." * add.blank if FUNCTION if.digit duplicate "0" = swap duplicate "1" = swap duplicate "2" = swap duplicate "3" = swap duplicate "4" = swap duplicate "5" = swap duplicate "6" = swap dupl...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.