Pith. sign in

REVIEW 3 major objections 6 minor 31 references

Improved Mixing and Pressure Loss Formulations for Gas Network Optimization

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper claims that a previously non-viable discrete MINLP for gas network validation can be made fast and fully reliable by tightening it with cuts and replacing the pressure-loss term with a flow-splitting model.

desk verdict Clean fs pressure-loss derivation and large speedups, but the 100/0 global-optimality claim needs a feasibility audit before it's convincing. read the letter →

arxiv 2501.11608 v1 pith:RRAUNMDA submitted 2025-01-20 math.OC

Geonhee Kim , Christopher Lourenco , Daphne Skipper , Luze Xu This is my paper · ORCID
classification math.OC MSC 90C1190C2690C30
keywords gasnetworkoptimizationnominationvalidationmixed-integernonlinearprogrammingmixingpressurelossflowsplittingMcCormickcutsglobal
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

This paper takes aim at the nomination-validation problem: decide whether a proposed set of gas supplies and demands can be routed through an existing network while respecting flow balance, gas mixing, heat-power bounds, and pipe pressure limits. Earlier work had concluded that the continuous nonlinear (NLP) formulation of this problem was far better than the discrete mixed-integer (MINLP) one, which global solvers could rarely crack. The authors claim that this conclusion was a formulation artifact, not an inherent property: adding three classes of valid cuts and a pressure-loss model built on the flow-splitting variables makes the MINLP competitive. On 100 instances of a 582-node realistic network, the improved MINLP solved every instance to global optimality within the one-hour limit, with a mean time of 21.1 seconds and a slowest time of 142.7 seconds, while the baseline MINLP timed out on 24 instances and averaged 750.6 seconds on the rest. If this carries over to other networks, operators can use a globally solvable discrete model instead of a continuous one that is either very fast or times out.

What carries the argument

The load-bearing object is the flow-splitting decomposition of every pipe: the signed flow is $q_{u,v}=\beta_{u,v}-\gamma_{u,v}$, with $\beta_{u,v}$ and $\gamma_{u,v}$ the forward and backward parts. Because the mixing constraints already introduce these variables, the paper reuses them to rewrite $|q_{u,v}|$ as $\beta_{u,v}+\gamma_{u,v}$ (discrete) or $2\beta_{u,v}-q_{u,v}$ (continuous), turning the nonsmooth absolute value in the friction pressure-loss law into a smooth quadratic expression. On the discrete side, three cut families, the McCormick inequalities, the flow-direction cuts, and the bilinear-term bounds, tighten the integer relaxation. These two ingredients together, not any single constraint, are what make the MINLP's branch-and-bound search fast enough to certify global optimality on every tested instance.

What would settle it

Run the cut-strengthened MINLP with the flow-splitting pressure-loss model on the full set of nomination files for the same 582-node network and on the 4197-node network, using the same one-hour limit; if a nontrivial fraction of instances times out or the mean speedup over the baseline drops far below a factor of 35, the paper's central claim would be refuted. A second check is to solve the same 100 instances with a different global MINLP solver to see whether the improvement is an artifact of one solver's branch-and-bound implementation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the MINLP's poor showing in earlier comparisons came from a loose formulation and an expensive pressure-loss term, not from the intrinsic difficulty of discrete flow-splitting. The authors add McCormick inequalities on the four bilinear products in the mixing constraints, flow-direction cuts on the binary variables that forbid all-in or all-out flow patterns at nodes, and direct upper bounds on the bilinear products keyed to the binary variables. They then derive a flow-splitting pressure-loss model that uses the forward and backward flow variables as a smooth stand-in for the absolute value of flow, matching the most accurate piecewise friction model (HP-PC) asymptotically and at zero flow while remaining quadratic. In the MINLP, the cuts alone cut average solve time by roughly 60 percent; the flow-splitting pressure-loss model then cuts it by another 93 percent, and the combination solves all 100 test instances. A simpler signed-quadratic variant is even faster in the MINLP but mislabels 48 of 100 NLP instances as infeasible, so the paper recommends the flow-splitting model as the best balance of accuracy, speed, and reliability.

Load-bearing premise

The speedup and 100 percent stability were measured on 100 instances of one 582-node network with a single global solver, and the 99 non-baseline nomination files are described only as randomly selected with no seed or selection procedure; if those instances are easier or more homogeneous than other real networks, the factor-of-35 improvement may not transfer.

Editorial extensions

If this is right

  • The MINLP with cuts and the flow-splitting pressure-loss model is no longer dominated by the NLP: it solved all 100 test instances while the baseline NLP timed out on 19 and the baseline MINLP timed out on 24.
  • The cuts are universally beneficial for the MINLP, with average speedups of roughly 2.4 times for the square-root model, 5.0 times for the flow-splitting model, and 10.4 times for the simpler PKr model, but they are harmful for the NLP, where the McCormick inequalities increased average solve time by a factor of 5.1.
  • The flow-splitting pressure-loss model also stabilizes the NLP, roughly halving the number of timeouts compared with the baseline (11 versus 19), even though it cannot match the speed of the best NLP on instances that solve quickly.
  • The simpler PKr pressure-loss model is the fastest MINLP variant, with a mean solve time of 15.6 seconds, but it should not be used with the NLP because it caused 48 instances to be mislabeled infeasible.

Reading between the lines

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

  • Beyond the paper: the flow-splitting pressure-loss trick should transfer to any network-flow model that already splits flows into forward and backward variables and has a nonsmooth signed loss term, such as water-distribution or heat-network models, since the same substitution of $\beta+\gamma$ for $|q|$ gives a smooth quadratic loss there.
  • Beyond the paper: the sharp contrast in how cuts affect the MINLP versus the NLP suggests the gains come from tighter relaxations feeding discrete branching, while the extra constraints steepen the continuous problem; a testable extension is to apply the cuts only in the branch-and-bound relaxation while solving the NLP subproblems without them.
  • Beyond the paper: because all tests use one global solver and one 582-node network, the practical recommendation would be stronger if replicated with a second global MINLP solver and with the larger 4197-node network, which the authors themselves leave for future work.
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 / 6 minor

Summary. The paper improves two gas network optimization models from Hante and Schmidt by adding three classes of cuts to the MINLP and by proposing two new pressure-loss approximations that exploit flow-splitting variables: a high-accuracy "fs" model tuned to the HP-PC pressure-loss law, and a simpler smooth PKr model. After deriving parameter choices for the fs model to match HP-PC asymptotically and at zero flow, the authors report computational experiments on 100 GasLib-582 instances with BARON. The headline results are a factor-35 mean speedup for the MINLP, 100/100 instances solved within 2.5 minutes using the fs model with all cuts, and improved stability of the NLP variant.

Significance. If the computational claims withstand scrutiny, the paper makes a useful practical contribution: it shows that a strengthened MINLP formulation can be competitive with NLP formulations for nomination validation with gas mixing, it provides two smooth pressure-loss models with simple algebraic forms, and it ships publicly available data and model files, which is a reproducibility strength. The algebraic derivation of the fs model in Theorem 5.2 is clean, and the performance-profile comparisons are informative. However, the numerical verification has a serious gap: BARON mislabels 10 uncut fs instances as infeasible while the cut version solves all 100, which calls for independent confirmation before the global-optimality claims can be accepted.

major comments (3)
  1. [Section 6.4, Table 5] The table reports that the MINLP with phi_fs and no cuts solved 90 instances and mislabeled 10 as infeasible, while the same model with cuts (20)-(22) solved all 100. Because the cuts are valid inequalities, every feasible point of the cut model is feasible for the uncut model, so the 10 "infeasible" outcomes cannot be genuine infeasibilities. The paper acknowledges the mislabeling in the text but does not address its implications. This is not a cosmetic issue: it shows that BARON can return an incorrect status on these models, so the claim that all 100 instances were solved to global optimality in the cut version is not independently established. I ask for a cross-validation with a second global solver, or a documented feasibility and optimality audit with explicit tolerances, for the key model configurations, and for a discussion of how the reported factor-of-35 speedup is affected by this verification gap.
  2. [Section 5.4.1, Eq. (30)] Theorem 5.2 defines d_tilde by (30) but does not establish that d_tilde is positive and finite. Substituting the definitions gives b_tilde = (ln rho + 1) t^2, which is negative whenever k/D < 3.71/e, the same regime covered by Theorem 5.1, and the sign of the denominator 64 eta A omega - 2 t D Lambda is parameter-dependent. If d_tilde is nonpositive, the denominators beta+gamma+d_tilde in (32) and 2 beta - q + d_tilde in (33) can vanish for admissible flows, introducing a singularity in the proposed pressure-loss model. The theorem should include explicit conditions on pipe parameters that guarantee d_tilde > 0, or the computational study should verify the sign of d_tilde for every pipe in GasLib-582. In addition, the assertion that (phi_fs)''(0) is "very small in magnitude" is not quantified; using the formula 2 Lambda (1 - b_tilde / d_tilde^2) with representative parameter values can give large values, so this claim needs support.
  3. [Section 4.2] The flow-direction cuts (21) are described as valid, but no proof is provided. Because the binary variables d are not uniquely determined when the corresponding flow is zero, validity requires an argument that every feasible point admits an assignment of d satisfying (21). This is easy to supply but is load-bearing: the comparison in Table 5 between the uncut and cut models assumes that these cuts cannot exclude feasible points. Please add a short proof or a reference for the validity of (21).
minor comments (6)
  1. [Section 1.2] Typo: "proprosed" should be "proposed".
  2. [Section 6.3] The text says 99 nomination files were "randomly selected," but no seed or selection procedure is given; if the exact instance list is already in the public repository, please state this explicitly to make the selection reproducible.
  3. [Section 3.2.1] In the sentence introducing (17), "nodes u in A" should read "nodes u in V", since u is a node index.
  4. [Section 6.1] Typo: "bilinaer" should be "bilinear".
  5. [Section 5.4.2 and elsewhere] The name "Prandlt-Karman" is misspelled; it should be "Prandtl-Karman".
  6. [Table 5] The final column heading "Infeasible" is potentially misleading; consider using "reported infeasible" or "misclassified as infeasible" to match the discussion in Section 6.4.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the new pressure-loss surrogates are matched to the externally defined HP-PC model and the reported speedups are empirical comparisons against an external baseline, so the central claims do not reduce to their own inputs.

full rationale

The paper's central contributions are (i) valid cuts for the MINLP, (ii) two pressure-loss surrogates, and (iii) a computational comparison. None of these presuppose the paper's conclusions. The fs model is explicitly derived as a smooth replacement for |q| in a surrogate of the HP-PC model: the parameters in Theorem 5.2 are chosen so that phi_fs agrees with phi_HP-PC asymptotically and at q=0 (matching value and first derivative, and second derivative in the asymptotic case). This is fitting a surrogate to an externally defined physics model, not fitting a parameter to the solver runtimes or to the nominal feasibility outcomes; the runtime speedups in Table 5 are then measured empirically against the [15] baseline, which is an external paper by disjoint authors (Hante and Schmidt). The PKr variant is explicitly identified as 'equivalent to the existing PKr pressure loss approximation', so no known model is being renamed as a new derivation. The paper does contain an internal verification flag: Table 5 reports MINLP with phi_fs and no cuts as 90 solved / 10 infeasible while the same model with cuts solves all 100, and the paper itself calls these 'mislabeled as infeasible.' Since the cuts are claimed valid, this inconsistency points to a solver reliability issue that deserves a feasibility audit, but it is not circularity: no target quantity (e.g., 'solved to global optimality' or 'speedup factor') is used as an input to any derivation. The weaknesses about instance selection and single-solver testing are generalizability concerns, not circularity.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical entities or fitted constants are introduced. The new cuts and pressure-loss formulas are derived from existing models; the main epistemic inputs are the gas-network idealizations inherited from [15,29], the HP-PC/sqrt pressure-loss benchmark, and trust in BARON's global solves.

free parameters (3)
  • Heat-power bound factors for exit nodes = P_u = 0.9 * q_u * H_m and P_u = 1.1 * q_u * H_m
    Chosen in Section 6.3 to create test instances; taken from [15]. These shape the test set but are not fitted to the speedup claim.
  • Complementarity relaxation parameter epsilon = 1e-6
    Fixed in (15a) following [15]; affects NLP complementarity formulation and the reported NLP stability comparison.
  • Sqrt pressure-loss smoothing parameters d_hat, e_hat = Positive root of (29), with d_hat = e_hat; e.g., 0.49794 for the sample pipe in Section 5.5
    Modeler-defined in [2] and set in Section 5.3 to balance square roots and match the HP-PC derivative at zero; inherited by the baseline and graphical comparisons.
assumptions (6)
  • domain assumption Stationary, isothermal flow; pipes have zero slope and a single, a priori unknown flow direction; all molar masses are equal so gas density is constant.
    Inherited from [15,29] in Section 2.1; the mixing and pressure-loss models are built on these idealizations.
  • domain assumption HP-PC model (25)-(26) is the ground-truth pressure-loss law, and the sqrt model (27) is a faithful smooth approximation of it.
    Sections 5.2-5.3; the fs model is designed to match HP-PC/sqrt, so the accuracy claim inherits their validity.
  • domain assumption The flow-splitting variables satisfy beta * gamma = 0 (binary in MINLP, complementarity in NLP), so |q| = beta + gamma exactly.
    Section 3.1; both new pressure-loss formulas and the mixing model rely on this identity.
  • standard math McCormick inequalities (20) are valid convex relaxations of the bilinear mixing terms.
    Section 4.1; standard result from [19].
  • domain assumption Flow-direction cuts (21) are valid for every feasible flow: each non-supply node has an incoming flow and each non-demand node has an outgoing flow.
    Section 4.2; the paper cites [13] for similar cuts but does not prove validity for this model.
  • domain assumption BARON version 23.3.11 with default settings correctly solves models to global optimality or correctly declares infeasibility.
    Section 6.2; Table 5 records some instances mislabeled as infeasible, so this assumption fails for some model variants.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Improved Mixing and Pressure Loss Formulations for Gas Network Optimization." pith.science (2026). https://pith.science/paper/RRAUNMDA

@misc{pith2026250111608,
  author       = {Pith},
  title        = {Pith review of: Improved Mixing and Pressure Loss Formulations for Gas Network Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RRAUNMDA}},
  note         = {Machine review of arXiv:2501.11608}
}
read the original abstract

Non-convex, nonlinear gas network optimization models are used to determine the feasibility of flows on existing networks given constraints on network flows, gas mixing, and pressure loss along pipes. This work improves two existing gas network models: a discrete mixed-integer nonlinear program (MINLP) that uses binary variables to model positive and negative flows, and a continuous nonlinear program (NLP) that implements complementarity constraints with continuous variables. We introduce cuts to expedite the MINLP and we formulate two new pressure loss models that leverage the flow-splitting variables: one that is highly accurate and another that is simpler but less accurate. In computational tests using the global solver BARON our cuts and accurate pressure loss improves: (1) the average run time of the MINLP by a factor of 35, (2) the stability of the MINLP by solving every tested instance within 2.5 minutes (the baseline model timed out on 25% of instances), (3) the stability of the NLP by solving more instances than the baseline. Our simpler pressure loss model further improved run times in the MINLP (by a factor of 48 versus the baseline MINLP), but was unstable in the context of the NLP.

Figures

Figures reproduced from arXiv: 2501.11608 by the authors.

Figure 1
Figure 1. Pressure loss models for a sample GasLib582 Pipe [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Baseline NLP model dominates baseline MINLP [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Effects of cuts on the computational performance of the NLP and MINLP. [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Performance Profile All Pressure Loss Models [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 30 canonical work pages

  1. [1]

    arXiv preprint arXiv:2112.08872 (2021)

    Bestuzheva, K., Besançon, M., Chen, W.K., Chmiela, A., Donkiewicz, T., van Doornmalen, J., Eifler, L., Gaul, O., Gamrath, G., Gleixner, A., et al.: The SCIP optimization suite 8.0. arXiv preprint arXiv:2112.08872 (2021)

    arXiv 2021

  2. [2]

    Optimization and Engineering10(1), 43–73 (2009)

    Burgschweiger,J.,Gnädig,B.,Steinbach,M.C.:Optimizationmodelsforoperativeplanningindrink- ing water networks. Optimization and Engineering10(1), 43–73 (2009)

    work page 2009

  3. [3]

    Dolan,E.D.,Moré,J.J.:Benchmarkingoptimizationsoftwarewithperformanceprofiles.Mathematical programming 91, 201–213 (2002)

    work page 2002

  4. [4]

    ORSA Journal on computing6(2), 207–216 (1994) 26 Kim et

    Drud, A.S.: Conopt—a large-scale grg code. ORSA Journal on computing6(2), 207–216 (1994) 26 Kim et. al

    work page 1994

  5. [5]

    SPE Drilling & Completion21(03), 212–215 (2006)

    Farshad, F.F., Rieke, H.H.: Surface-roughness design values for modern pipes. SPE Drilling & Completion21(03), 212–215 (2006)

    work page 2006

  6. [6]

    McGraw-Hill Educa- tion (2002)

    Finnemore, E.J., Franzini, J.B.: Fluid mechanics with engineering applications. McGraw-Hill Educa- tion (2002)

    work page 2002

  7. [7]

    AT & T Bell Laboratories Murray Hill, NJ (1987)

    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A mathematical programming language. AT & T Bell Laboratories Murray Hill, NJ (1987)

    work page 1987

  8. [8]

    In: Evaluating gas network capacities, pp

    Fügenschuh, A., Geißler, B., Gollmer, R., Morsi, A., Rövekamp, J., Schmidt, M., Spreckelsen, K., Steinbach, M.C.: Physical and technical fundamentals of gas networks. In: Evaluating gas network capacities, pp. 17–43, Chapter 2. SIAM (2015)

    work page 2015

Show all 31 references
  1. [9]

    In: Evaluating Gas Network Capacities, pp

    Geißler, B., Martin, A., Morsi, A., Schewe, L.: The MILP-relaxation approach. In: Evaluating Gas Network Capacities, pp. 103–122, Chapter 6. SIAM (2015)

    2015

  2. [10]

    Computers & Chemical Engineering82, 303–317 (2015)

    Geißler, B., Morsi, A., Schewe, L., Schmidt, M.: Solving power-constrained gas transportation prob- lems using an mip-based alternating direction method. Computers & Chemical Engineering82, 303–317 (2015)

    2015

  3. [11]

    SIAM review47(1), 99–131 (2005)

    Gill,P.E.,Murray,W.,Saunders,M.A.:Snopt:Ansqpalgorithmforlarge-scaleconstrainedoptimiza- tion. SIAM review47(1), 99–131 (2005)

    2005

  4. [12]

    ORSAJournalonComputing 7(2),205–217(1995)

    Greenberg,H.J.:Analyzingthepoolingproblem. ORSAJournalonComputing 7(2),205–217(1995)

    1995

  5. [13]

    Networks 79(1), 83–104 (2022)

    Habeck,O.,Pfetsch,M.E.:Combinatorialacyclicitymodelsforpotential-basedflows. Networks 79(1), 83–104 (2022)

    2022

  6. [14]

    Logistics operations and management pp

    Hamedi, M., Farahani, R.Z., Esmaeilian, G., et al.: Optimization in natural gas network planning. Logistics operations and management pp. 393–420 (2011)

    2011

  7. [15]

    EURO Journal on Computational Optimization7(3), 299–323 (2019)

    Hante, F.M., Schmidt, M.: Complementarity-based nonlinear programming techniques for optimal mixing in gas networks. EURO Journal on Computational Optimization7(3), 299–323 (2019)

    2019

  8. [16]

    SIAM (2015)

    Koch, T., Hiller, B., Pfetsch, M.E., Schewe, L.: Evaluating gas network capacities. SIAM (2015)

    2015

  9. [17]

    Cambridge University Press (1996)

    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical programs with equilibrium constraints. Cambridge University Press (1996)

    1996

  10. [18]

    Mathematical programming105, 563–582 (2006)

    Martin, A., Möller, M., Moritz, S.: Mixed integer models for the stationary case of gas network optimization. Mathematical programming105, 563–582 (2006)

    2006

  11. [19]

    Mathematical programming10(1), 147–175 (1976)

    McCormick,G.P.:Computabilityofglobalsolutionstofactorablenonconvexprograms:Parti—convex underestimating problems. Mathematical programming10(1), 147–175 (1976)

    1976

  12. [20]

    Journal of Global Optimization59(2-3), 503–526 (2014)

    Misener, R., Floudas, C.A.: Antigone: algorithms for continuous/integer global optimization of non- linear equations. Journal of Global Optimization59(2-3), 503–526 (2014)

    2014

  13. [21]

    Murtagh, B.A., Saunders, M.A., et al.: MINOS 5.0 user’s guide, vol. 83. Systems Optimization Laboratory, Department of Operations Research, Stanford ... (1983)

    1983

  14. [22]

    OGIL MUSZ, Tud, Kuzl., Budapest pp

    Papay, J.: Atermelestechnologiai parameterek valtozasa a gazlelepk muvelese soran. OGIL MUSZ, Tud, Kuzl., Budapest pp. 267–273 (1968)

    1968

  15. [23]

    Optimization Methods and Software30t(1), 15–53 (2015)

    Pfetsch, M.E., Fügenschuh, A., Geißler, B., Geißler, N., Gollmer, R., Hiller, B., Humpola, J., Koch, T., Lehmann, T., Martin, A., et al.: Validation of nominations in gas network optimization: models, methods, and solutions. Optimization Methods and Software30t(1), 15–53 (2015)

    2015

  16. [24]

    Applied Energy147, 536–555 (2015)

    Ríos-Mercado,R.Z.,Borraz-Sánchez,C.:Optimizationproblemsinnaturalgastransportationsystems: A state-of-the-art review. Applied Energy147, 536–555 (2015)

    2015

  17. [25]

    Journal of Global Optimization 8, 201–205 (1996)

    Sahinidis, N.V.: Baron: A general purpose global optimization software package. Journal of Global Optimization 8, 201–205 (1996)

    1996

  18. [26]

    McGraw-Hill Education (2002)

    Saleh, J.M.: Fluid flow handbook. McGraw-Hill Education (2002)

    2002

  19. [27]

    Data2(4) (2017)

    Schmidt, M., Aßmann, D., Burlacu, R., Humpola, J., Joormann, I., Kanelakis, N., Koch, T., Oucherif, D., Pfetsch, M.E., Schewe, L., Schwarz, R., Sirvent, M.: Gaslib—a library of gas network instances. Data2(4) (2017). DOI 10.3390/data2040040. URLhttps://www.mdpi.com/2306-5729/2/4/40

    2017 doi

  20. [28]

    Optimization and Engineering16(1), 131–164 (2015)

    Schmidt, M., Steinbach, M.C., Willert, B.M.: High detail stationary optimization models for gas networks. Optimization and Engineering16(1), 131–164 (2015)

    2015

  21. [29]

    Optimization and Engineering17(2), 437–472 (2016)

    Schmidt, M., Steinbach, M.C., Willert, B.M.: High detail stationary optimization models for gas networks: validation and results. Optimization and Engineering17(2), 437–472 (2016)

    2016

  22. [30]

    Mathematical programming106, 25–57 (2006)

    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming106, 25–57 (2006)

    2006

  23. [31]

    Ziena Optimization, Inc.[en ligne] disponible sur http://www

    Waltz, R.A., Nocedal, J.: Knitro 2.0 user’s manual. Ziena Optimization, Inc.[en ligne] disponible sur http://www. ziena. com (September, 2010)7, 33–34 (2004)

    2004

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.