Pith. sign in

REVIEW 4 major objections 3 minor 2 cited by

Strong Formulations for Hybrid System Control

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The paper establishes that the single-period two-mode hybrid control problem has a convex hull in the original variables, captured by two new cut families that, applied period-wise, make the n-period problem substantially faster to solve.

desk verdict A solid 1D convexification result with a load-bearing gap: the multi-dimensional cut validity is unproven and the cut formulas as written are invalid. read the letter →

arxiv 2412.11541 v1 pith:HLQ4IERZ submitted 2024-12-16 math.OC

Jisun Lee , Hyungki Im , Alper Atamtürk This is my paper · ORCID
classification math.OC MSC 90C1190C20
keywords hybridsystemcontrolmixed-integerquadraticprogrammingconvexhulldescriptionvalidcutsdisjunctiveperspectivereformulationmodelpredictivepower-splitelectricvehicle
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 attempts to establish that the n-period hybrid control problem, a mixed-integer quadratic program with convex quadratic costs and linear dynamics, can be solved far faster by strengthening its relaxation with valid cuts that live entirely in the original variable space. The authors first prove a convex hull description for the single-period two-mode subproblem, expressed through two new cut families: linear feasibility cuts and nonlinear cuts, the latter implemented in practice as gradient cuts. They then generalize these cuts to multi-dimensional states and controls, multiple modes, and nonzero constant terms in the dynamics, and apply them one period at a time to the full horizon. A sympathetic reader would care because the approach avoids the auxiliary variables of extended-space disjunctive formulations, and the reported experiments show substantially smaller branch-and-bound effort, with about a tenfold average speedup on a power-split hybrid electric vehicle energy-management benchmark.

What carries the argument

The load-bearing object is a one-dimensional projection problem $\tau(x_1,x_2,y,z)=\min_p (x_2-ap)^2$ over bounds $p$ derived from the disjunctive convex hull representation in an extended space. Projecting out the auxiliary variable $p$ produces two cut families: feasibility cuts, which are linear inequalities ensuring the projection is feasible, and a nonlinear cut, a convex piecewise-quadratic inequality that reduces to the perspective cut when $x_2$ lies inside the projected interval and strengthens it otherwise. In computations the nonlinear cut is implemented through its gradient linear underestimators, giving convex quadratic cuts that can be added to a MIQP solver without auxiliary variables. The multi-dimensional extension decomposes coordinate-wise when the quadratic cost matrix $Q_2$ is diagonal, and the multi-mode extension uses a two-way disjunction on the index set of modes.

What would settle it

Fix state dimension $dx=2$ and run the With-Cuts gradient model and the SPP model on the same random instances for horizons $n=40,50,60$ with a one-hour time limit; if WC-G's solve time grows past SPP's as $n$ increases, the claim that original-space cuts give a practical speedup over extended-space formulations for long horizons is false.

Watch

Extended reading notes

Core claim

The central claim is that the epigraph of a single-period hybrid control problem with convex quadratic cost and linear dynamics has a convex hull that can be described, after projecting out auxiliary variables, by two classes of valid cuts in the original variable space: feasibility cuts that enforce the bounds needed for the projection to be feasible, and a nonlinear cut that improves on the perspective cut by exploiting the linear dynamics constraint $x_2 = ax_1 + by + cz$. The same projection argument is extended, with a diagonal quadratic cost matrix assumption for the multi-dimensional case, to multi-mode and multi-dimensional problems, and the cuts are generated independently for each period of the $n$-period problem. The paper reports that the resulting With-Cuts formulation substantially reduces relaxation gaps and computation times relative to the plain MIQP, and that in the vehicle study the gradient-cut version is about ten times faster on average, with worst-case times that remain manageable within the sampling period.

Load-bearing premise

The n-period extension generates cuts for each period independently, ignoring the coupling between consecutive periods, so the relaxation gap can grow with the horizon; the multi-dimensional cut derivation also assumes the quadratic cost matrix $Q_2$ is diagonal, and either assumption, if it fails in practice, would undercut the practical advantage the paper claims.

Editorial extensions

If this is right

  • The single-period two-mode problem has an explicit convex hull in the original variables, so a solver can apply the two cut families without adding auxiliary variables.
  • Because each cut is generated from the linear dynamics constraint, the resulting relaxation is at least as strong as the perspective cut and is strictly stronger whenever the state variable falls outside the projected interval.
  • For high state dimension ($dx \ge 4$) the With-Cuts model beats the SPP extended-space model in computation time, while for small $dx$ SPP remains faster, so the two formulations are complementary rather than one dominating.
  • Applying the cuts period by period keeps the variable count small; in the vehicle application the gradient-cut model runs about ten times faster on average than the plain MIQP and finishes within the MPC sampling time.

Reading between the lines

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

  • A natural extension not pursued in the paper is to choose the two-way mode split $K_1$ adaptively inside branch-and-bound. Because any nonempty $K_1$ yields valid cuts, a dynamic separation rule could tighten the relaxation exactly where the current period-wise cuts are weak.
  • The period-wise construction ignores coupling between consecutive periods, so a hybrid strategy that uses SPP-style coupling for small state dimensions or long horizons and original-space cuts for large state dimensions would plausibly outperform either method alone; the paper's own $dx=2$ results hint at this.
  • Because the multi-dimensional derivation assumes the quadratic cost matrix $Q_2$ is diagonal, a concrete stress test is to rotate the coordinates so that $Q_2$ becomes dense; the resulting gap between the cut relaxation and the true convex hull would show how much of the strengthening depends on that assumption.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper studies an n-period hybrid control problem (HCP) formulated as a mixed-integer quadratic program with convex quadratic cost and linear dynamics. The authors first derive, for a single-period two-mode one-dimensional problem, an extended-space convex hull representation and then project out the auxiliary variable to obtain feasibility cuts and nonlinear/gradient cuts. They claim that these cuts are generalized to multi-dimensional state/control spaces, to multi-mode problems via a two-way disjunction, and then applied period-by-period to the full n-period problem. Computational experiments on synthetic instances and a power-split hybrid electric vehicle energy-management problem report substantially reduced solution times, especially for the gradient-cut variant WC-G.

Significance. If the generalized cuts are valid, the paper offers a practically attractive alternative to extended-space formulations because it keeps the number of variables small while improving the continuous relaxation. The one-dimensional derivation in Section III-A is careful and largely coherent, and the computational comparison against the shortest-path formulation of Marcucci et al. is informative. However, the multi-dimensional and multi-mode extensions are asserted rather than proved, and the relationship between the implemented models and the claimed 'convex hull in the original variable space' is not fully clear. The significance is therefore conditional on closing the proof gaps, but the underlying idea is valuable and the experimental evidence is promising.

major comments (4)
  1. [Abstract and Section IV] The abstract states that the paper gives a convex hull description 'in the original variable space' and that the cuts are generalized to the multi-dimensional, multi-mode case. However, Section IV describes WC-NL as adding 'nonlinear cuts as in (3a)', and (3a) is the extended-space inequality with the auxiliary variable p from Proposition 1. Only the gradient-cut model WC-G appears to operate purely in the original variables. This discrepancy should be resolved: either state explicitly that the exact hull is only obtained before linearization, or clarify what is actually implemented in each model.
  2. [Section III-B, Eqs. (10)-(11)] The validity of the multi-dimensional cuts is the load-bearing step for the numerical claims, but it is not proved. After relaxing the projection problem by replacing the bound constraints with S2, by dropping sigma in C(A), and by decomposing coordinate-wise, the paper asserts that 'the same cut-generation process can be employed.' The text does not demonstrate that the elementwise max/min definitions of ell and u produce a feasible box that is a relaxation of the true projection's feasible set in the direction needed for tau' <= tau, nor does it prove that the resulting cut is valid for (7) and, after period-wise application, for (1). Please provide a formal statement and proof analogous to Corollary 5 for the multi-dimensional case.
  3. [Corollary 6 and Table II] The displayed cuts use the reference value (\bar{x}_2 - \bar{\sigma})^2 in the first quadratic term, making that term constant, while the remaining terms depend on x2, y, and z. As written, this is not a well-defined valid inequality for (2) unless it is only the point value used to construct the gradient cut in (6). The same ambiguity appears in the 'Cut' row of Table II, where \bar{x}_{2,i} appears in the leading term. Please clarify which object is added to the solver and prove that it is a valid underestimator of the projected nonlinear cut.
  4. [Section III-D / Appendix C] Proposition 9 is stated for the multi-mode case, and Corollary 10 depends on it, but the proof in Appendix C gives details for cases (A) and (B) and only says that (C)-(H) 'can also be proved by negating each condition.' Since these feasibility cuts are the foundation for generating the Table II cuts at arbitrary relaxation points, the missing cases should be supplied explicitly or the proof should be completed in the appendix.
minor comments (3)
  1. [Section I and Eq. (2)] There is a typo 'singe-period' in the introduction, and the constraint in (2) writes 'q2x2^2 + ry2 + z' where the control term appears to be r y^2; please check the notation and make it consistent with the later expressions such as Corollary 5.
  2. [End of Section III-A] The sentence 'Note that all results in this section hold for dy >= 1 as well' is not proved and is not immediate because y enters through a quadratic term y^T R y; either provide the argument or soften the claim.
  3. [Section III-E and Figures 3-4] The paper acknowledges that generating cuts period-by-period ignores coupling between periods, and Figure 3 shows that the WC-G relaxation gap grows with n for dx=2 and some instances time out for n >= 40. The abstract and conclusion should state more precisely that the practical advantage is dimension-dependent rather than claiming a uniform substantial reduction for all n-period instances.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cut derivations start from the disjunctive definition of the feasible set and are not fitted or self-citation-forced.

full rationale

The paper's central derivation begins with the definition of the single-period feasible set as a union of mode-specific convex sets and applies disjunctive programming, convex combination, and projection. Proposition 1 and Corollary 3 are derived from that definition rather than from any assumed target result, and the nonlinear cut in Corollary 5 is obtained by explicitly solving the projection problem (4). The gradient cut in Corollary 6 follows from convexity of the resulting function, so it is not a fitted quantity renamed as a prediction. There are no calibrated parameters, no term fitted to the reported benchmarks, and no external claim that is load-bearing: the perspective reformulation citations [44], [45] appear as background and as a comparison point, and the remark that the new cut 'enhances' that reformulation is supported by the preceding derivation rather than imported from those references. The multi-dimensional generalization in Section III-B does replace the exact projection by a coordinate-wise relaxation and the multi-period extension in Section III-E generates cuts period-by-period while conceding a larger relaxation gap, but these are potential correctness or strength concerns, not circularity: they do not make the claimed result equivalent to its own input by construction. Similarly, the ambiguity about barred values in Corollary 6 and Table II is a clarity or validity issue, not a circular reduction. The paper is self-contained against the external SPP benchmark, and the self-citations are not used to force the main claim. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted to data or introduced ad hoc; all cuts are derived. The axioms are standard convex analysis, the positive semidefinite structure of the problem class, and the domain approximation needed for the vehicle case study.

assumptions (4)
  • domain assumption Cost matrices Q_t, R_t, S_t are symmetric positive semidefinite and Q2 is diagonal in the multi-dimensional derivation.
    Invoked in problem (1) and Section III-B; convexity and coordinate-wise decomposition of the projection depend on it.
  • standard math Disjunctive programming projection principle: the convex hull of a union of convex sets is the projection of the convex combination of points in those sets.
    Basis of Proposition 1 and Corollary 8.
  • domain assumption State bounds and mode-dependent control bounds define the feasible region; the bounds are finite so the projection problem (4) is well posed.
    Used in the projection problem (4) and the feasibility cuts; without finite bounds the projection bounds are undefined.
  • domain assumption The linearized PS-HEV model (19) is an adequate local approximation of the nonlinear powertrain model (18).
    Section V-B introduces this to make the vehicle problem fit the HCP form; the computational claims for the vehicle depend on the linearization quality.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Strong Formulations for Hybrid System Control." pith.science (2026). https://pith.science/paper/HLQ4IERZ

@misc{pith2026241211541,
  author       = {Pith},
  title        = {Pith review of: Strong Formulations for Hybrid System Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HLQ4IERZ}},
  note         = {Machine review of arXiv:2412.11541}
}
abstract

We study the mixed-integer quadratic programming formulation of an $n$-period hybrid control problem with a convex quadratic cost function and linear dynamics. We first give the convex hull description of the single-period, two-mode problem in the original variable space through two new classes of valid cuts. These cuts are then generalized to the single-period, multi-mode, multi-dimensional case and applied to solve the general $n$-period hybrid control problem. Computational experiments demonstrate the effectiveness of the proposed strong formulations derived through the cut generation process in the original variable space. These formulations yield a substantial reduction in computational effort for synthetic test instances and instances from the energy management problem of a power-split hybrid electric vehicle.

Figures

Figures reproduced from arXiv: 2412.11541 by the authors.

Figure 1
Figure 1. Relaxation gap as a function of dx [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Computational time as a function of dx. binary variables. Neither of the two versions of the WC models outperforms the other consistently, but WC-G with gradient cuts has a shorter computational time on average in most cases. We further examine the performance of WC-G and SPP models as a function of n for dx = 2 and dx = 4 with a time limit of one hour. The results are summarized in Figures 3 and 4. For dx = 2, the … view at source ↗
Figure 3
Figure 3. Performance of WC-G and SPP as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Performance of WC-G and SPP as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Rank-one convexification for quadratic optimization problems with step function penalties

    math.OC 2025-04 conditional novelty 7.0 of 10

    An explicit convex hull for rank-one quadratic optimization with step function penalties is derived, enabling stronger copositive and SDP relaxations and a robust convex SVM surrogate.

  2. Mixed Discrete and Continuous Planning using Shortest Walks in Graphs of Convex Sets

    cs.RO 2025-07 conditional novelty 6.0 of 10

    Shortest walks in graphs of convex sets, guided by SDP-computed cost-to-go lower bounds, provide a unified approximate planner for robot motion, skill chaining, and hybrid control.

Reference graph

Works this paper leans on

47 extracted references · 45 canonical work pages · cited by 2 Pith papers

  1. [1]

    Borrelli, A

    F. Borrelli, A. Bemporad, and M. Morari, Pre- dictive Control for Linear and Hybrid Systems . Cambridge University Press, 2017

    work page 2017

  2. [2]

    Computational complexity of the capacitated lot size problem,

    G. R. Bitran and H. H. Yanasse, “Computational complexity of the capacitated lot size problem,” Management Science , vol. 28, no. 10, pp. 1174– 1186, 1982

    work page 1982

  3. [3]

    The explicit linear quadratic regu- lator for constrained systems,

    A. Bemporad, M. Morari, V . Dua, and E. N. Pistikopoulos, “The explicit linear quadratic regu- lator for constrained systems,” Automatica, vol. 38, no. 1, pp. 3–20, 2002

    work page 2002

  4. [4]

    Dynamic programming for constrained optimal control of discrete-time linear hybrid sys- tems,

    F. Borrelli, M. Baoti ´c, A. Bemporad, and M. Morari, “Dynamic programming for constrained optimal control of discrete-time linear hybrid sys- tems,” Automatica, vol. 41, no. 10, pp. 1709–1721, 2005

    work page 2005

  5. [5]

    A multiparametric programming approach for mixed- integer quadratic engineering problems,

    V . Dua, N. A. Bozinis, and E. N. Pistikopoulos, “A multiparametric programming approach for mixed- integer quadratic engineering problems,” Comput- ers & Chemical Engineering , vol. 26, no. 4-5, pp. 715–733, 2002

    work page 2002

  6. [6]

    Multi-parametric toolbox (MPT),

    M. Kvasnica, P. Grieder, M. Baoti ´c, and M. Morari, “Multi-parametric toolbox (MPT),” 2004, available: http://control.ee.ethz.ch/œmpt/

    work page 2004

  7. [7]

    Improved complexity analysis of branch and bound for hybrid MPC,

    D. Axehill and M. Morari, “Improved complexity analysis of branch and bound for hybrid MPC,” in 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 4216–4222

    work page 2010

  8. [8]

    A multiresolution approximation method for fast explicit model predictive control,

    S. Summers, C. N. Jones, J. Lygeros, and M. Morari, “A multiresolution approximation method for fast explicit model predictive control,” 15 IEEE Transactions on Automatic Control , vol. 56, no. 11, pp. 2530–2541, 2011

    work page 2011

Show all 47 references
  1. [9]

    A parametric branch and bound ap- proach to suboptimal explicit hybrid MPC,

    D. Axehill, T. Besselmann, D. M. Raimondo, and M. Morari, “A parametric branch and bound ap- proach to suboptimal explicit hybrid MPC,” Auto- matica, vol. 50, no. 1, pp. 240–246, 2014

    2014

  2. [10]

    Learning binary warm starts for multiparametric mixed-integer quadratic programming,

    D. Masti and A. Bemporad, “Learning binary warm starts for multiparametric mixed-integer quadratic programming,” in 18th European Control Confer- ence (ECC) . IEEE, 2019, pp. 1494–1499

    2019

  3. [11]

    Safe and near-optimal policy learning for model predic- tive control using primal-dual neural networks,

    X. Zhang, M. Bujarbaruah, and F. Borrelli, “Safe and near-optimal policy learning for model predic- tive control using primal-dual neural networks,” in 2019 American Control Conference (ACC) . IEEE, 2019, pp. 354–359

    2019

  4. [12]

    Near-optimal rapid MPC using neural net- works: A primal-dual policy learning framework,

    ——, “Near-optimal rapid MPC using neural net- works: A primal-dual policy learning framework,” IEEE Transactions on Control Systems Technology , vol. 29, no. 5, pp. 2102–2114, 2020

    2020

  5. [13]

    Fast non-parametric learning to accelerate mixed-integer program- ming for hybrid model predictive control,

    J.-J. Zhu and G. Martius, “Fast non-parametric learning to accelerate mixed-integer program- ming for hybrid model predictive control,” IF AC- PapersOnLine, vol. 53, no. 2, pp. 5239–5245, 2020

    2020

  6. [14]

    Learning mixed- integer convex optimization strategies for robot planning and control,

    A. Cauligi, P. Culbertson, B. Stellato, D. Bertsimas, M. Schwager, and M. Pavone, “Learning mixed- integer convex optimization strategies for robot planning and control,” in 59th IEEE Conference on Decision and Control (CDC) , 2020, pp. 1698– 1705

    2020

  7. [15]

    CoCo: Online mixed-integer control via supervised learning,

    A. Cauligi, P. Culbertson, E. Schmerling, M. Schwager, B. Stellato, and M. Pavone, “CoCo: Online mixed-integer control via supervised learning,” IEEE Robotics and Automation Letters , vol. 7, no. 2, pp. 1447–1454, 2021

    2021

  8. [16]

    Online mixed- integer optimization in milliseconds,

    D. Bertsimas and B. Stellato, “Online mixed- integer optimization in milliseconds,” INFORMS Journal on Computing , vol. 34, no. 4, pp. 2229– 2248, 2022

    2022

  9. [17]

    Learning disagreement regions with deep neural networks to reduce practical complex- ity of mixed-integer MPC,

    A. Chakrabarty, R. Quirynen, D. Romeres, and S. Di Cairano, “Learning disagreement regions with deep neural networks to reduce practical complex- ity of mixed-integer MPC,” in IEEE International Conference on Systems, Man, and Cybernetics (SMC), 2021, pp. 3238–3244

    2021

  10. [18]

    Learning for online mixed-integer model predic- tive control with parametric optimality certificates,

    L. Russo, S. H. Nair, L. Glielmo, and F. Borrelli, “Learning for online mixed-integer model predic- tive control with parametric optimality certificates,” IEEE Control Systems Letters , vol. 7, pp. 2215– 2220, 2023

    2023

  11. [19]

    A dual gradient pro- jection quadratic programming algorithm tailored for model predictive control,

    D. Axehill and A. Hansson, “A dual gradient pro- jection quadratic programming algorithm tailored for model predictive control,” in 47th IEEE Con- ference on Decision and Control , 2008, pp. 3057– 3064

    2008

  12. [20]

    Embedded mixed- integer quadratic optimization using accelerated dual gradient projection,

    V . V . Naik and A. Bemporad, “Embedded mixed- integer quadratic optimization using accelerated dual gradient projection,” IF AC-PapersOnLine, vol. 50, no. 1, pp. 10 723–10 728, 2017

    2017

  13. [21]

    A numerically stable solver for positive semidefinite quadratic programs based on nonnegative least squares,

    A. Bemporad, “A numerically stable solver for positive semidefinite quadratic programs based on nonnegative least squares,” IEEE Transactions on Automatic Control , vol. 63, no. 2, pp. 525–531, 2017

    2017

  14. [22]

    A numerically robust mixed-integer quadratic programming solver for embedded hybrid model predictive control,

    A. Bemporad and V . V . Naik, “A numerically robust mixed-integer quadratic programming solver for embedded hybrid model predictive control,” IF AC- PapersOnLine, vol. 51, no. 20, pp. 412–417, 2018

    2018

  15. [23]

    OSQP: An operator splitting solver for quadratic programs,

    B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP: An operator splitting solver for quadratic programs,” Mathematical Program- ming Computation , vol. 12, no. 4, pp. 637–672, 2020

    2020

  16. [24]

    Embedded mixed-integer quadratic optimization using the OSQP solver,

    B. Stellato, V . V . Naik, A. Bemporad, P. Goulart, and S. Boyd, “Embedded mixed-integer quadratic optimization using the OSQP solver,” in 2018 Eu- ropean Control Conference (ECC) . IEEE, 2018, pp. 1536–1541

    2018

  17. [25]

    Warm start of mixed- integer programs for model predictive control of hybrid systems,

    T. Marcucci and R. Tedrake, “Warm start of mixed- integer programs for model predictive control of hybrid systems,” IEEE Transactions on Automatic Control, vol. 66, no. 6, pp. 2433–2448, 2020

    2020

  18. [26]

    BnB-DAQP: A mixed-integer QP solver for embedded applica- tions,

    D. Arnstr ¨om and D. Axehill, “BnB-DAQP: A mixed-integer QP solver for embedded applica- tions,” IF AC-PapersOnLine, vol. 56, no. 2, pp. 7420–7427, 2023

    2023

  19. [27]

    A dual active-set solver for embedded quadratic pro- gramming using recursive LDL T updates,

    D. Arnstr ¨om, A. Bemporad, and D. Axehill, “A dual active-set solver for embedded quadratic pro- gramming using recursive LDL T updates,” IEEE Transactions on Automatic Control , vol. 67, no. 8, pp. 4362–4369, 2022

    2022

  20. [28]

    Early termination of convex QP solvers in mixed-integer programming for real-time decision making,

    J. Liang, S. Di Cairano, and R. Quirynen, “Early termination of convex QP solvers in mixed-integer programming for real-time decision making,” IEEE Control Systems Letters , vol. 5, no. 4, pp. 1417– 1422, 2020

    2020

  21. [29]

    FORCES: Fast optimization for real-time control on embedded systems,

    A. Domahidi, “FORCES: Fast optimization for real-time control on embedded systems,” 2012

    2012

  22. [30]

    Embedded optimization for mixed logical dynamical systems,

    D. Frick, A. Domahidi, and M. Morari, “Embedded optimization for mixed logical dynamical systems,” Computers & Chemical Engineering , vol. 72, pp. 21–33, 2015

    2015

  23. [31]

    A structure exploiting branch-and-bound algorithm for mixed-integer model predictive control,

    P. Hespanhol, R. Quirynen, and S. Di Cairano, “A structure exploiting branch-and-bound algorithm for mixed-integer model predictive control,” in 18th European Control Conference (ECC). IEEE, 2019, pp. 2763–2768

    2019

  24. [32]

    Tailored presolve techniques in branch-and-bound method for fast 16 mixed-integer optimal control applications,

    R. Quirynen and S. Di Cairano, “Tailored presolve techniques in branch-and-bound method for fast 16 mixed-integer optimal control applications,” Opti- mal Control Applications and Methods , vol. 44, no. 6, pp. 3139–3167, 2023

    2023

  25. [33]

    Convex relaxations for mixed integer predictive control,

    D. Axehill, L. Vandenberghe, and A. Hansson, “Convex relaxations for mixed integer predictive control,” Automatica, vol. 46, no. 9, pp. 1540–1545, 2010

    2010

  26. [34]

    Mixed-integer for- mulations for optimal control of piecewise-affine systems,

    T. Marcucci and R. Tedrake, “Mixed-integer for- mulations for optimal control of piecewise-affine systems,” in Proceedings of the 22nd ACM Interna- tional Conference on Hybrid Systems: Computation and Control, 2019, pp. 230–239

    2019

  27. [35]

    A more scalable mixed- integer encoding for metric temporal logic,

    V . Kurtz and H. Lin, “A more scalable mixed- integer encoding for metric temporal logic,” IEEE Control Systems Letters , vol. 6, pp. 1718–1723, 2021

    2021

  28. [36]

    Shortest paths in graphs of convex sets,

    T. Marcucci, J. Umenberger, P. Parrilo, and R. Tedrake, “Shortest paths in graphs of convex sets,” SIAM Journal on Optimization , vol. 34, no. 1, pp. 507–532, 2024

    2024

  29. [37]

    Piecewise affine modeling and constrained optimal control for a pneumatic artificial muscle,

    G. Andrikopoulos, G. Nikolakopoulos, I. Arvani- takis, and S. Manesis, “Piecewise affine modeling and constrained optimal control for a pneumatic artificial muscle,” IEEE Transactions on Industrial Electronics, vol. 61, no. 2, pp. 904–916, 2013

    2013

  30. [38]

    Feedback design for multi-contact push recovery via LMI approxima- tion of the piecewise-affine quadratic regulator,

    W. Han and R. Tedrake, “Feedback design for multi-contact push recovery via LMI approxima- tion of the piecewise-affine quadratic regulator,” in IEEE-RAS 17th International Conference on Humanoid Robotics (Humanoids) , 2017, pp. 842– 849

    2017

  31. [39]

    Footstep planning on uneven terrain with mixed-integer convex optimiza- tion,

    R. Deits and R. Tedrake, “Footstep planning on uneven terrain with mixed-integer convex optimiza- tion,” in 2014 IEEE-RAS International Conference on Humanoid Robots , 2014, pp. 279–286

    2014

  32. [40]

    Simultaneous contact, gait, and motion plan- ning for robust multilegged locomotion via mixed- integer convex optimization,

    B. Aceituno-Cabezas, C. Mastalli, H. Dai, M. Foc- chi, A. Radulescu, D. G. Caldwell, J. Cappelletto, J. C. Grieco, G. Fern ´andez-L´opez, and C. Sem- ini, “Simultaneous contact, gait, and motion plan- ning for robust multilegged locomotion via mixed- integer convex optimizatio...

    2017

  33. [41]

    Supermodularity and valid inequalities for quadratic optimization with indicators,

    A. Atamt ¨urk and A. G ´omez, “Supermodularity and valid inequalities for quadratic optimization with indicators,” Mathematical Programming, vol. 201, no. 1, pp. 295–338, 2023

    2023

  34. [42]

    2x2- convexifications for convex quadratic optimization with indicator variables,

    S. Han, A. G ´omez, and A. Atamt ¨urk, “2x2- convexifications for convex quadratic optimization with indicator variables,” Mathematical Program- ming, vol. 202, no. 1, pp. 95–134, 2023

    2023

  35. [43]

    On the convex hull of convex quadratic optimization problems with indicators,

    L. Wei, A. Atamt ¨urk, A. G ´omez, and S. K ¨uc ¸¨ukyavuz, “On the convex hull of convex quadratic optimization problems with indicators,” Mathematical Programming , vol. 204, no. 1, pp. 703–737, 2024

    2024

  36. [44]

    A strong conic quadratic reformulation for machine- job assignment with controllable processing times,

    M. S. Akt ¨urk, A. Atamt ¨urk, and S. G ¨urel, “A strong conic quadratic reformulation for machine- job assignment with controllable processing times,” Operations Research Letters , vol. 37, no. 3, pp. 187–191, 2009

    2009

  37. [45]

    Perspective reformu- lation and applications,

    O. G ¨unl¨uk and J. Linderoth, “Perspective reformu- lation and applications,” in Mixed Integer Nonlin- ear Programming. Springer, 2012, pp. 61–89

    2012

  38. [46]

    Drake: Model-based design and verification for robotics,

    R. Tedrake and the Drake Development Team, “Drake: Model-based design and verification for robotics,” 2019, available: https://drake.mit.edu

    2019

  39. [47]

    MPC-based energy management of a power-split hybrid electric vehicle,

    H. Borhan, A. Vahidi, A. M. Phillips, M. L. Kuang, I. V . Kolmanovsky, and S. Di Cairano, “MPC-based energy management of a power-split hybrid electric vehicle,” IEEE Transactions on Control Systems Technology, vol. 20, no. 3, pp. 593–603, 2011

    2011

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.