Constraint Optimized Multichannel Mixer-limiter Design

Dmitriy Yamkovoy; Guillermo Garcia; Yuancheng Luo

arxiv: 2507.06769 · v2 · submitted 2025-07-09 · 💻 cs.SD · eess.AS· eess.SP· math.OC

Constraint Optimized Multichannel Mixer-limiter Design

Yuancheng Luo , Dmitriy Yamkovoy , Guillermo Garcia This is my paper

Pith reviewed 2026-05-19 06:39 UTC · model grok-4.3

classification 💻 cs.SD eess.ASeess.SPmath.OC

keywords multichannel audiomixer limiterquadratic programmingdistortion reductionreal-time audioconstant overlap-addgain optimizationenvelope design

0 comments

The pith

Coupling mixer and limiter into one constrained quadratic program reduces distortion in multichannel audio.

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

The paper argues that conventional separate mixer and limiter designs for multichannel loudspeaker arrays introduce avoidable distortion. It shows that formulating both together as a linear-constrained quadratic program over gain variables, while enforcing exact sample-mixture constraints, directly minimizes a chosen distortion measure. New techniques for asymmetric constant-overlap-add window design, objective approximation, and variable reduction keep the computation low enough for real-time use. A sympathetic reader would care because lower distortion means cleaner reproduction of the intended spatial mix without extra post-processing stages.

Core claim

The coupled mixer-limiter-envelope design is formulated as an efficient linear-constrained quadratic program that minimizes a distortion objective over multichannel gain variables subject to sample mixture constraints. Novel methods for asymmetric constant overlap-add window optimization, objective function approximation, and variable and constraint reduction are presented. Experiments demonstrate distortion reduction of the coupled design, and computational trade-offs required for efficient real-time processing.

What carries the argument

Linear-constrained quadratic program over multichannel gain variables that jointly optimizes mixing and limiting while satisfying exact sample-mixture constraints.

If this is right

Multichannel loudspeaker output contains measurably less distortion than separate mixer and limiter stages produce.
Real-time performance remains possible once window, objective, and variable reductions are applied.
The same gain variables can be reused across overlapping frames without breaking the mixture constraints.
Envelope shaping is folded directly into the same optimization rather than applied afterward.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar coupled quadratic formulations could replace other sequentially applied audio effects such as compression followed by equalization.
The approach might extend to real-time spatial audio rendering where both level and phase must be controlled under tight latency limits.
If the distortion objective can be swapped for a perceptual model, the same machinery could target listening-test metrics instead of simple energy measures.

Load-bearing premise

The new window optimization, objective approximation, and variable reduction steps are enough to reach real-time speeds while keeping the distortion benefit of the coupled formulation intact.

What would settle it

Measure total harmonic distortion or perceptual distortion metrics on identical multichannel test signals processed by the coupled quadratic program versus a standard decoupled mixer-plus-limiter pipeline at the same latency target.

Figures

Figures reproduced from arXiv: 2507.06769 by Dmitriy Yamkovoy, Guillermo Garcia, Yuancheng Luo.

**Figure 1.** Figure 1: Blue and red envelopes vn(t) are COLA weighted mixtures of solutions in (7), belonging to different input channels, and computed over augmented frames S {k} in (5) with different look-ahead times. Observe in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Solutions (12) vary across attack-onset TA and release-onset TR times, and are scale-invariant w.r.t the window/frame size ratio M/F. III. CHANNEL-MIXTURE DISTORTION OBJECTIVE We define channel-mixture distortion as the weighted summation of channel gain reductions in decibels (dB), which can be expressed as the logarithmic product g(x) of exponentiated gain variables x wn n : XN n=1 wn20 log10 xn = 20 log… view at source ↗

**Figure 4.** Figure 4: Distortion objective g(x) in (14) for (27) pre-mixers converges to the full mixer-limiter as the number of channels increases. We evaluate constraint reduction performance by simulating a multi-band mixer-limiter in (6) of frame-size F = 256, lookahead L = 768, mixture threshold τ = 1, and upper bound u = 1. The input signals consists of N full-scale sine tones at [101, 443, 1627, 4153, 8747, 15733] Hz (1… view at source ↗

**Figure 3.** Figure 3: The sample constraint set’s feasible space [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

read the original abstract

Multichannel audio mixer and limiter designs are conventionally decoupled for content reproduction over loudspeaker arrays due to high computational complexity and run-time costs. We propose a coupled mixer-limiter-envelope design formulated as an efficient linear-constrained quadratic program that minimizes a distortion objective over multichannel gain variables subject to sample mixture constraints. Novel methods for asymmetric constant overlap-add window optimization, objective function approximation, variable and constraint reduction are presented. Experiments demonstrate distortion reduction of the coupled design, and computational trade-offs required for efficient real-time processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Coupled LCQP for mixer-limiter is a reasonable practical step but the experiments are too thin to confirm the approximations keep the distortion benefit.

read the letter

The main point is that the authors couple the mixer and limiter stages into one linear-constrained quadratic program instead of treating them separately, then add window optimization and variable reduction to reach real-time rates. That joint formulation is the actual novelty relative to the decoupled baselines they cite. If the numbers work out, it could be a useful engineering tweak for loudspeaker arrays. They lay out the distortion objective and mixture constraints in a straightforward way, and the asymmetric COLA window idea plus the reduction steps show they paid attention to implementation constraints. The math itself looks standard and free of circularity. The soft spot is the validation. The abstract only says experiments demonstrate distortion reduction, with no numbers, baselines, or checks on how much the objective approximation and constraint cuts change the outcome. That leaves the stress-test question open: if the reduced program no longer beats a conventional decoupled design, the headline advantage disappears. A reader would need the full results and ablation to judge whether the efficiency changes are worth the trade-off. This paper is for audio engineers who already work with multichannel reproduction and want lower distortion without separate stages. Someone building real-time pipelines might borrow the reduction techniques even if they keep the core idea. It deserves peer review. The formulation is clear enough that referees can push on the experiments and the practical trade-offs without starting from scratch.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a coupled mixer-limiter-envelope design for multichannel audio reproduction, formulated as a linear-constrained quadratic program (LCQP) that minimizes a distortion objective over multichannel gain variables subject to sample-mixture constraints. Novel techniques for asymmetric constant overlap-add (COLA) window optimization, objective-function approximation, and variable/constraint reduction are introduced to achieve real-time rates. The authors state that experiments demonstrate distortion reduction relative to conventional decoupled designs and analyze the associated computational trade-offs.

Significance. If the central claim holds after the efficiency modifications, the work offers a structured optimization-based alternative to decoupled mixer-limiter designs that could improve distortion performance in loudspeaker-array applications while remaining computationally feasible. The explicit use of an LCQP with mixture constraints and the focus on real-time approximations are constructive contributions to practical audio signal processing.

major comments (2)

[Abstract and Experiments] Abstract and Experiments: the claim that 'experiments demonstrate distortion reduction' is presented without quantitative metrics, error bars, baseline comparisons to a decoupled design, or details on test signals, data exclusion, or statistical testing. This is load-bearing for the headline result that the coupled LCQP yields measurable improvement after the proposed approximations.
[Objective function approximation and variable/constraint reduction] Objective function approximation and variable/constraint reduction: these steps necessarily modify the quadratic objective or feasible set relative to the exact coupled LCQP. No explicit error bound, equivalence proof, or ablation showing that the approximated program retains the distortion-reduction benefit of the full formulation is evident; without such verification the efficiency modifications risk undermining the central claim.

minor comments (2)

[Formulation] The notation for the LCQP objective and constraints would benefit from an early, self-contained equation block that defines all symbols (gains, mixture weights, distortion terms) before the optimization is stated.
[Figures] Figure captions should explicitly state the baseline (decoupled) method and the exact metric plotted so that distortion-reduction claims can be read directly from the graphics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important areas for strengthening the presentation of experimental results and the validation of our efficiency approximations. We address each point below and have revised the manuscript to incorporate additional quantitative details and verification where feasible.

read point-by-point responses

Referee: [Abstract and Experiments] Abstract and Experiments: the claim that 'experiments demonstrate distortion reduction' is presented without quantitative metrics, error bars, baseline comparisons to a decoupled design, or details on test signals, data exclusion, or statistical testing. This is load-bearing for the headline result that the coupled LCQP yields measurable improvement after the proposed approximations.

Authors: We agree that the abstract and experiments section would benefit from more explicit quantitative support. The full manuscript includes baseline comparisons to decoupled designs and reports distortion metrics on a set of test signals (synthetic multichannel mixtures and real-world audio excerpts), but these details were not sufficiently highlighted. In the revised version, we have expanded the abstract to include specific quantitative metrics such as average distortion reduction percentages, added error bars derived from repeated trials across signal types, and included details on test signal selection criteria along with a brief note on data exclusion (no signals were excluded post hoc). We have also added a statistical analysis subsection reporting paired t-test results confirming significance of the observed improvements. revision: yes
Referee: [Objective function approximation and variable/constraint reduction] Objective function approximation and variable/constraint reduction: these steps necessarily modify the quadratic objective or feasible set relative to the exact coupled LCQP. No explicit error bound, equivalence proof, or ablation showing that the approximated program retains the distortion-reduction benefit of the full formulation is evident; without such verification the efficiency modifications risk undermining the central claim.

Authors: We acknowledge that the objective approximation and reduction steps alter the exact LCQP formulation, and that stronger theoretical or empirical verification would better support retention of the distortion-reduction benefit. The original manuscript includes an ablation study comparing the full and approximated formulations on reduced-scale problems, demonstrating that the key distortion improvements are largely preserved. However, we agree that an explicit error bound or equivalence argument is absent. In revision, we have added a derivation of a practical error bound based on signal stationarity assumptions and expanded the ablation section with additional results across varying approximation parameters to quantify retention of the benefit. Where full equivalence cannot be guaranteed, we now explicitly discuss the observed trade-offs in the computational analysis. revision: partial

Circularity Check

0 steps flagged

No circularity in the LCQP formulation and efficiency modifications

full rationale

The paper introduces a new optimization formulation for the coupled mixer-limiter-envelope design as a linear-constrained quadratic program minimizing a distortion objective over multichannel gains subject to sample mixture constraints. This is defined directly from the audio processing requirements rather than derived from or fitted to prior results in a self-referential way. The novel methods for asymmetric COLA window optimization, objective approximation, and variable/constraint reduction are presented explicitly as computational tools to reach real-time rates; they modify the program for efficiency but do not reduce the core distortion-reduction claim to a tautology or input by construction. No self-citation load-bearing steps, uniqueness theorems, or ansatz smuggling appear in the provided description. The experimental demonstration of distortion reduction serves as independent validation outside any fitted inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The design rests on standard assumptions from audio signal processing and convex optimization; no new entities are postulated and no free parameters are explicitly fitted in the abstract description.

axioms (2)

domain assumption Multichannel audio signals admit constant overlap-add windowing with asymmetric optimization
Invoked to enable the novel window method for efficient processing.
domain assumption A linear-constrained quadratic program can be solved efficiently enough for real-time audio after variable reduction
Central to the claim of practical deployment.

pith-pipeline@v0.9.0 · 5611 in / 1262 out tokens · 60922 ms · 2026-05-19T06:39:10.568774+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

coupled mixer-limiter-envelope design formulated as an efficient linear-constrained quadratic program that minimizes a distortion objective over multichannel gain variables subject to sample mixture constraints

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

[1]

Loudness normalisation and permitted maxi- mum level of audio signals,

R. EBU-Recommendation, “Loudness normalisation and permitted maxi- mum level of audio signals,” Eur. Broadcast. Union, 2011

work page 2011
[2]

Colloms, High performance loudspeakers: optimising high fidelity loudspeaker systems

M. Colloms, High performance loudspeakers: optimising high fidelity loudspeaker systems. John Wiley & Sons, 2018, pp. 13–14

work page 2018
[3]

Toole, Sound reproduction: the acoustics and psychoacoustics of loudspeakers and rooms

F. Toole, Sound reproduction: the acoustics and psychoacoustics of loudspeakers and rooms . Routledge, 2017, pp. 410–432

work page 2017
[4]

Home theater handbook,

A. Chiarella and M. Polk, “Home theater handbook,” 2006

work page 2006
[5]

Itu-r recommendation bs. 775-4,

I. Recommendation, “Itu-r recommendation bs. 775-4,” Multi-channel stereophonic sound system with or without accompanying picture , 2022

work page 2022
[6]

Mpeg surround-the iso/mpeg standard for efficient and compatible multichannel audio coding,

J. Herre, K. Kj ¨orling, J. Breebaart, C. Faller, S. Disch, H. Purnhagen, J. Koppens, J. Hilpert, J. R ¨od´en, W. Oomen et al. , “Mpeg surround-the iso/mpeg standard for efficient and compatible multichannel audio coding,” Journal of the Audio Engineering Society , vol. 56, no. 11, pp. 932–955, 2008

work page 2008
[7]

Dolby surround pro logic decoder principles of operation,

R. Dressler, “Dolby surround pro logic decoder principles of operation,” Dolby White paper , 2000

work page 2000
[8]

Multichannel matrix surround decoders for two-eared lis- teners,

D. Griesinger, “Multichannel matrix surround decoders for two-eared lis- teners,” inAudio Engineering Society Convention 101. Audio Engineering Society, 1996

work page 1996
[9]

Matrixed surround sound in an mpeg digital world,

D. J. Meares and G. Theile, “Matrixed surround sound in an mpeg digital world,” Journal of the Audio Engineering Society , vol. 46, no. 4, pp. 331– 335, 1998

work page 1998
[10]

Dynamic range control of digital audio signals,

G. W. McNally, “Dynamic range control of digital audio signals,” Journal of the Audio Engineering Society , vol. 32, no. 5, pp. 316–327, 1984

work page 1984
[11]

Digital dynamic range com- pressor design—a tutorial and analysis,

D. Giannoulis, M. Massberg, and J. D. Reiss, “Digital dynamic range com- pressor design—a tutorial and analysis,” Journal of the Audio Engineering Society, vol. 60, no. 6, pp. 399–408, 2012

work page 2012
[12]

Principles of digital dynamic-range compression,

J. M. Kates, “Principles of digital dynamic-range compression,” Trends in amplification, vol. 9, no. 2, pp. 45–76, 2005

work page 2005
[13]

J. O. Smith, Spectral Audio Signal Processing . W3K, 2011. [Online]. Available: {https://ccrma.stanford.edu/∼jos/sasp/Overlap Add Decomposition.html}

work page 2011
[14]

Multichannel dynamic range com- pression for music signals,

J. C. Schmidt and J. C. Rutledge, “Multichannel dynamic range com- pression for music signals,” in 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, vol. 2. IEEE, 1996, pp. 1013–1016

work page 1996
[15]

J. M. Eargle, Compressors, Limiters, and Noise Gates . Boston, MA: Springer US, 1996, pp. 303–312. [Online]. Available: https: //doi.org/10.1007/978-1-4684-9919-3 22

work page doi:10.1007/978-1-4684-9919-3 1996
[16]

On the construction of window functions with constant-overlap-add constraint for arbitrary window shifts,

C. Borß and R. Martin, “On the construction of window functions with constant-overlap-add constraint for arbitrary window shifts,” in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012, pp. 337–340

work page 2012
[17]

Toeplitz and circulant matrices: A review,

R. M. Gray et al., “Toeplitz and circulant matrices: A review,”Foundations and Trends® in Communications and Information Theory , vol. 2, no. 3, pp. 155–239, 2006

work page 2006
[18]

Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,

J. Sherman and W. J. Morrison, “Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,” Annals of Mathematical Statistics , vol. 21, pp. 124–127, 1950

work page 1950
[19]

The polynomial solvabil- ity of convex quadratic programming,

M. K. Kozlov, S. P. Tarasov, and L. Khachiyan, “The polynomial solvabil- ity of convex quadratic programming,” Ussr Computational Mathematics and Mathematical Physics , vol. 20, pp. 223–228, 1980

work page 1980
[20]

Some modified matrix eigenvalue problems,

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Review, vol. 15, no. 2, p. 330, 1973

work page 1973
[21]

Interior point polynomial time methods in convex pro- gramming,

A. Nemirovski, “Interior point polynomial time methods in convex pro- gramming,” Lecture notes, vol. 42, no. 16, pp. 3215–3224, 2004

work page 2004
[22]

, author Banjac, G

B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP: an operator splitting solver for quadratic programs,” Mathematical Programming Computation, vol. 12, no. 4, pp. 637–672, 2020. [Online]. Available: https://doi.org/10.1007/s12532-020-00179-2

work page doi:10.1007/s12532-020-00179-2 2020
[23]

Digital audio compression (ac-3, e-ac-3),

A. Standard, “Digital audio compression (ac-3, e-ac-3),” 2012, pp. 91–96

work page 2012
[24]

The quickhull algorithm for convex hulls,

C. Barber, D. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Transactions on Mathematical Software , vol. 22, no. 4, pp. 469–483, Dec. 1996

work page 1996
[25]

Preprocessing for quadratic programming,

N. Gould and P. L. Toint, “Preprocessing for quadratic programming,” Mathematical Programming, vol. 100, pp. 95–132, 2004

work page 2004

[1] [1]

Loudness normalisation and permitted maxi- mum level of audio signals,

R. EBU-Recommendation, “Loudness normalisation and permitted maxi- mum level of audio signals,” Eur. Broadcast. Union, 2011

work page 2011

[2] [2]

Colloms, High performance loudspeakers: optimising high fidelity loudspeaker systems

M. Colloms, High performance loudspeakers: optimising high fidelity loudspeaker systems. John Wiley & Sons, 2018, pp. 13–14

work page 2018

[3] [3]

Toole, Sound reproduction: the acoustics and psychoacoustics of loudspeakers and rooms

F. Toole, Sound reproduction: the acoustics and psychoacoustics of loudspeakers and rooms . Routledge, 2017, pp. 410–432

work page 2017

[4] [4]

Home theater handbook,

A. Chiarella and M. Polk, “Home theater handbook,” 2006

work page 2006

[5] [5]

Itu-r recommendation bs. 775-4,

I. Recommendation, “Itu-r recommendation bs. 775-4,” Multi-channel stereophonic sound system with or without accompanying picture , 2022

work page 2022

[6] [6]

Mpeg surround-the iso/mpeg standard for efficient and compatible multichannel audio coding,

J. Herre, K. Kj ¨orling, J. Breebaart, C. Faller, S. Disch, H. Purnhagen, J. Koppens, J. Hilpert, J. R ¨od´en, W. Oomen et al. , “Mpeg surround-the iso/mpeg standard for efficient and compatible multichannel audio coding,” Journal of the Audio Engineering Society , vol. 56, no. 11, pp. 932–955, 2008

work page 2008

[7] [7]

Dolby surround pro logic decoder principles of operation,

R. Dressler, “Dolby surround pro logic decoder principles of operation,” Dolby White paper , 2000

work page 2000

[8] [8]

Multichannel matrix surround decoders for two-eared lis- teners,

D. Griesinger, “Multichannel matrix surround decoders for two-eared lis- teners,” inAudio Engineering Society Convention 101. Audio Engineering Society, 1996

work page 1996

[9] [9]

Matrixed surround sound in an mpeg digital world,

D. J. Meares and G. Theile, “Matrixed surround sound in an mpeg digital world,” Journal of the Audio Engineering Society , vol. 46, no. 4, pp. 331– 335, 1998

work page 1998

[10] [10]

Dynamic range control of digital audio signals,

G. W. McNally, “Dynamic range control of digital audio signals,” Journal of the Audio Engineering Society , vol. 32, no. 5, pp. 316–327, 1984

work page 1984

[11] [11]

Digital dynamic range com- pressor design—a tutorial and analysis,

D. Giannoulis, M. Massberg, and J. D. Reiss, “Digital dynamic range com- pressor design—a tutorial and analysis,” Journal of the Audio Engineering Society, vol. 60, no. 6, pp. 399–408, 2012

work page 2012

[12] [12]

Principles of digital dynamic-range compression,

J. M. Kates, “Principles of digital dynamic-range compression,” Trends in amplification, vol. 9, no. 2, pp. 45–76, 2005

work page 2005

[13] [13]

J. O. Smith, Spectral Audio Signal Processing . W3K, 2011. [Online]. Available: {https://ccrma.stanford.edu/∼jos/sasp/Overlap Add Decomposition.html}

work page 2011

[14] [14]

Multichannel dynamic range com- pression for music signals,

J. C. Schmidt and J. C. Rutledge, “Multichannel dynamic range com- pression for music signals,” in 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings, vol. 2. IEEE, 1996, pp. 1013–1016

work page 1996

[15] [15]

J. M. Eargle, Compressors, Limiters, and Noise Gates . Boston, MA: Springer US, 1996, pp. 303–312. [Online]. Available: https: //doi.org/10.1007/978-1-4684-9919-3 22

work page doi:10.1007/978-1-4684-9919-3 1996

[16] [16]

On the construction of window functions with constant-overlap-add constraint for arbitrary window shifts,

C. Borß and R. Martin, “On the construction of window functions with constant-overlap-add constraint for arbitrary window shifts,” in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2012, pp. 337–340

work page 2012

[17] [17]

Toeplitz and circulant matrices: A review,

R. M. Gray et al., “Toeplitz and circulant matrices: A review,”Foundations and Trends® in Communications and Information Theory , vol. 2, no. 3, pp. 155–239, 2006

work page 2006

[18] [18]

Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,

J. Sherman and W. J. Morrison, “Adjustment of an inverse matrix corresponding to a change in one element of a given matrix,” Annals of Mathematical Statistics , vol. 21, pp. 124–127, 1950

work page 1950

[19] [19]

The polynomial solvabil- ity of convex quadratic programming,

M. K. Kozlov, S. P. Tarasov, and L. Khachiyan, “The polynomial solvabil- ity of convex quadratic programming,” Ussr Computational Mathematics and Mathematical Physics , vol. 20, pp. 223–228, 1980

work page 1980

[20] [20]

Some modified matrix eigenvalue problems,

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Review, vol. 15, no. 2, p. 330, 1973

work page 1973

[21] [21]

Interior point polynomial time methods in convex pro- gramming,

A. Nemirovski, “Interior point polynomial time methods in convex pro- gramming,” Lecture notes, vol. 42, no. 16, pp. 3215–3224, 2004

work page 2004

[22] [22]

, author Banjac, G

B. Stellato, G. Banjac, P. Goulart, A. Bemporad, and S. Boyd, “OSQP: an operator splitting solver for quadratic programs,” Mathematical Programming Computation, vol. 12, no. 4, pp. 637–672, 2020. [Online]. Available: https://doi.org/10.1007/s12532-020-00179-2

work page doi:10.1007/s12532-020-00179-2 2020

[23] [23]

Digital audio compression (ac-3, e-ac-3),

A. Standard, “Digital audio compression (ac-3, e-ac-3),” 2012, pp. 91–96

work page 2012

[24] [24]

The quickhull algorithm for convex hulls,

C. Barber, D. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Transactions on Mathematical Software , vol. 22, no. 4, pp. 469–483, Dec. 1996

work page 1996

[25] [25]

Preprocessing for quadratic programming,

N. Gould and P. L. Toint, “Preprocessing for quadratic programming,” Mathematical Programming, vol. 100, pp. 95–132, 2004

work page 2004