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Convexification of Multi-period Quadratic Programs with Indicators

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Pith's one-line read The paper proves that after eliminating state variables via linear dynamics, the mixed-integer epigraph of a multi-period quadratic program with indicator variables has an exact closed convex hull described by one positive semidefinite…

desk verdict Solid theory paper: exact convex hull for block-factorizable MIQPs, with correct math and a useful shortest-path algorithm; the computational section oversells by omitting the existing DP baseline. read the letter →

arxiv 2412.17178 v1 pith:QJERSUWD submitted 2024-12-22 math.OC

Jisun Lee , Andrés Gómez , Alper Atamtürk This is my paper · ORCID
classification math.OC MSC 90C1190C2090C2590C3515A09
keywords mixed-integerquadraticprogrammingindicatorvariablesconvexhullblock-factorizablematricessecond-orderconeshortestpathreformulationmulti-periodoptimizationhybridsystemcontrol
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 sets out to prove that a broad family of multi-period quadratic programs with fixed-charge indicators, where a state is driven by linear dynamics and an action either is zero or triggers a fixed cost, can be convexified exactly. The route is to eliminate the state variables: the resulting quadratic cost matrix has a special factorizable, or block-factorizable, structure. The paper derives a closed form for the inverse of every principal submatrix of such a matrix as a short sum of low-rank matrices indexed by the arcs of a directed acyclic graph. Combining that with a general hull characterization of quadratic sets with indicators yields an exact description of the mixed-integer epigraph in an extended space, using one positive semidefinite condition and O($n^{2}$) linear path constraints, and the description is shown to be second-order cone representable. If correct, the paper turns a class of problems that branch-and-bound solves poorly into one with a strong exact relaxation that commercial conic solvers handle quickly, and, without side constraints, into a shortest-path problem solvable in polynomial time.

What carries the argument

The engine is a rank decomposition of $Q^{{-1}}$, and of every principal-submatrix inverse, into O($n^{2}$) low-rank pieces indexed by DAG arcs. For scalar factorizable Q, each piece is Λ[i→j] = (u_j/(u_i(u_j v_i - u_i v_j))) (e_i - (u_i/u_j)e_j)(e_i - (u_i/u_j)e_j)^T, and the inverse of every principal submatrix uses only the pieces whose indices are consecutive in the active set; hence the entire family of inverse points is encoded by the path polytope (14). The block version replaces scalars by d×d matrices, with Λ[i→j] = (E_i - E_j $U_j^{{-T}}$ U_i^T)(U_i V_i^T - U_i $U_j^{{-1}}$ V_j U_i^T)^{-1}(E_i - E_j $U_j^{{-T}}$ U_i^T)^T. The positive semidefinite block matrix [W x; x^T τ] enforces τ ≥ x^T Q x whenever W is the inverse of Q[S], and because W is a nonnegative combination of positive semidefinite pieces, the whole hull is SOCP-representable.

What would settle it

Generate a random side-constraint-free instance of the original form with n=10, nonsingular A_i and positive definite P_i, compute the value returned by the shortest-path algorithm of Proposition 9, and compare it with exhaustive enumeration of all $2^{10}$ indicator sets, each a small convex quadratic program; any gap between the two optimal values would refute the exactness claim.

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Extended reading notes

Core claim

For a positive definite block-factorizable matrix Q = U • V, the central result (Theorem 8) is that the closed convex hull of X^B_Q = {(x,z,τ): τ ≥ x^T Q x, x[i](1-z_i)=0, z∈{0,1}^n} is exactly the set of (x,z,τ) for which there are nonnegative arc weights w and a matrix W satisfying the path-conservation equations (14a), the coupling z_ℓ = sum of incoming arc weights (14b), the identity W = Σ Λ[i→j] w_{ij} (18), and the positive semidefinite condition [[W,x],[x^T,τ]] ⪰ 0. The matrices Λ[i→j] are fixed rank-d matrices built from Q's factors; they are precisely the summands in the closed-form inverse of the principal submatrix Q[S] for the set S of active periods. Thus the hard part of the general hull theorem, the convex hull of inverse-submatrix points, becomes the vertex-arc incidence polytope of paths in a DAG, and the overall hull is polynomially sized. From this description the paper derives a tight SOCP formulation (Proposition 10) and, for the side-constraint-free version, a shortest-path algorithm (Proposition 9).

Load-bearing premise

The construction depends on the projected cost matrix Q being block-factorizable, which for the original multi-period problem holds when each transition matrix A_i is nonsingular, each weight matrix P_i is positive definite, and the state dimension does not change between periods; it also imports the general hull characterization of quadratic sets with indicators as a black box.

Editorial extensions

If this is right

  • For problems of the studied form with additional side constraints, the exact hull of the quadratic-plus-indicator part gives a SOCP relaxation with O(n^2) conic constraints that can be embedded in branch-and-bound; the paper's experiments show nearly zero root gaps and only a handful of branch-and-bound nodes.
  • In the absence of side constraints, the problem is solvable as a shortest path on a directed acyclic graph in O(n^2 · π(d)) time, where π(d) is the cost of a d×d matrix inversion and multiplication, and in O(n^2) time in the scalar case.
  • The rank-d decomposition of principal-submatrix inverses gives a finite catalog of O(n^2) precomputable matrices, so the convex hull description can be written down without solving any auxiliary optimization problem.
  • The settings covered include calcium-imaging deconvolution and hybrid-vehicle path following; in both case studies the MISOCP model solves instances that time out as plain MIQPs, although the conic model occasionally suffers numerical errors.
  • Because the description exactly convexifies the quadratic-plus-indicator part, adding any additional constraints from the feasible region's side constraints preserves the tightness of that core relaxation; the paper's constrained experiments confirm the resulting relaxation stays strong.

Reading between the lines

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

  • An extension the paper does not pursue: the same arc-based inverse decomposition may apply to banded or tree-structured cost matrices beyond the tridiagonal-inverse class, potentially yielding compact hulls for graph-structured multi-period problems.
  • We infer from the numerical results that the practical bottleneck has shifted from relaxation quality to numerical stability of the conic solver; the reported infeasible and suboptimal cases suggest that scaled or alternative SOCP representations could be as valuable as the hull itself.
  • The shortest-path reformulation has a natural linear-programming dual with arc potentials, and that dual may yield a combinatorial separation routine for adding cuts to problems with side constraints; this is implicit in the path polytope but not developed in the paper.
  • For singular transition matrices or time-varying state dimensions, block-factorizability fails; one could test whether a perturbed or limiting formulation recovers the hull approximately, but the exact statement as written stops at the nonsingular case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper studies a multi-period mixed-integer quadratic program with linear state dynamics and indicator variables on the controls. It projects out the state variables and observes that the resulting quadratic cost matrix is (block-)factorizable when the transition matrices are nonsingular and the weighting matrices are positive definite. For this class it derives a closed-form expression for the inverses of all principal (block-)submatrices as a sum of low-rank matrices indexed by consecutive pairs in the support. Using the Wei--Atamturk--Gomez--Kucukyavuz hull characterization, it then gives an exact closed convex hull of the mixed-integer epigraph in an extended space, expressed by a positive semidefinite constraint and an O(n^2)-sized network-flow path polytope. It further shows that the hull is SOCP-representable and that, without side constraints, the problem reduces to a shortest path problem on a DAG solvable in O(n^2 * pi(d)) time. Two computational case studies, calcium-imaging deconvolution and a hybrid electric vehicle path-following problem, demonstrate the practical impact and also report numerical robustness issues of the conic formulations.

Significance. If the results are correct, this is a substantial contribution to the convexification literature for MIQPs with indicators. The paper identifies a structured class -- (block-)factorizable cost matrices -- for which the exact convex hull, which for general positive definite Q requires an exponential-size polytope in the Wei et al. representation, collapses to a compact polynomial-size description based on a totally unimodular network polytope. The inverse formulas for block-factorizable matrices and their principal submatrices are new and are used in a clean way. The shortest path algorithm for the side-constraint-free case is a genuine polynomial-time result for a nontrivial class of multi-period MIQPs. The paper is also honest about limitations: it explicitly states the nonsingularity/positive-definiteness assumptions under which the block structure exists, it clearly notes where the exact hull does not apply (e.g., the nonnegativity constraint in calcium deconvolution), and it reports numerical failures of the SOCP models.

minor comments (6)
  1. [Section 2, Proposition 2] In the displayed representation, the variable tau is not explicitly constrained to be nonnegative and the dimension of z is not stated; please make clear that z in R^n and tau in R_+, or note that these are implied by the PSD block and the definition of P^B_T.
  2. [Section 3.2, Proposition 5] The proof that all extreme points of (14) are integral is compressed: after projecting out z and W, one obtains a network-flow polytope in w only, but the argument should explicitly state that the linear map from w to (z,W) is injective, so integrality of w transfers to the original extreme points; as written, the claim that an integral optimal solution exists does not by itself rule out nonintegral extreme points of the extended polytope.
  3. [Section 4.1, Proposition 9] The reduction from (20) to (21) is terse at the point where the paper states that z* is binary in extreme optimal solutions; since (20) is an unbounded SDP set, arguing directly that the value function over P^B_Q is linear and therefore minimized at an integral extreme point of the bounded polytope P^B_Q would be more precise and would avoid relying on extreme points of the unbounded set.
  4. [Section 4.2, Proposition 10] Variables tau_{0j} and h_{0j} are declared for 0 <= i < j <= n+1 but are never used in the conic constraints, which begin at i=1; please restrict the index range or explicitly state that the 0-index variables are unused.
  5. [Notation, Section 1] The convention '0f(x/0)' should be written as '0 * f(x/0)' or with a space, since as printed it can be misread as a function name.
  6. [Appendix C, proof of Proposition 7] The telescoping step in the computation of Inverse of Xi uses symmetry of U_i V_i^T for all i, not only for i=n as the final sentence suggests; since this symmetry is implied by Assumption 2, it would be helpful to state it explicitly before the computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper specializes an independent convex-hull theorem to block-factorizable matrices via an explicit low-rank inverse decomposition.

full rationale

The derivation chain is self-contained relative to its stated assumptions. The paper first projects out the state variables under the explicit scope conditions of nonsingular A_i and positive definite P_i, obtaining a block-factorizable cost matrix Q. The convex-hull description in Theorem 8 combines Proposition 1, imported from Wei et al. [56], with a new, explicit description of the polyhedron P^B_Q. Although [56] shares two co-authors with the present paper, Proposition 1 is a general, parameter-free characterization for arbitrary positive definite T and binary indicator sets; it does not assume factorizability or the target block structure, so it qualifies as independent support under the stated rules. The genuinely new content is the closed-form inverse representation: Corollary 1 and Corollary 2 follow from Meurant's standard tridiagonal inverse formula and from Proposition 7, whose proof is given from scratch in Appendix C. Observation 1 and Observation 2 establish that only O(n^2) rank-one/rank-d matrices are needed to represent every principal-submatrix inverse, and the path constraints (14a)-(14d) together with (18) are shown in Propositions 4 and 5 to describe exactly the convex hull of the relevant extreme points, with integrality following from total unimodularity of the network-flow matrix. Proposition 10 then applies the standard Nesterov-Nemirovskii conic reformulation, and Proposition 9 reduces the unconstrained case to a shortest path problem by substituting the same inverse decomposition into the objective. No parameter is fitted to data and then reported as a prediction; no result is assumed in the form it is derived; and the singular-A_i limitation is an explicit assumption of the problem class, not a hidden premise. Therefore the central claim is not circular.

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

No parameters are fitted to data anywhere in the paper; the only inputs are the problem data and the structural assumptions on Q. The axioms listed are the imported theorems (Wei et al., Meurant, Nesterov-Nemirovskii) and the domain assumptions (nonsingular A_i, positive definite P_i) that the central claim rests on. The paper introduces no new physical or mathematical entities; the path variables w are internal to the extended formulation.

assumptions (6)
  • standard math Wei et al. (2024) convex hull characterization for general PSD T with indicators (Proposition 1)
    Black-box theorem used to reduce the convex hull to the polyhedron P_T. It is a general parameter-free derivation, not specific to factorizable matrices.
  • standard math Inverses of factorizable matrices are tridiagonal with the Meurant rank-one decomposition (Proposition 3, from [47])
    Basis for the rank-one or rank-d decomposition of the inverse; cited from Meurant's book.
  • domain assumption Q is positive definite if and only if Assumptions 1 and 2 hold (Propositions 11 and 12)
    The paper considers positive definite factorizable matrices; the assumptions are proven equivalent.
  • domain assumption Nonsingular A_i and positive definite P_i in the original multi-period problem (1)
    Ensures the projected cost matrix is positive definite and block-factorizable; stated in the problem definition.
  • standard math Nesterov-Nemirovskii conic reformulation for sums of rank-one PSD matrices (Proposition 10)
    Used to convert the SDP in the convex hull into SOCP constraints; cited from [48].
  • standard math Totally unimodular network matrices give integral extreme points of the path polytope (Proposition 5)
    The proof of integrality of the extended formulation (14) relies on total unimodularity of node-arc incidence matrices.

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Cite this review

Pith. "Pith review of Convexification of Multi-period Quadratic Programs with Indicators." pith.science (2026). https://pith.science/paper/QJERSUWD

@misc{pith2026241217178,
  author       = {Pith},
  title        = {Pith review of: Convexification of Multi-period Quadratic Programs with Indicators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QJERSUWD}},
  note         = {Machine review of arXiv:2412.17178}
}
abstract

We study a multi-period convex quadratic optimization problem, where the state evolves dynamically as an affine function of the state, control, and indicator variables in each period. We begin by projecting out the state variables using linear dynamics, resulting in a mixed-integer quadratic optimization problem with a (block-) factorizable cost matrix. We discuss the properties of these matrices and derive a closed-form expression for their inverses. Employing this expression, we construct a closed convex hull representation of the epigraph of the quadratic cost over the feasible region in an extended space. Subsequently, we establish a tight second-order cone programming formulation with $\mathcal{O}(n^2)$ conic constraints. We further propose a polynomial-time algorithm based on a reformulation of the problem as a shortest path problem on a directed acyclic graph. To illustrate the applicability of our results across diverse domains, we present case studies in statistical learning and hybrid system control.

Figures

Figures reproduced from arXiv: 2412.17178 by the authors.

Figure 1
Figure 1. Directed acyclic graph G: arc (i, j) represents rank-one matrix Λi→j . Proposition 4. Given any z ∈ {0,1} n , the unique solution (w,W ) to (14) satisfies W = Qˆ −1 S , where S = {ℓ ∈ [n] : zℓ = 1}. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Computational time of MIQP, SOCP, and SPP on relaxed calcium deconvolution. [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

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Reference graph

Works this paper leans on

60 extracted references · 57 canonical work pages

  1. [1]

    Improved algorithms for economic lot size problems

    Alok Aggarwal and James K Park. Improved algorithms for economic lot size problems. Operations Research, 41(3):549–571, 1993

    work page 1993

  2. [2]

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

    M Selim Aktürk, Alper Atamtürk, and Sinan Gürel. A strong conic quadratic reformulation for machine-job assignment with controllable processing times. Operations Research Letters, 37(3):187– 191, 2009

    work page 2009

  3. [3]

    Quadratic optimization with switching variables: the convex hull for n = 2

    Kurt M Anstreicher and Samuel Burer. Quadratic optimization with switching variables: the convex hull for n = 2. Mathematical Programming, 188(2):421–441, 2021

    work page 2021

  4. [4]

    Strong formulations for quadratic optimization with M-matrices and indicator variables

    Alper Atamtürk and Andrés Gómez. Strong formulations for quadratic optimization with M-matrices and indicator variables. Mathematical Programming, 170(1):141–176, 2018

    work page 2018

  5. [5]

    Supermodularity and valid inequalities for quadratic optimiza- tion with indicators

    Alper Atamtürk and Andrés Gómez. Supermodularity and valid inequalities for quadratic optimiza- tion with indicators. Mathematical Programming, 201(1):295–338, 2023

    work page 2023

  6. [6]

    Sparse and smooth signal estimation: Convexifi- cation of ℓ0-formulations

    Alper Atamtürk, Andrés Gómez, and Shaoning Han. Sparse and smooth signal estimation: Convexifi- cation of ℓ0-formulations. Journal of Machine Learning Research, 22(52):1–43, 2021

    work page 2021

  7. [7]

    Alper Atamtürk and Juan C. Muñoz. A study of the lot-sizing polytope. Mathematical Programming, 99:443–465, 2004

    work page 2004

  8. [8]

    A parametric branch and bound approach to suboptimal explicit hybrid MPC

    Daniel Axehill, Thomas Besselmann, Davide Martino Raimondo, and Manfred Morari. A parametric branch and bound approach to suboptimal explicit hybrid MPC. Automatica, 50(1):240–246, 2014

    work page 2014

Show all 60 references
  1. [9]

    Improved complexity analysis of branch and bound for hybrid MPC

    Daniel Axehill and Manfred Morari. Improved complexity analysis of branch and bound for hybrid MPC. In 49th IEEE Conference on Decision and Control (CDC), pages 4216–4222. IEEE, 2010

    2010

  2. [10]

    Convex relaxations for mixed integer predictive control

    Daniel Axehill, Lieven Vandenberghe, and Anders Hansson. Convex relaxations for mixed integer predictive control. Automatica, 46(9):1540–1545, 2010

    2010

  3. [11]

    Submodular functions: from discrete to continuous domains

    Francis Bach. Submodular functions: from discrete to continuous domains. Mathematical Program- ming, 175:419–459, 2019. 24

    2019

  4. [12]

    Brève communication

    Jacques Baranger and M Duc-Jacquet. Brève communication. Matrices tridiagonales symétriques et matrices factorisables. Revue française d’informatique et de recherche opérationnelle. Série rouge, 5(R3):61–66, 1971

    1971

  5. [13]

    Strong formulations for multi-item capacitated lot sizing

    Imre Barany, Tony J Van Roy, and Laurence A Wolsey. Strong formulations for multi-item capacitated lot sizing. Management Science, 30(10):1255–1261, 1984

    1984

  6. [14]

    Control of systems integrating logic, dynamics, and con- straints

    Alberto Bemporad and Manfred Morari. Control of systems integrating logic, dynamics, and con- straints. Automatica, 35(3):407–427, 1999

    1999

  7. [15]

    The explicit linear quadratic regulator for constrained systems

    Alberto Bemporad, Manfred Morari, Vivek Dua, and Efstratios N Pistikopoulos. The explicit linear quadratic regulator for constrained systems. Automatica, 38(1):3–20, 2002

    2002

  8. [16]

    A new perspective on low-rank optimiza- tion

    Dimitris Bertsimas, Ryan Cory-Wright, and Jean Pauphilet. A new perspective on low-rank optimiza- tion. Mathematical Programming, 202(1):47–92, 2023

    2023

  9. [17]

    A parametric approach for solving convex quadratic optimization with indicators over trees

    Aaresh Bhathena, Salar Fattahi, Andrés Gómez, and Simge Küçükyavuz. A parametric approach for solving convex quadratic optimization with indicators over trees. arXiv preprint arXiv:2404.08178, 2024

    2024 arXiv

  10. [18]

    Computational complexity of the capacitated lot size problem

    Gabriel R Bitran and Horacio H Yanasse. Computational complexity of the capacitated lot size problem. Management Science, 28(10):1174–1186, 1982

    1982

  11. [19]

    On mathematical programming with indicator constraints

    Pierre Bonami, Andrea Lodi, Andrea Tramontani, and Sven Wiese. On mathematical programming with indicator constraints. Mathematical Programming, 151(1):191–223, 2015

    2015

  12. [20]

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

    Hoseinali Borhan, Ardalan Vahidi, Anthony M Phillips, Ming L Kuang, Ilya V Kolmanovsky, and Stefano Di Cairano. MPC-based energy management of a power-split hybrid electric vehicle. IEEE Transactions on Control Systems Technology, 20(3):593–603, 2011

    2011

  13. [21]

    Dynamic programming for constrained optimal control of discrete-time linear hybrid systems

    Francesco Borrelli, Mato Baoti´ c, Alberto Bemporad, and Manfred Morari. Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica, 41(10):1709–1721, 2005

    2005

  14. [22]

    Predictive Control for Linear and Hybrid systems

    Francesco Borrelli, Alberto Bemporad, and Manfred Morari. Predictive Control for Linear and Hybrid systems. Cambridge University Press, 2017

    2017

  15. [23]

    Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices

    Sebastien Bossu. Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices. Linear Algebra and its Applications, 2024

    2024

  16. [24]

    Single-source shortest paths

    Thomas H Cormen. Single-source shortest paths. Introduction to Algorithms, 2001

    2001

  17. [25]

    On valid inequalities for quadratic programming with contin- uous variables and binary indicators

    Hongbo Dong and Jeff Linderoth. On valid inequalities for quadratic programming with contin- uous variables and binary indicators. In International Conference on Integer Programming and Combinatorial Optimization, pages 169–180. Springer, 2013

    2013

  18. [26]

    A multiparametric programming approach for mixed-integer quadratic engineering problems

    Vivek Dua, Nikolaos A Bozinis, and Efstratios N Pistikopoulos. A multiparametric programming approach for mixed-integer quadratic engineering problems. Computers & Chemical Engineering, 26(4-5):715–733, 2002. 25

    2002

  19. [27]

    A simple forward algorithm to solve general dynamic lot sizing models with n periods in O (n logn) or O (n) time

    Awi Federgruen and Michal Tzur. A simple forward algorithm to solve general dynamic lot sizing models with n periods in O (n logn) or O (n) time. Management Science, 37(8):909–925, 1991

    1991

  20. [28]

    Perspective cuts for a class of convex 0–1 mixed integer programs

    Antonio Frangioni and Claudio Gentile. Perspective cuts for a class of convex 0–1 mixed integer programs. Mathematical Programming, 106(2):225–236, 2006

    2006

  21. [29]

    Outlier detection in time series via mixed-integer conic quadratic optimization

    Andrés Gómez. Outlier detection in time series via mixed-integer conic quadratic optimization. SIAM Journal on Optimization, 31(3):1897–1925, 2021

    1925

  22. [30]

    Real-time solution of quadratic optimization problems with banded matrices and indicator variables

    Andrés Gómez, Shaoning Han, and Leonardo Lozano. Real-time solution of quadratic optimization problems with banded matrices and indicator variables. arXiv preprint arXiv:2405.03051, 2024

    2024 arXiv

  23. [31]

    Perspective reformulations of mixed integer nonlinear programs with indicator variables

    Oktay Günlük and Jeff Linderoth. Perspective reformulations of mixed integer nonlinear programs with indicator variables. Mathematical Programming, 124(1):183–205, 2010

    2010

  24. [32]

    Perspective reformulation and applications

    Oktay Günlük and Jeff Linderoth. Perspective reformulation and applications. In Mixed Integer Nonlinear Programming, pages 61–89. Springer, 2012

    2012

  25. [33]

    2x2-convexifications for convex quadratic optimization with indicator variables

    Shaoning Han, Andrés Gómez, and Alper Atamtürk. 2x2-convexifications for convex quadratic optimization with indicator variables. Mathematical Programming, 202(1):95–134, 2023

    2023

  26. [34]

    An efficient algorithm for image segmentation, markov random fields and related problems

    Dorit S Hochbaum. An efficient algorithm for image segmentation, markov random fields and related problems. Journal of the ACM (JACM), 48(4):686–701, 2001

    2001

  27. [35]

    Dynamic path planning and trajectory tracking using MPC for satellite with collision avoidance

    Qinglei Hu, Jingjie Xie, and Chenliang Wang. Dynamic path planning and trajectory tracking using MPC for satellite with collision avoidance. ISA transactions, 84:128–141, 2019

    2019

  28. [36]

    Exact spike train inference viaℓ0 optimization

    Sean Jewell and Daniela Witten. Exact spike train inference viaℓ0 optimization. The annals of applied statistics, 12(4):2457, 2018

    2018

  29. [37]

    Fast nonconvex deconvolution of calcium imaging data

    Sean W Jewell, Toby Dylan Hocking, Paul Fearnhead, and Daniela M Witten. Fast nonconvex deconvolution of calcium imaging data. Biostatistics, 21(4):709–726, 2020

    2020

  30. [38]

    Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot

    Scott Kuindersma, Robin Deits, Maurice Fallon, Andrés Valenzuela, Hongkai Dai, Frank Permenter, Twan Koolen, Pat Marion, and Russ Tedrake. Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot. Autonomous Robots, 40:429–455, 2016

    2016

  31. [39]

    A more scalable mixed-integer encoding for metric temporal logic

    Vince Kurtz and Hai Lin. A more scalable mixed-integer encoding for metric temporal logic. IEEE Control Systems Letters, 6:1718–1723, 2021

    2021

  32. [40]

    Facets and algorithms for capacitated lot sizing

    Janny MY Leung, Thomas L Magnanti, and Rita Vachani. Facets and algorithms for capacitated lot sizing. Mathematical Programming, 45:331–359, 1989

    1989

  33. [41]

    Polyhedral analysis of quadratic optimization problems with stieltjes matrices and indicators

    Peijing Liu, Alper Atamtürk, Andrés Gómez, and Simge Küçükyavuz. Polyhedral analysis of quadratic optimization problems with stieltjes matrices and indicators. arXiv preprint arXiv:2404.04236, 2024

    2024 arXiv

  34. [42]

    A graph-based decomposition method for convex quadratic optimization with indicators

    Peijing Liu, Salar Fattahi, Andrés Gómez, and Simge Küçükyavuz. A graph-based decomposition method for convex quadratic optimization with indicators. Mathematical Programming, 200(2):669– 701, 2023. 26

    2023

  35. [43]

    Mixed-integer formulations for optimal control of piecewise-affine systems

    Tobia Marcucci and Russ Tedrake. Mixed-integer formulations for optimal control of piecewise-affine systems. In Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control, pages 230–239, 2019

    2019

  36. [44]

    Shortest paths in graphs of convex sets

    Tobia Marcucci, Jack Umenberger, Pablo Parrilo, and Russ Tedrake. Shortest paths in graphs of convex sets. SIAM Journal on Optimization, 34(1):507–532, 2024

    2024

  37. [45]

    Mixed-integer quadratic program trajectory generation for heterogeneous quadrotor teams

    Daniel Mellinger, Alex Kushleyev, and Vijay Kumar. Mixed-integer quadratic program trajectory generation for heterogeneous quadrotor teams. In 2012 IEEE international conference on robotics and automation, pages 477–483. IEEE, 2012

    2012

  38. [46]

    A review on the inverse of symmetric tridiagonal and block tridiagonal matrices

    Gérard Meurant. A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM Journal on Matrix Analysis and Applications, 13(3):707–728, 1992

    1992

  39. [47]

    Direct and iterative methods for linear systems

    Gérard Meurant. Direct and iterative methods for linear systems. 2024

    2024

  40. [48]

    Interior-point Polynomial Algorithms in Convex Program- ming

    Yurii Nesterov and Arkadii Nemirovskii. Interior-point Polynomial Algorithms in Convex Program- ming. SIAM, 1994

    1994

  41. [49]

    Valid inequalities and separation for capacitated economic lot sizing

    Yves Pochet. Valid inequalities and separation for capacitated economic lot sizing. Operations Research Letters, 7(3):109–115, 1988

    1988

  42. [50]

    Optimal trajectory planning for autonomous driving integrating logical constraints: An MIQP perspective

    Xiangjun Qian, Florent Altché, Philipp Bender, Christoph Stiller, and Arnaud de La Fortelle. Optimal trajectory planning for autonomous driving integrating logical constraints: An MIQP perspective. In 2016 IEEE 19th international conference on intelligent transportation system...

    2016

  43. [51]

    Real-time mixed-integer quadratic pro- gramming for vehicle decision-making and motion planning

    Rien Quirynen, Sleiman Safaoui, and Stefano Di Cairano. Real-time mixed-integer quadratic pro- gramming for vehicle decision-making and motion planning. IEEE Transactions on Control Systems Technology, 2024

    2024

  44. [52]

    On the inverse of band matrices

    P Rózsa. On the inverse of band matrices. Integral Equations and Operator Theory, 10:82–95, 1987

    1987

  45. [53]

    A multiresolution approxima- tion method for fast explicit model predictive control

    Sean Summers, Colin N Jones, John Lygeros, and Manfred Morari. A multiresolution approxima- tion method for fast explicit model predictive control. IEEE Transactions on Automatic Control , 56(11):2530–2541, 2011

    2011

  46. [54]

    Economic lot sizing: An O (n logn) algorithm that runs in linear time in the Wagner-Whitin case

    Albert Wagelmans, Stan Van Hoesel, and Antoon Kolen. Economic lot sizing: An O (n logn) algorithm that runs in linear time in the Wagner-Whitin case. Operations Research, 40(1-supplement-1):S145– S156, 1992

    1992

  47. [55]

    Dynamic version of the economic lot size model

    Harvey M Wagner and Thomson M Whitin. Dynamic version of the economic lot size model. Management Science, 5(1):89–96, 1958

    1958

  48. [56]

    On the convex hull of convex quadratic optimization problems with indicators

    Linchuan Wei, Alper Atamtürk, Andrés Gómez, and Simge Küçükyavuz. On the convex hull of convex quadratic optimization problems with indicators. Mathematical Programming, 204(1):703–737, 2024

    2024

  49. [57]

    Quadratic convex reformulations for semicon- tinuous quadratic programming

    Baiyi Wu, Xiaoling Sun, Duan Li, and Xiaojin Zheng. Quadratic convex reformulations for semicon- tinuous quadratic programming. SIAM Journal on Optimization, 27(3):1531–1553, 2017. 27

    2017

  50. [58]

    Real-time detection of clus- tered events in video-imaging data with applications to additive manufacturing

    Hao Yan, Marco Grasso, Kamran Paynabar, and Bianca Maria Colosimo. Real-time detection of clus- tered events in video-imaging data with applications to additive manufacturing. IISE Transactions, 54(5):464–480, 2022

    2022

  51. [59]

    Efficient suboptimal solutions of switched lqr problems

    Wei Zhang, Alessandro Abate, and Jianghai Hu. Efficient suboptimal solutions of switched lqr problems. In 2009 American Control Conference, pages 1084–1091, 2009

    2009

  52. [60]

    i −1X τ=1 ³ Πi −τ−1 t =1 Ai −t ´ x[τ] + g i #⊤ P i

    Xiaojin Zheng, Xiaoling Sun, and Duan Li. Improving the performance of MIQP solvers for quadratic programs with cardinality and minimum threshold constraints: A semidefinite program approach. INFORMS Journal on Computing, 26(4):690–703, 2014. 28 Appendix A. Reformulation of mu...

    2014

Pith tools

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