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Global and local approaches for the minimization of a sum of pointwise minima of convex functions

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Pith's one-line read Relaxed alternating minimization, applied to sums of pointwise minima of convex functions, has every accumulation point of its iterates be a critical point, and a compact big-M mixed-integer convex program gives exact global and local…

desk verdict Solid extension of clipped-convex to sums of minima with a compact MICP and a useful family of local heuristics; the main convergence theorem has a missing boundedness assumption that is easy to fix. read the letter →

arxiv 2501.19372 v1 pith:RDW355YK submitted 2025-01-31 math.OC

Guillaume Van Dessel , François Glineur This is my paper · ORCID
classification math.OC MSC 90C2690C1190C25
keywords sumofpointwiseminimaconvexfunctionsclippednonconvexoptimizationalternatingminimizationmixed-integerprogrammingbig-Mreformulationlocaloptimalitycertificationpiecewise-linearregression
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

The paper studies problem (SMC): minimize a convex main function plus an average of N terms, where each term is the pointwise minimum of finitely many convex component functions. This covers truncated losses, piecewise-linear regression, and restricted facility location, and the problem is NP-hard in general. The authors' central claim is that a family of relaxed alternating minimization (r-AM) heuristics, built from a bi-convex reformulation and safeguarded exploration of the weights, converges in the sense that every cluster point of its iterates is a critical point of the nonconvex problem. The same reformulations yield a compact big-M mixed-integer convex program for global solution, plus a localized version that can certify local optimality of a candidate or produce a strictly better point. A sympathetic reader would care because this gives hard nonconvex problems a local method with formal first-order guarantees and a practical route to check optimality locally.

What carries the argument

The load-bearing objects are the bi-convex reformulation (BIC-SMC), in which $F(x)=\min_Q \bar F(x,Q)$ over a product of simplices, and the gain $G^*(x_+,Q)=\bar F(x_+,Q)-\min_{\tilde Q}\bar F(x_+,\tilde Q)$, which measures how much can still be gained at fixed $x$ by reoptimizing the weights. Lemma 3.2's exploration bound fixes the exploration ratios so that $\bar F(x_+,Q)-\bar F(x_+,Q_+)\ge (1-C)G^*(x_+,Q)$; this single inequality creates the sufficient descent that powers the convergence proof. The third object is the big-M epigraph reformulation, where binary variables select the active component of each term and $S$-bounds $M^{(s)}_{l_+,l}$ majorize component differences; the local version restricts binary variables to the degeneracy set $I(\hat x)$, making local optimality checks scale with the degeneracy factor, the number of component combinations active at the candidate.

What would settle it

Run r-AM on a bounded, fully-active instance from Proposition 1.1 with $d=N=1$, $n_s=2$, $X=[-10,10]$, and fixed $C_k=1/2$, starting from a weight vector at a point where the gain $G^*$ is strictly positive. Evaluate the gain at every accumulation point of the resulting sequence; the theorem predicts $G^*=0$ at each one, so discovering a single cluster point with strictly positive gain would disprove Theorem 3.3.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the nonconvex objective can be rewritten as a bi-convex function of the variable $x$ and of weight vectors $Q$ in a product of simplices: for fixed weights the problem is convex, and for fixed $x$ the optimal weights identify the active components. Classical alternating minimization (AM) is the special case that always moves all weight to an active component. The paper shows that replacing the exact weight minimization by a convex combination of AM weights and an exploration candidate, with the mixing ratio set by Lemma 3.2 so that a fixed fraction $(1-C_k)$ of the current gain is retained, preserves sufficient descent. Summing these descents yields an $O(1/((1-\bar C)K))$ bound on the best gain and, by continuity of the gain, implies that every cluster point of $\{x_k\}$ is critical (Theorem 3.3). A parallel contribution is an equivalent big-M mixed-integer convex program for (SMC) that uses one set of binary selection variables per term and no extra copies of $x$, together with a localized version whose solution either certifies local optimality of a candidate or produces a strictly better point.

Load-bearing premise

The load-bearing premise is that the sequence of r-AM iterates stays bounded, or otherwise has at least one cluster point; the claim 'every cluster point is critical' is only meaningful and only proved under that condition, yet boundedness is not listed among the theorem's assumptions.

Editorial extensions

If this is right

  • For any instance satisfying the stated assumptions, every r-AM method with any exploration candidate produces iterates whose cluster points are critical, so the heuristics carry first-order stationarity guarantees rather than only empirical support.
  • The gain bound (66) gives an explicit iteration complexity: reaching gain below $\delta$ requires at most $(\hat F-F^*)/((1-\bar C)\delta)$ iterations, each costing one convex solve and one component evaluation.
  • The compact big-M MICP introduces $O(N\max_s n_s)$ binary variables and no extra copies of $x$, in contrast to the perspective reformulation of [6] that uses $N$ extra copies of the decision variable; this makes global and local mixed-integer solving more tractable.
  • Corollary 2.4 converts local optimality certification into a computation: solving the localized problem (45) on a neighbourhood either certifies the candidate or returns a strictly better point from which to restart local search.

Reading between the lines

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

  • The convergence theorem's unstated reliance on boundedness of the iterates suggests that, for unbounded feasible sets, practitioners should add an explicit bound or safeguard; otherwise the phrase 'every cluster point is critical' is vacuous when no cluster point exists.
  • The numerical gap between softmin r-AM and AM grows with the number of components per term, which suggests that r-AM's benefit over AM scales with the amount of near-degeneracy in the component values rather than with problem dimension alone.
  • The local optimality certificate could be turned into a practical branch-and-search loop: run a local method, certify or improve on its output with the localized MICP, and repeat; this loop's performance against a global solver is a natural benchmark the paper does not run.
  • Because the big-M constants in the local model only need to be valid on a neighbourhood, progressively enlarging that neighbourhood gives a computable, though not guaranteed, way to distinguish local minima from mere critical points.
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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

5 major / 5 minor

Summary. The paper studies the minimization of a sum of pointwise minima of convex functions (SMC), a nonsmooth nonconvex problem that includes clipped convex regression and facility location as special cases. The authors propose two complementary instruments. First, a global approach: a compact big-M mixed-integer convex program (MICP) reformulation of SMC, avoiding the many copies of the primal variable required by the perspective reformulation, plus a local variant that replaces the objective by a reduced model using only active components near a candidate point. Second, a local approach: a bi-convex reformulation (BIC-SMC) of SMC, from which a family of relaxed alternating minimization (r-AM) methods is derived; the r-AM methods differ from standard alternating minimization only in the Q-update, and include a safeguarded exploration step governed by Lemma 3.2. The central theoretical result, Theorem 3.3, states that every cluster point of the r-AM sequence is a critical point of SMC. The paper also reports numerical experiments on piecewise-linear regression and restricted facility location, where r-AM variants (SM and MM) outperform AM and DCA in most tested settings, and a procedure for certifying local optimality on neighbourhoods using the local MICP model.

Significance. If the convergence theorem and the local-optimality certification procedure are correct, the paper makes a useful contribution: it provides a flexible family of first-order methods for a broad class of nonsmooth nonconvex problems, with a proof that cluster points satisfy first-order necessary conditions, and it gives a practical MICP-based way to check and possibly escape from critical points. The numerical evidence that the proposed r-AM variants improve over AM and DCA on realistic benchmarks is valuable, and the authors state that code and a package are publicly available. The paper also contains clearly useful reformulation machinery (big-M bounds, local model, bi-convex model) with several examples where the bounds are computed in closed form. However, the main convergence theorem has a load-bearing gap related to existence of cluster points and an unstated continuity assumption, and the local-optimality certification counts in Section 4.3 appear to overstate what a timeout outcome alone can certify. With these points fixed, the paper would make a solid contribution to nonsmooth nonconvex optimization.

major comments (5)
  1. [Theorem 3.3 (Section 3.4)] The proof of part (ii) fixes an arbitrary cluster point (x⊥, Q⊥) of the sequence {(x_k, Q_k)} and uses an infinite subsequence converging to it, but no stated assumption guarantees that such a cluster point exists. X is not assumed compact, and the component functions and main function are only assumed closed, proper and convex on Rd; Assumption 1 only requires a deterministic argmin oracle, which does not prevent the oracle from returning a divergent sequence of minimizers as Q varies. If the sequence has no cluster point, the statement 'every cluster point is critical' is vacuously true and the advertised convergence guarantee is empty. The theorem should either add an explicit boundedness or coercivity assumption ensuring that the iterates remain in a compact set, or state the conclusion conditionally on the existence of a cluster point.
  2. [Lemma 3.2, Eq. (63)] The exploration rule (63) contains a division by the quantity ⟨q̂_+^{(s)} − (q*)^{(s)}, h^{(s)}(x_+)⟩, which can be zero even when q̂_+^{(s)} ≠ (q*)^{(s)}, for instance when all components of h^{(s)}(x_+) are equal. The paper's footnote only specifies the case q̂_+^{(s)} = (q*)^{(s)}, in which ε^{(s)} is set to 1; the zero-denominator case with q̂_+^{(s)} ≠ (q*)^{(s)} is left undefined. This is load-bearing because the sufficient decrease bound used in Theorem 3.3 relies on the inequality from Lemma 3.2 holding for every iteration. A complete rule for this case must be provided and the sufficient decrease inequality verified.
  3. [Theorem 3.3 proof, continuity of G*] The proof of part (ii) asserts that G*(x,Q) is continuous in (x,Q) ∈ X × Q, and uses this continuity to transfer a positive gain at the cluster point to infinitely many nearby iterates. This continuity does not follow from the stated assumptions: closedness and convexity of the extended-valued functions h̄ and h_l imply continuity only on the relative interiors of their domains, not on the whole feasible set X, and X has not been assumed compact. Either a continuity assumption on the component functions over X must be added explicitly, or the proof must justify the continuity from the problem data and the chosen topology.
  4. [Section 4.3, local optimality certification counts] The procedure reports as 'local optimality certifications' the number of trials in which the solver did not find a point x_new with F(x_new) ≤ F(x̂) − δ_glob within the 120-second time limit. This outcome does not by itself certify local optimality of x̂: a timeout also produces that outcome without solving (45) to global optimality. The counts (e.g., 5/5 for several datasets) therefore depend on solver time limits and do not, as reported, establish that the candidates are local minimizers. The authors should record the solver termination status (proven optimal versus stopped at node limit/time limit) and count as certifications only the cases where (45) was solved to proven global optimality.
  5. [Proposition 1.6] Proposition 1.6 is stated as applying 'verbatim' from the authors' prior work [11] (Proposition 1.4). Since the proof is not reproduced and the proposition is used as a basis for the local model in Lemma 2.3 and Corollary 2.4, it would strengthen the paper to either state the precise assumptions under which [11, Proposition 1.4] applies to the present setting (including the role of X and the continuity of the components) or give a short proof in the appendix.
minor comments (5)
  1. [Table 1] The κ_k expressions in Table 1 are ambiguous; for example, 'κ_k = (3/2)^k 3/4' is not a standard notation. Please write the schedule explicitly, e.g., κ_k = (3/2)^{3k/4}, and clarify the role of k in each schedule.
  2. [Section 2.3] The statement that zero gain implies criticality is named Lemma 2.6 in the Contributions list (Section 1.2) but Proposition 2.6 in Section 2.3 and later in the proof of Theorem 3.3; the numbering should be harmonized.
  3. [Equation (31)] Equation (31) writes '−M(s)_l+,l ← min_u ...' with a negative sign on the left-hand side, which is confusing. Please define the sign convention explicitly and check whether the formula yields a valid upper bound M(s)_l+,l.
  4. [Algorithm 1] The stopping criterion in Algorithm 1 uses the quantity υ − F(x_k), where υ stores F̄(x_k, Q_k) after the update, but F(x_k) is a separate function evaluation. The pseudocode should state that this evaluation is performed at each iteration and clarify the relationship between F(x_k) and the stored value υ, especially in the first iteration where υ = ∞.
  5. [Throughout] There are several typographical slips, for example 'for for every' in equation (69), 'Baratt-Boyd' in Table 1 versus 'Barratt' in the text, and 'minimization of (45)' in Section 4.3 where the minimization problem is (44) or (45) depending on the surrounding discussion. These do not affect the mathematics but should be corrected in a final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the r-AM convergence and MICP exactness proofs are derived from explicit assumptions; the only minor issues are an auxiliary self-citation and unstated regularity hypotheses in Theorem 3.3.

full rationale

I examined the derivation chain. The big-M MICP exactness (Section 2.1, Eq. (40)) is proved from the S-bounds inequalities (38)-(39); no parameter is fitted to the conclusion. The local MICP (Lemma 2.3) rests on Proposition 1.6, which is imported verbatim from the authors' own prior work [11]; this is a self-citation, but the cited proposition is a parameter-free continuity/active-set lemma and is not an ansatz or uniqueness claim, so it does not reduce Lemma 2.3 to its own input. The bi-convex reformulation (49) and the gain/criticality link (Proposition 2.6, Appendix B) are standard subgradient calculations, not definitions of the conclusion. The r-AM convergence proof (Theorem 3.3) is a genuine decrease argument from Lemma 3.2; however, the proof silently uses continuity of G* and assumes the sequence has a cluster point (e.g., bounded X or coercive/level-bounded objective), which are not in Assumptions 1-2 and are not proved. That is a correctness/completeness gap, not a circularity: the conclusion 'every cluster point is critical' is vacuous without an existence hypothesis, and the continuity hypothesis is extra. The numerical section tunes hyperparameters on the same benchmark datasets (Section 4, 'some fine tuning has been attempted'), which mildly weakens the empirical claim, but the paper does not present that as a derived prediction. No equation or fitted parameter is renamed as a first-principles result. Score 2 reflects the one minor non-load-bearing self-citation and the mild tuning concern; the central theoretical derivation is self-contained.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central theoretical results rely on standard convex analysis, two explicit domain assumptions (exact oracle and S-bounds), and a self-cited lemma about active sets. The practical experiments introduce tuned hyperparameters and ad hoc neighborhood radii. No new theoretical entities (particles, forces, etc.) are postulated.

free parameters (2)
  • hyperparameter schedule {kappa_k} for r-AM candidates (SM, MM, BB) = e.g., kappa_k = (3/2)^k * 3/4 for SM, kappa_k = k^{2/3} for MM
    The schedules in Table 1 were tuned on the same benchmark datasets. The empirical superiority of SM/MM over AM may depend on these choices. The theoretical convergence result is independent of kappa, but the practical performance comparison is not.
  • neighbourhood radii R and R_infinity in Section 4.3 = R = 10^{-1}, R_infinity = 2*10^{-1}
    These radii define the local optimization regions for certification. They are chosen ad hoc and affect the success of local optimality certification.
assumptions (4)
  • domain assumption Exact minimization oracle for convex subproblems (Assumption 1)
    The r-AM algorithm and Theorem 3.3 require access to exact minimizers of the convex problems (50). In practice, iterative solvers provide approximate solutions, which could affect the convergence guarantee.
  • domain assumption Existence of S-bounds for S superset of X (Assumption 2)
    The global MICP formulation (41) is valid only if big-M constants satisfying (28) are available for the whole feasible set. Computing these is itself a hard problem in general; the paper lists tractable special cases but the general assumption is restrictive.
  • ad hoc to paper Proposition 1.4 from the authors' prior paper [11] on local stability of active sets (used as Proposition 1.6)
    The local optimality conditions (Lemma 2.3) critically rely on the claim that active sets only shrink in a neighborhood, which is cited verbatim from the authors' own earlier work without proof in this paper. This is a self-referential dependency that is plausible but not independently verified here.
  • ad hoc to paper Boundedness of iterates or existence of cluster points in Theorem 3.3
    The proof of Theorem 3.3 assumes that the sequence {x_k} has a cluster point (via first-countable arguments), but no boundedness or compactness assumption is stated. This is an unflagged assumption needed for the conclusion to be non-vacuous.

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Pith. "Pith review of Global and local approaches for the minimization of a sum of pointwise minima of convex functions." pith.science (2026). https://pith.science/paper/RDW355YK

@misc{pith2026250119372,
  author       = {Pith},
  title        = {Pith review of: Global and local approaches for the minimization of a sum of pointwise minima of convex functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDW355YK}},
  note         = {Machine review of arXiv:2501.19372}
}
abstract

Numerous machine learning and industrial problems can be modeled as the minimization of a sum of $N$ so-called clipped convex functions (SCC), i.e. each term of the sum stems as the pointwise minimum between a constant and a convex function. In this work, we extend this framework to capture more problems of interest. Specifically, we allow each term of the sum to be a pointwise minimum of an arbitrary number of convex functions, called components, turning the objective into a sum of pointwise minima of convex functions (SMC). Problem (SCC) is NP-hard, highlighting an appeal for scalable local heuristics. In this spirit, one can express (SMC) objectives as the difference between two convex functions to leverage the possibility to apply (DC) algorithms to compute critical points of the problem. Our approach relies on a bi-convex reformulation of the problem. From there, we derive a family of local methods, dubbed as relaxed alternating minimization (r-AM) methods, that include classical alternating minimization (AM) as a special case. We prove that every accumulation point of r-AM is critical. In addition, we show the empirical superiority of r-AM, compared to traditional AM and (DC) approaches, on piecewise-linear regression and restricted facility location problems. Under mild assumptions, (SCC) can be cast as a mixed-integer convex program (MICP) using perspective functions. This approach can be generalized to (SMC) but introduces many copies of the primal variable. In contrast, we suggest a compact big-M based (MICP) equivalent formulation of (SMC), free of these extra variables. Finally, we showcase practical examples where solving our (MICP), restricted to a neighbourhood of a given candidate (i.e. output iterate of a local method), will either certify the candidate's optimality on that neighbourhood or providing a new point, strictly better, to restart the local method.

Figures

Figures reproduced from arXiv: 2501.19372 by the authors.

Figure 1
Figure 1. F(x1, x2) = min{(x1 − 3)2 + 1 3 (x2 + 3)2 , (x1 + 3)2 + 1 6 x 2 2 , 15} + min{(x2 − 2x1 + 1)2 , |x1 + 2|} Remark 2. On [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. value functions — C = 0 → sum of maximums, C = 1 → sum of minimums Based on (40), we can formulate the following parametric mixed-integer convex program. For every x ∈ X and C ∈ [0, 1], F ∗ (C) = minx∈X VC(x) where VC(x) := min (η (1),...,η(N)) ∈ RN h¯(x) + 1 N X N s=1 η (s) (41) s.t. t (s) ∈ {0, 1} ns , Xns l=1 t (s) l = 1 ∀s ∈ [N] h (s) l+ (x) ≤ η (s) + C · Xns l=1 t (s) l M (s) l+,l ∀s ∈ [N], ∀l+ ∈ [ns] One obtai… view at source ↗
Figure 3
Figure 3. A first glimpse of relaxed Alternating Minimization (r-AM) methods 3.3. Q-updates We explain now the threefold weight update mechanism in r-AM methods. Let x+ ∈ X , we will extensively use the shorthand h (s) (x+) := h (s) 1 (x+), . . . , h(s) ns (x+)  ∈ R (ns) ∀s ∈ [N]. Informally, Q-updates in r-AM are simply convex combinations between the weights associated with AM, i.e. (59), and sound candidate weights that t… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Piecewise-Linear Regression: (time-value) and (value distribution) plots 29 [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]
Figure 5
Figure 5. Figure 5: Restricted Facility Location: (value distribution) Λ = 0 (left) and Λ = 10 (right) 32 [PITH_FULL_IMAGE:figures/full_fig_p032_5.png]

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Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [11]

    G. V. Dessel and F. Glineur. Global minimization of a pointwise minimum of a finite collection of functions. arXiv preprint arXiv:2412.04625 , 2024

    work page Pith review arXiv 2024

  2. [1]

    F. J. A. Artacho, R. Campoy, and P. T. Vuong. The Boosted DC algorithm for linearly constrained DC programming. Set-Valued and Variational Analysis, 30:1265 – 1289, 2022

    work page 2022

  3. [2]

    Arthur and S

    D. Arthur and S. Vassilvitskii. k-means++: the advantages of careful seeding. In Proceed- ings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms , SODA ’07, pages 1027–1035, USA, 2007. Society for Industrial and Applied Mathematics

    work page 2007

  4. [3]

    Bagirov and S

    A. Bagirov and S. M. Karasozen B. Discrete gradient method: derivative-free method for nonsmooth optimization. Journal of Optimization Theory and Applications , 137(2):317– 334, 2008

    work page 2008

  5. [4]

    Bagirov, S

    A. Bagirov, S. Taheri, N. Karmitsa, N. Sultanova, and S. Asadi. Robust piecewise lin- ear L1-regression via nonsmooth DC optimization. Optimization Methods and Software , 37(4):1289–1309, 2022

    work page 2022

  6. [5]

    Bagirov and J

    A. Bagirov and J. Zhang. Hybrid simulating annealing method and discrete gradient method for global optimization. Proceedings of Industrial Mathematics Symposium, 2003

    work page 2003

  7. [6]

    Barratt, G

    S. Barratt, G. Angeris, and S. Boyd. Minimizing a sum of clipped convex functions. Optimization Letters, 14:2443–2459, 2020

    work page 2020

  8. [7]

    S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning, 3:1–122, 2011

    work page 2011

Show all 36 references
  1. [8]

    H. Chen, H. He, Y. Wang, and al. An efficient alternating minimization method for fourth degree polynomial optimization. Journal of Global Optimization , 82:82–103, 2022

    2022

  2. [9]

    Y. Cui, T. Chang, M. Hong, and J.-S. Pang. A study of piecewise linear-quadratic pro- grams. Journal of Optimization Theory and Applications , 186:523–553, 2020

    2020

  3. [10]

    Dao and M

    M. Dao and M. Tam. Union averaged operators with applications to proximal algorithms for min-convex functions. Journal of Optim. Theory and Applications , 181:61–94, 2019

    2019

  4. [12]

    Diamond and S

    S. Diamond and S. P. Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of machine learning research : JMLR , 17, 2016

    2016

  5. [13]

    Dias and L

    G. Dias and L. Liberti. Exploiting symmetries in mathematical programming via orbital independence. Ann. Oper. Res., 298:149–182, 2021

    2021

  6. [14]

    L. Ding, Z. Chen, X. Wang, and W. Yin. Efficient algorithms for sum-of-minimum opti- mization. In International Conference on Machine Learning , 2024

    2024

  7. [15]

    Grippo and M

    L. Grippo and M. Sciandrone. On the convergence of the block nonlinear Gauss-Seidel method under convex constraints. Oper. Res. Lett., 26:127–136, 2000

    2000

  8. [16]

    H. W. Hamacher and S. Nickel. Multifacility and Restricted Location Problems , pages 2409–2413. Springer US, Boston, MA, 2009

    2009

  9. [17]

    Ho-Nguyen and F

    N. Ho-Nguyen and F. Kilin¸ c-Karzan. A second-order cone based approach for solving the trust-region subproblem and its variants. SIAM Journal on Optimization , 27(3):1485– 1512, 2017

    2017

  10. [18]

    C. Huang. Relu networks are universal approximators via piecewise linear or constant functions. Neural Computations., 32(11):2249–2278, 2020

    2020

  11. [19]

    Jain and A

    P. Jain and A. Tewari. Alternating minimization for regression problems with vector- valued outputs. In Proceedings of the 29th International Conference on Neural Informa- tion Processing Systems - Volume 1 , NIPS’15, pages 1126–1134, Cambridge, MA, USA,

  12. [20]

    Kronqvist, D

    J. Kronqvist, D. E. Bernal, A. Lundell, and I. E. Grossmann. A review and comparison of solvers for convex MINLP. Optimization and Engineering , 20:397–455, 2018. 35

    2018

  13. [21]

    H. A. Le Thi and T. Pham Dinh. DC programming and DCA: thirty years of develop- ments. Math. Program., 169:5–68, 2018

    2018

  14. [22]

    L. Liberti. Symmetry in Mathematical Programming, pages 263–283. Springer New York, 2012

    2012

  15. [23]

    Liu and H

    T.-Y. Liu and H. Jiang. Minimizing sum of truncated convex functions and its applica- tions. Journal of Computational and Graphical Statistics , 28(1):1–10, 2019

    2019

  16. [24]

    Mosek modeling cookbook v.3.3.0

    MOSEK Aps. Mosek modeling cookbook v.3.3.0. https://docs.mosek.com/ MOSEKModelingCookbook-a4paper.pdf/, 2024. [Online; accessed 29-June-2024]

    2024

  17. [25]

    Moulay, V

    E. Moulay, V. L´ echapp´ e, and F. Plestan. Properties of the sign gradient descent algo- rithms. Inf. Sci. , 492(C):29–39, aug 2019

    2019

  18. [26]

    Ong and L

    C. Ong and L. An. Learning sparse classifiers with difference of convex functions algo- rithms. Optimization Methods and Software , 28(4):830–854, 2013

    2013

  19. [27]

    Piccialli, A

    V. Piccialli, A. M. Sudoso, and A. Wiegele. SOS-SDP: An exact solver for minimum sum-of-squares clustering. INFORMS Journal on Computing , 34(4):2144–2162, 2022

    2022

  20. [28]

    F. M. Pirani and F. Ulus. Algorithms for DC programming via polyhedral approximations of convex functions. arXiv preprint arXiv:2309.05487 , 2023

    2023 arXiv

  21. [29]

    A. R. R. Ghosh and J. Zhang. Optimization approach for clustering datasets with weights. Optimization Methods and Software , 20(2-3):335–351, 2005

    2005

  22. [30]

    Rubinov, N

    A. Rubinov, N. Soukhoroukova, and J. Ugon. Minimization of the Sum of Minima of Convex Functions and Its Application to Clustering , pages 409–434. Springer US, Boston, MA, 2005

    2005

  23. [31]

    K. Sabo, R. Scitovski, ˇS. Ungar, and Z. Tomljanovi´ c. A method for searching for a glob- ally optimal k-partition of higher-dimensional datasets. Journal of Global Optimization , 89:633–653, 2024

    2024

  24. [32]

    M. Song, Y. Xia, and J. Zheng. On globally solving nonconvex trust region subproblem via projected gradient method. arXiv preprint arXiv:2207.05597 , 2022

    2022 arXiv

  25. [33]

    Baron user manual v.2024.5.8

    The Optimization Firm. Baron user manual v.2024.5.8. https://minlp-downloads. nyc3.cdn.digitaloceanspaces.com/docs/baron%20manual.pdf, 2024. [Online; ac- cessed 29-June-2024]

    2024

  26. [34]

    T. Tran, K. Figenschou, and P.-T. Vuong. The Boosted DC algorithm for clustering with constraints. arXiv preprint arXiv:2310.14148 , 2023

    2023 arXiv

  27. [35]

    Trespalacios and I

    F. Trespalacios and I. E. Grossmann. Improved big-M reformulation for generalized disjunctive programs. Computers & Chemical Engineering , 76:98–103, 2015

    2015

  28. [36]

    Zuo and Y

    X. Zuo and Y. Jiang. Solution methodologies for minimizing a sum of pointwise minima of two functions. Optimization Letters, 17:75–87, 2023. 36 Appendix A. (SMC) formulation of Piecewise-Linear L1-Regression As such, (14) is a nonsmooth nonconvex optimization problem. One can ...

    2023

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.