Minima of Functions of Several Variables with Inequalities as Side Conditions

By W. Karush. 2014

820 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Numerical Analysis

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Citations in the corpus

Algorithmic recourse under imperfect causal knowledge: a probabilistic approach (2020, background): “6 In particular, we first rewrite the constrained optimisation problem in unconstrained form with Lagrangian [23, 28]: L(✓, ) := costF(a) + thresh(a) EXd(I)|✓ ⇥ h xnd(I),✓,Xd(I) ⇤ .”
Hinge-Loss Markov Random Fields and Probabilistic Soft Logic (2015, methodology): “Proof We prove the lemma via the Karush-Kuhn-Tucker (KKT) conditions (Karush, 1939; Kuhn and Tucker, 1951).”
On label dependence and loss minimization in multi-label classification (2012, neutral): “To show that one of these two probability distributions defines the maximum of the linear program, we verify the Karush-Kuhn-Tucker conditions (Karush 1939; Kuhn and Tucker 1951) for a given value of t .”
Consistent and Accurate Frequency Oracles under Local Differential Privacy (2019, methodology): “In Appendix A, we use the KKT condition [21, 22] to solve optimization problem in Eq.”
Incremental Support Vector Learning for Ordinal Regression (2015, methodology): “From the KKT theorem [5], we obtain the following KKT conditions:”
Incremental learning for ν-Support Vector Regression (2015, methodology): “Then by the KKT theorem (Karush, 1939), the first-order derivative ofW leads to the following KKT conditions:”
Hinge-Loss Markov Random Fields and Probabilistic Soft Logic (2015, methodology): “Proof We prove the lemma via the Karush-Kuhn-Tucker (KKT) conditions (Karush, 1939; Kuhn and Tucker, 1951).”
beta-VAE: Learning Basic Visual Concepts with a Constrained Variational Framework (2016, background): “2 as a Lagrangian under the KKT conditions (Kuhn & Tucker, 1951; Karush, 1939), we obtain: where the KKT multiplier β is the regularisation coefﬁcient that constrains the capacity of the latent information channel z and puts implicit independence pressure on the learnt posterior due to the…”
Scaling Distributed Machine Learning with the Parameter Server (2014, neutral): “6A user-defined Karush-Kuhn-Tucker (KKT) filter [26].”
CALM: Consistent Adaptive Local Marginal for Marginal Release under Local Differential Privacy (2018, background): “Formally, we have the following problem: [22, 23], we can derive the solution: , by taking the partial derivative of L for each of w i , we have w i = − µ 2 C i .”
A General-Purpose Software Framework for Dynamic Optimization (Een algemene softwareomgeving voor dynamische optimalisatie) (2013, background): “AD Algorithmic differentiation BDF Backward differentiation formulas CCS Compressed column storage CPU Central processing unit CRS Compressed row storage DAE Differential-algebraic equation END External numerical differentiation GN Gauss-Newton GPU Graphics processing unit IND Internal numerical dif”