Implicit Functions and Solution Mappings

By Asen L. Dontchev, R. Tyrrell Rockafellar. 2009

1,154 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Computational Theory and Mathematics, Mathematical Physics

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Citations in the corpus

Meta-Learning With Differentiable Convex Optimization (2019, background): “, [12, 7, 2]) on the optimality (KKT) conditions to obtain the necessary gradients.”
On the Differentiability of the Solution to Convex Optimization Problems (2018, methodology): “(6) Then the implicit function theorem is as follows, repeated from Dontchev and Rockafellar [2].”
OptNet: Differentiable Optimization as a Layer in Neural Networks (2017, methodology): “In this paper, we use implicit differentiation (Dontchev & Rockafellar, 2009; Griewank & Walther, 2008) and techniques from matrix differential calculus (Magnus & Neudecker, 1988) to derive the gradients from the KKT matrix of the problem.”
DeepEMD: Differentiable Earth Mover's Distance for Few-Shot Learning (2020, methodology): “To embed the optimization problem into the model for end-to-end training, we apply the implicit function theorem [11], [12], [13] to form the Jacobian”
Tracking Moving Agents via Inexact Online Gradient Descent Algorithm (2017, background): “Time-varying optimization problems have classically been studied in the context of optimal control [33], target tracking [34], [35], non-stationary optimization [36], and parametric programming [37].”
Openness stability and implicit multifunction theorems: Applications to variational systems (2011, methodology): “For this, we use some ideas we have previously developed in [6].”
Second-order growth, tilt stability, and metric regularity of the subdifferential (2013, neutral): “, [9, 11, 17].”
Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods (2016, methodology): “ prox-linear methods combine the two. In the language of variational analysis, the existence of an error bound is exactly “metric subregularity” of the gradient-like mapping; see Dontchev-Rockafellar [42]. We will show that subregularity of the gradient-like mapping is equivalent to subregularity of”
METRIC REGULARITY—A SURVEY PART 1. THEORY (2016, methodology): “A number of characterizations of strong regularity can be found in [35].”
Metric regularity and Lipschitzian stability of parametric variational systems (2010, methodology): “In [5, 10, 12, 25], the reader can find more information and discussion on the calmness property with recent results and applications.”
Partially Does It: Towards Scene-Level FG-SBIR with Partial Input (2022, methodology): “7 using Dini classical implicit function theorem [21] as,”
Lyusternik-Graves theorem and fixed points (2011, methodology): “For a comprehensive treatment of these developments together with historical remarks, see the recent book [9], as well as the earlier survey [14].”
Differentiable Convex Optimization Layers (2019, methodology): “By applying the implicit function theorem [36, 34] to the optimality conditions of a cone program, it is possible to compute its derivative D s ( A, b, c ) .”
Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods (2016, background): “…n → 2 R m is said to be metrically subregular [12, Page 183] at ¯ x for ¯ y if ¯ y ∈ G (¯ x ), and there exist a constant κ > 0, a neighborhood U of ¯ x and a neighborhood V of ¯ y such that dist( x, A property closely related to metric subregularity is the notion of calmness (see [12, Page 182]).”
THE IMPLICIT AND THE INVERSE FUNCTION THEOREMS: EASY PROOFS (2012, neutral): “[4] A.”
DeepEMD: Few-Shot Image Classification With Differentiable Earth Mover’s Distance and Structured Classifiers (2020, methodology): “To embed the optimization problem into the model for end-to-end training, we can apply the implicit function theorem [3,8,22] to form the Jacobian matrix of the optimal optimization variable with respect to the problem parameters [3].”