Convex Analysis and Nonlinear Optimization: Theory and Examples. Jonathan M. Borwein and Adrian S. Lewis, Springer, New York, 2000

By Michael J. Todd. 2003

403 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis, Mathematical Optimization

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Numerical Analysis

In Mathematical Optimization

Citations in the corpus

Block-Coordinate Frank-Wolfe Optimization for Structural SVMs (2012, methodology): “For a more detailed explanation of Fenchel duality, we refer the reader to the standard literature, e.g. (Borwein & Lewis, 2006, Theorem 3.3.5).”
Reinforcement Learning via Fenchel-Rockafellar Duality (2020, methodology): “In this section, we present one of the most general and useful tools associated with Fenchel conjugates, known as the Fenchel-Rockafellar duality (Rockafellar, 1970; Borwein and Lewis, 2010).”
Introduction to Online Convex Optimization (2016, background): “On the other side, there are the numerous introductory essays to convex optimization and convex analysis, to name a few (23; 78; 76; 77; 21; 92).”
Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning (2014, positive): “A necessary ﬁrst-order condition for θ to be a local minimum of f is to have ∇ f ( θ, θ ′ − θ ) ≥ 0 for all θ ′ in Θ (see, e.g., [7]). is stationary In other words, there is no feasible descent direction θ ′ − θ and θ a point.”
From optimal transport to generative modeling: the VEGAN cookbook (2017, background): “…that F ( Q ) = 0 if and only if P Z = Q Z , and for any λ > 0, consider the following relaxed unconstrained version of W † c ( P X , P G ): It is well known [13] that under mild conditions adding a penalty as in (12) is equivalent to adding a constraint of the form F ( Q ) ≤ µ λ for some µ λ > 0.”
Structured Variable Selection with Sparsity-Inducing Norms (2009, background): “…ˆ r ∈ R p deﬁned as In addition, the solution ˆ w satisﬁes Ω ∗ [ L ( ˆ w )] ≤ µ. Proof The problem min w ∈ R p L ( w ) + µ Ω( w ) = min w ∈ R p F ( w ) being convex, the directional derivative optimality condition are necessary and sufﬁcient (Borwein and Lewis, 2006, Propositions 2.1.1-2.1.2).”
Learning Classifiers with Fenchel-Young Losses: Generalized Entropies, Margins, and Algorithms (2018, background): “As the name indicates, this family of loss functions is grounded in the Fenchel-Young inequality (Borwein and Lewis, 2010, Proposition 3.3.4) (3) The inequality, together with well-known properties of convex conjugates, imply the following results.”
Loss Functions for Top-k Error: Analysis and Insights (2015, background): “The primal and Fenchel dual [7] objective functions are given as”
Proximal Methods for Hierarchical Sparse Coding (2010, neutral): “Otherwise, combining the deﬁnitions of t g , z g and the fact that k k ∗ ≤ (Borwein and Lewis, 2006), that is, for all y such that k y k ∗ ≤ t , ( w − k ) ⊤ ( y − k ) ≤ 0.”
A Survey on Demand Response Programs in Smart Grids: Pricing Methods and Optimization Algorithms (2015, background): “The set of variables is known as the design vector, while the function is termed as the objective function [161].”
Task-Driven Dictionary Learning (2010, neutral): “A classical result (see [58], page 238)”
Graph Implementations for Nonsmooth Convex Programs (2008, neutral): “For the basics of convex analysis, see, e.g., [BV04, Roc70, vT84, Ber03, BL00 ].”
Taking Advantage of Sparsity in Multi-Task Learning (2009, background): “where ∂ denotes the subdifferential (see, for example, [4] for more information on convex analysis).”
Sparse Modeling for Image and Vision Processing (2014, background): “Given some data x, matrix D, and parameter µ > 0, we indeed know from Lagrange multiplier theory [see, e.g., Borwein and Lewis, 2006, Boyd and Vandenberghe, 2004] that for all solution α? of (1.9), there exists a parameter λ ≥ 0 such that α? is also a solution of (1.10).”