Convex Optimization

By Stephen P. Boyd, Lieven Vandenberghe. 2004

4,794 citations recorded by the source corpus.

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

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Citations in the corpus

Discrete Distribution Estimation under Local Privacy (2016, methodology): “From (Boyd & Vandenberghe, 2004), the solution to this problem is given by ˆ p i Given the T i ’s, p is computed according to the empirical estimator.”
Adaptive Bound Optimization for Online Convex Optimization (2010, background): “…setting, it is well known that convergence rates can often be dramatically improved through the use of preconditioning, accomplished by an appropriate change of coordinates taking into account both the shape of the objective function and the feasible region [Boyd and Vandenberghe, 2004].”
OptNet: Differentiable Optimization as a Layer in Neural Networks (2017, background): “To see this, note that if we let I be the set where λ > 0 , then the matrix is non-singular (scaling the second block by D ( λ (cid:63) ) − 1 gives a standard KKT system (Boyd & Vandenberghe, 2004, Section 10.4), which is nonsingular for invertible Q and [ G T”
Stochastic Primal-Dual Q-Learning (2018, positive): “In particular, consider the Lagrangian function h i Using the standard results in convex optimization theories, LPs satisfy the Slater’s condition (Boyd and Vandenberghe, 2004, Chapter 5), and by the strong duality, the min-max problem satisﬁes min is the Lagrangian multiplier.”
Robustness and Regularization of Support Vector Machines (2008, background): “n f∗(y) ≥ 0, since f(0) = 0. Furthermore, f∗(0) = sup x ∈Rn(0 ⊤x − f(x)) = −inf x Rn f(x) = 0 completes the proof of the ﬁrst part. (ii) For g(·) closed and convex, g(·) = (g(·)∗)∗ (Rockafellar, 1970;Boyd and Vandenberghe, 2004). The second part follows from the ﬁrst part by setting f(·) = g∗(·). Th”
The Dantzig selector: Statistical estimation when P is much larger than n (2005, background): “(1.11) Here and below, we will refer to (1.11) as the ideal MSE.”
Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n) (2013, methodology): “2 Analysis of Newton step Self-concordance has been traditionally used in the analysis of Newton’s method (see [29, 24]).”
Optimization with Sparsity-Inducing Penalties (2011, methodology): “ such that ιΩ∗(z) is equal to 0 if Ω∗(z) ≤1 and +∞otherwise. Then, we have the equality ιΩ∗ = supw∈Rp[zTw−Ω(w)]. This result is well-known and appears in several text books (e.g., see Example 3.26 in [25]); for the sake of completeness, we give below of proof of this equality: Proposition 1.2 (Fench”
Tree-Guided Group Lasso for Multi-Task Regression with Structured Sparsity (2009, methodology): “As suggested in Chen et al. (2011), a back-tracking line search can be used to determine the step size for large J [Boyd and Vandenberghe (2004)].”
Learning with Submodular Functions: A Convex Optimization Perspective (2011, methodology): “By strong convex duality (which applies because Slater’s condition [30] is satisfied), we can invert the “min” and “max” operations and get: g∗∗(w) = min δ∈[0,1]p max s∈Rp s⊤w − δ⊤|s|+ f(δ)”