Optimization by Vector Space Methods

By D. Luenberger. 1968

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Citations in the corpus

Certifying Some Distributional Robustness with Principled Adversarial Training (2017, methodology): “As taking P = Q yields W c ( Q, Q ) = 0, Slater’s condition holds and we may apply standard (inﬁnite dimensional) duality results [42, Thm. 8.7.1] to obtain sup Now, noting that for any M ∈ Π( P, Q ) we have fdP = f ( x ) dM ( x, z ), we have that the rightmost quantity in the preceding display…”
Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates (2013, background): “This objective is Fréchet differentiable, and as a consequence, the necessary and sufficient conditions for optimality [19] of f̂ are that”
Robust kernel density estimation (2008, background): “al of Tat x2Xwith incremental h2Xis dened as T(x;h) = lim !0 T(x+ h) T(x) : If T(x 0;h) is dened for all h2X, a necessary condition for Tto have a minimum at x 0 is that T(x 0;h) = 0 for all h2X(Luenberger, 1997). From this optimality principle, we have the following lemma. Lemma 1. Suppose assumpti”
Economic-emission dispatch problem: A semi-definite programming approach (2014, methodology): “By fixing the degree k, the optimal polynomial which gives the best approximating error for the exponential function in the emission function can be computed using the Gram-Schmidt procedure [52] or equivalently by the least square error approximation.”
Control theory for linear systems (2002, background): “16 were given in Luenberger [116] and Ackermann [1], respectively.”
Convex Computation of the Region of Attraction of Polynomial Control Systems (2012, background): “The cone K (cid:48) is equipped with the weak-* topology; see [29, Section 5.10].”
Some Notes on Alternating Optimization (2002, methodology): “More typically, we resort to computational methods such as gradient-based descent in all variables [12-14]; and gradient-based alternating descent (AO) on subsets of the variables [15-17], A relatively new set of interesting techniques that eschew the use of optimality conditions from calculus are b”
Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation (2004, methodology): “formulated as follows [DS83, Lue69 ]: Solve the Newton equation”
Finite element error analysis for state-constrained optimal control of the Stokes equations (2008, unknown): “…of the support functional: since the interior of K is not empty by Assumption 2.3, the supporting hyperplane theorem implies int K = µ∈V * ∞ , µ =0 {v ∈ V ∞ | µ , v V * ∞ ,V∞ s(µ)}, (5.1) where s : V * ∞ → R denotes the support functional, i.e. s(µ) = sup v∈K µ , v V * ∞ ,V∞ (see, e.g., [28]).”