The Theory of Max-Min and its Application to Weapons Allocation Problems

By J. Danskin. 1967

Subfields: Mathematical Optimization, Game & Decision Theory

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Citations in the corpus

Composing graphical models with neural networks for structured representations and fast inference (2016, background): “when argmaxy f(x, y) has more than one element, see Danskin [27], Fiacco [28, Section 2.”
Envelope Theorems for Arbitrary Choice Sets (2002, background): “Aversion of this result was first obtained by Danskin (1967) .”
Regularity of quasiconvex envelopes (2000, background): “5 (in all dimensions m, n ≥ 1). However, it is possible to show by adapting an argument from [43] (see also [ 17 ] and [18]) that, under quite general conditions, F has finite one-sided directional derivatives with respect to u everywhere.”
Optimization Problems with Perturbations: A Guided Tour (1998, background): “By Danskin’s theorem, h(x) is directionally differentiable and h′(x0, d) = max { d · ∇f(x0), max t∈T (x0) d · ∇gt(x0) } .”
Rényi Fair Inference (2019, methodology): “Since the optimization problem maxw fB(θ,w) is strongly concave in w, using Danskin’s theorem (see Danskin (1967) and Bertsekas (1971)), we conclude that the function gB(·) is differentiable.”
ON NONSMOOTH OPTIMALITY THEOREMS FOR ROBUST OPTIMIZATION PROBLEMS (2014, unknown): “1], [12]).”
Adversarial Robustness Against the Union of Multiple Perturbation Models (2019, background): “Adversarial training Although solving the min-max optimization problem may seem daunting, a classical result known as Danskin’s theorem [Danskin, 1967] says that the gradient of a maximization problem is equal to the gradient of the objective evaluated at the optimum.”
On the Convergence and Robustness of Adversarial Training (2021, methodology): “Due to the nonlinearities in DNNs such as ReLU (Nair & Hinton, 2010) and max-pooling functions, the exact assumptions of Danskin’s theorem (Danskin, 2012) do not hold.”
Generalized semi-infinite programming: A tutorial (2008, background): “For example, in standard semi-infinite programming the index set mapping Y( x)≡ Y is constant, and the Theorem of Danskin [ 15 ] then says that is directionally differentiable with”
ELLA: An Efficient Lifelong Learning Algorithm (2013, methodology): “C. ∇Lg(L) is Lipschitz on the space of latent model components L. Proof sketch: Part (A) can be easily shown using the fact that α is continuous and by applying a corollary of Theorem 4.1 as stated by Bonnans & Shapiro (1998) (originally shown by Danskin (1967)).”