On conjugate convex functions, Canadian - Fenchel - 1949 () Citation Context ... is invariant under the addition of an affine function to '. Theorem 2.5 (/ is a merit function for (2.1)). / is a gap function. Proof. In order to prove the theorem, we shall make use of the Fenchel...
Theory of submodular programs: A fenchel-type min-max theorem and subgradients of submodular functions We define a convex (or concave) conjugate function of a submodular (or supermodular) function and show a Fenchel-typemin-max theorem for submodular and ... Satoru,Fujishige - 《Mathematical Pro...
T., Level sets and continuity of conjugate convex functions,Trans. Amer. Math. Soc. 123 (1966), 46–63. Google Scholar Rocakafellar, R. T., Conjugates and Legendre transforms of convex functions,Canad. J. Math. 19 (1967a), 200–205. Google Scholar Rockafellar, R. T., Duality ...
Seeger . “On conjugate functions, subgradients, and directional derivatives of a class of optimality criteria in experimental design”. Statistics, Vol. 22 (1991), in press.T. Hoang and A. Seeger, On conjugate functions, subgradients, and directional derivatives of a class of optimality ...
A duality theory using conjugate functions is established for mathematical programs that involve the composition of two convex functions. This generalizes our earlier work in quadratic and composite geometric programs. A specific application to minimax programs is given. 关键词: Theoretical or Mathematical...
Keywords: non-convex problems; large-batch training; stochastic normalized gradient descent; momentumCite as: Zhao S-Y, Shi C-W, Xie Y-P, et al. Stochastic normalized gradient descent with momentum for large-batch training. Sci China Inf Sci, 2024, 67(11): 212101, doi: 10.1007/s11432-...
Under proper conditions, we show that our method is globally convergent for uniformly convex functions. We give a numerical comparison of the implementations of our method and two efficient hybrid CG methods proposed by Dai and Yuan using a set of unconstrained optimization test problems from the ...
Also it is easy to verify, using results given in Raiffa and Schlaifer [ 8] , that under a wide variety of models of k independent data generating processes and natural conjugate prior distributions, the posterior variance (or prior expectation of the posterior variance) of a linear ...
This is done using a Lipschitz extension method given by a convex combination of the McShane and the Whitney extensions of real functions defined on subspaces of (quasi-pseudo-)metric spaces. Roughly speaking, after fixing the graph that models the problem, we have to find a (quasi-)metric ...
In this paper, zero duality gap conditions in nonconvex optimization are investigated. It is considered that dual problems can be constructed with respect to the weak conjugate functions, and/or directly by using an augmented Lagrangian formulation. Both of these approaches and the related strong ...