In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as ∞-category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for (∞, 1)-categorie...

We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 − d inviscid Burgers equation. We first prove the existence of minimizers and, by a Γ-convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes...

In this paper, a new hybrid conjugate gradient algorithm is proposed for solving unconstrained optimization problems. This new method can generate sufficient descent directions unrelated to any line search. Moreover, the global convergence of the proposed method is proved under the Wolfe line search. Numerical experiments are also presented to show the efficiency of the proposed algorithm, espe...

Following the results obtained in Preord and in Cat, we characterize the effective étale-descent morphisms inM -Ord, the category ofM -ordered sets for a given monoid M . Furthermore we show that in M -Ord every effective descent morphism is effective for étale-descent (while the converse is false), and we generalize it to a more general context of relational algebras.