Asymptotic behavior of solutions: An application to stochastic NLP

In this article we study the consistency of optimal and stationary (KKT) points of a stochastic non-linear optimization problem involving expectation functionals, when the underlying probability distribution associated with the random variable is weakly approximated by a sequence of random probability measures. The optimization model includes constraints with expectation functionals those are not captured in direct application of the previous results on optimality conditions exist in the literature. We first study the consistency of stationary points of a general NLP problem with convex and locally Lipschitz data and then apply those results to the stochastic NLP problem and stochastic minimax problem. Moreover, we derive an exponential bound for such approximations using a large deviation principle.

     

Weighted heuristic search in networks

The evaluation function used in heuristic search algorithms commonly has the form , where n is any node in the network, is the cost of the best path currently known from the start node to n, and is the heuristic estimate associated with node n. A more general form of the evaluation function can be obtained by incorporating a weighting parameter α:

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, where 0≤ α ≤1. Such an evaluation function has been used in some recent experimental investigations of the 8-puzzle problem. In this paper a theoretical framework is developed for the analysis of the worst-case behavior of weighted heuristic search algorithms. A new algorithm is proposed whose worst-case performance characteristics are greatly superior to those of earlier algorithms in terms of the following two measures: how good is the solution, and how many nodes are expanded. Bounds are also obtained on some useful network parameters for both general and special types of heuristic estimate functions.     

Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.