An elementary approach to polynomial optimization on polynomial meshes

A polynomial mesh on a multivariate compact set or manifold is a sequence of finite norming sets for polynomials whose norming constant is independent of degree. We apply the recently developed theory of polynomial meshes to an elementary discrete approach for polynomial optimization on nonstandard domains, providing a rigorous (over)estimate of the convergence rate. Examples include surface/solid subregions of sphere or torus, such as caps, lenses, lunes, and slices

A new look at nonnegativity on closed sets and polynomial optimization

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ with support equal to K, are all positive semidefinite (if $K$ is compact $\mu$ is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with {\it no} lifting, of the cone of nonnegative polynomials of degree at most $d$. Wen used in polynomial optimization on certain simple closed sets \K (like e.g., the whole space $\R^n$, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

Distributed optimization over time-varying directed graphs

We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of $O(\ln(t)/\sqrt{t})$, where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes

Temporal Deformable Convolutional Encoder-Decoder Networks for Video Captioning

It is well believed that video captioning is a fundamental but challenging task in both computer vision and artificial intelligence fields. The prevalent approach is to map an input video to a variable-length output sentence in a sequence to sequence manner via Recurrent Neural Network (RNN). Nevertheless, the training of RNN still suffers to some degree from vanishing/exploding gradient problem, making the optimization difficult. Moreover, the inherently recurrent dependency in RNN prevents parallelization within a sequence during training and therefore limits the computations. In this paper, we present a novel design --- Temporal Deformable Convolutional Encoder-Decoder Networks (dubbed as TDConvED) that fully employ convolutions in both encoder and decoder networks for video captioning. Technically, we exploit convolutional block structures that compute intermediate states of a fixed number of inputs and stack several blocks to capture long-term relationships. The structure in encoder is further equipped with temporal deformable convolution to enable free-form deformation of temporal sampling. Our model also capitalizes on temporal attention mechanism for sentence generation. Extensive experiments are conducted on both MSVD and MSR-VTT video captioning datasets, and superior results are reported when comparing to conventional RNN-based encoder-decoder techniques. More remarkably, TDConvED increases CIDEr-D performance from 58.8% to 67.2% on MSVD.Comment: AAAI 201

Parameters identification of a cannon counter-recoil mechanism based on PSO and interval analysis theory

In this paper, two methods based on PSO algorithm and interval sequence conversion model or interval analysis theory are proposed to identify two kinds of uncertain parameters of a cannon counter-recoil mechanism during the manual recoil and forward moving process. Before identifying, some test data were obtained during the manual recoil process. Then, the uncertain parameters were described by interval number and a mathematical model about recoil process of a cannon was built. Taking the similarity degree of time-series data as an optimization objective function, Particle Swarm Optimization (PSO) algorithm was used to solve the deterministic optimization problem which was transformed by interval sequence conversion model, and the parameter identification of recuperator in the manual recoil process of a cannon was achieved. On this basis, combining PSO algorithm and Krawczyk algorithm of interval analysis theory, the uncertain parameters of recoil brake were identified. Finally, the results of identification proved that the above-mentioned two methods had a relatively high identification accuracy