We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

The ranking and selection problem is a well-known mathematical framework for the formal study of optimal information collection. Expected improvement (EI) is a leading algorithmic approach to this problem; the practical benefits of EI have repeatedly been demonstrated in the literature, especially in the widely studied setting of Gaussian sampling distributions. However, it was recently proved that some of the most well-known EI-type methods achieve suboptimal convergence rates. We investigate a recently proposed variant of EI (known as ‘complete EI’) and prove that, with some minor modifications, it can be made to converge to the rate-optimal static budget allocation without requiring any tuning.

We propose a new adaptive and composite Barzilai–Borwein (BB) step size by integrating the advantages of such existing step sizes. Particularly, the proposed step size is an optimal weighted mean of two classical BB step sizes and the weights are updated at each iteration in accordance with the quality of the classical BB step sizes. Combined with the steepest descent direction, the adaptive and composite BB step size is incorporated into the development of an algorithm such that it is efficient to solve large-scale optimization problems. We prove that the developed algorithm is globally convergent and it R-linearly converges when applied to solve strictly convex quadratic minimization problems. Compared with the state-of-the-art algorithms available in the literature, the proposed step size is more efficient in solving ill-posed or large-scale benchmark test problems.

This work is motivated by the question of whether there are spaces X for which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space ${\open R}\hbox{P}^m$, it is known that $\hbox{TC}^S ({\open R}\hbox{P}^{m})$ captures, with a few potential exceptional cases, the Euclidean embedding dimension of ${\open R}\hbox{P}^{m}$. We now show that, for all m≥1, $\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$ is characterized as the smallest positive integer n for which there is a symmetric ${\open Z}_{2}$-biequivariant map Sm×Sm→Sn with a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of ${\open R}\hbox{P}^{2^{e}}$ for e≥1. In particular, this leaves the torus S1×S1 as the only closed surface whose symmetric (symmetrized) TCS (TCΣ) invariant is currently unknown.

We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.

We present an extension of vendor-managed inventory (VMI) problems by considering advertising and pricing policies. Unlike the results available in the literature, the demand is supposed to depend on the retail price and advertising investment policies of the manufacturer and retailers, and is a random variable. Thus, the constructed optimization model for VMI supply chain management is a stochastic bi-level programming problem, where the manufacturer is the upper level decision-maker and the retailers are the lower-level ones. By the expectation method, we first convert the stochastic model into a deterministic mathematical program with complementarity constraints (MPCC). Then, using the partially smoothing technique, the MPCC is transformed into a series of standard smooth optimization subproblems. An algorithm based on gradient information is developed to solve the original model. A sensitivity analysis has been employed to reveal the managerial implications of the constructed model and algorithm: (1) the market parameters of the model generate significant effects on the decision-making of the manufacturer and the retailers, (2) in the VMI mode, much attention should be paid to the holding and shortage costs in the decision-making.

Selecting important variables and estimating coordinate covariation have received considerable attention in the current big data deluge. Previous work shows that the gradient of the regression function, the objective function in regression and classification problems, can provide both types of information. In this paper, an algorithm to learn this gradient function is proposed for nonidentical data. Under some mild assumptions on data distribution and the model parameters, a result on its learning rate is established which provides a theoretical guarantee for using this method in dynamical gene selection and in network security for recognition of malicious online attacks.

The error of a distributed algorithm for big data classification with a support vector machine (SVM) is analysed in this paper. First, the given big data sets are divided into small subsets, on which the classical SVM with Gaussian kernels is used. Then, the classification error of the SVM for each subset is analysed based on the Tsybakov exponent, geometric noise, and width of the Gaussian kernels. Finally, the whole error of the distributed algorithm is estimated in terms of the error of each subset.

The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

We solve the problem of concept learning using a semi-tensor product method. All possible hypotheses are expressed under the framework of a semi-tensor product. An algorithm is raised to derive the version space. In some cases, the new approach improves the efficiency compared to the previous approach.

As a new business form, the buy-online and pick-up-in-store (BOPS) mode allows consumers to pay for goods online and pick them up in a physical store. In this paper, an equilibrium model is constructed to formulate an optimal decision-making problem for online and offline retailers under the BOPS mode, where the online retailer determines the retail price of the goods and the consignment quantity in a physical store, while the offline retailer chooses the revenue share of profit by a consignment contract. Different to the existing models, the cost of overstocking and loss of understocking are incorporated into the profit function of the online retailer due to the randomness of demand. For the objective function of the offline retailer, the cross-sale quantity generated by the BOPS mode is taken into account. Then the game between the online and offline retailers is expressed as a stochastic Nash equilibrium model. Based on the analytic properties of the model, necessary conditions for the equilibrium solution are obtained. A case study and sensitivity analysis are employed to reveal the managerial implications of the model, which can provide a number of valuable suggestions on optimizing the strategies for the online and offline retailers under the BOPS mode.

In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case.

This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}n$ in time $2^{0.3774\, n}$ using memory $2^{0.2925\, n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.

We propose a modified projected Polak–Ribière–Polyak (PRP) conjugate gradient method, where a modified conjugacy condition and a method which generates sufficient descent directions are incorporated into the construction of a suitable conjugacy parameter. It is shown that the proposed method is a modification of the PRP method and generates sufficient descent directions at each iteration. With an Armijo-type line search, the theory of global convergence is established under two weak assumptions. Numerical experiments are employed to test the efficiency of the algorithm in solving some benchmark test problems available in the literature. The numerical results obtained indicate that the algorithm outperforms an existing similar algorithm in requiring fewer function evaluations and fewer iterations to find optimal solutions with the same tolerance.

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable X n,q : the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for X n,q . If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

As models for molecular evolution, immune response, and local search algorithms, various authors have used a stochastic process called the evolutionary walk, which goes as follows. Assign a random number to each vertex of the infinite N-ary tree, and start with a particle on the root. A step of the process consists of searching for a child with a higher number and moving the particle there; if no such child exists, the process stops. The average number of steps in this process is asymptotic, as N → ∞, to log N, a result first proved by Macken and Perelson. This paper relates the evolutionary walk to a process called random bisection, familiar from combinatorics and number theory, which can be thought of as a transformed Poisson process. We first give a thorough treatment of the exact walk length, computing its distribution, moments and moment generating function. Next we show that the walk length is asymptotically normally distributed. We also treat it as a mixture of Poisson random variables and compute the asymptotic distribution of the Poisson parameter. Finally, we show that in an evolutionary walk with uniform vertex numbers, the ‘jumps’, ordered by size, have the same asymptotic distribution as the normalized cycle lengths in a random permutation.

Kohonen self-organizing interval maps are considered. In this model a linear graph is embedded randomly into the unit interval. At each time a point is chosen randomly according to a fixed distribution. The nearest vertex and some of its nearby neighbors are moved closer to the point. These models have been proposed as models of learning in the audio-cortex. The models possess not only the structure of a Markov chain, but also the added structure of a random dynamical system. This structure is used to show that for a large class of these models, in a strong way, the initial conditions are unimportant and only the dynamics govern the future. A contractive condition is proven in spite of the fact that the maps are not continuous. This, in turn, shows that the Markov chain is uniformly ergodic.

A stochastic gradient descent method is combined with a consistent auxiliary estimate to achieve global convergence of the recursion. Using step lengths converging to zero slower than 1/n and averaging the trajectories, yields the optimal convergence rate of 1/√n and the optimal variance of the asymptotic distribution. Possible applications can be found in maximum likelihood estimation, regression analysis, training of artificial neural networks, and stochastic optimization.