Computing the dominant eigenpair of an essentially nonnegative tensor is an important topic in tensor analysis and computation because of the important applications in network resource allocations. In this paper, we present an improved homotopy-type method inspired by Chen et al. (2019) to compute the dominant eigenpair of an essentially nonnegative tensor. Based on an approximation technique, the proposed algorithm is suitable for both irreducible and reducible cases and the global convergence analysis is established. Finally, we implement the method via a prediction–correction approach for path following and some numerical results are reported to illustrate the efficiency of the proposed algorithm.

The progressive hedging algorithm (PHA) is an effective solution method for solving monotone stochastic variational inequalities (SVIs). However, this validity is based on the assumption of global maximal monotonicity. In this paper, we propose a localized PHA for solving nonmonotone SVIs and show that its validity is based on the weaker assumption of locally elicitable maximal monotonicity. Furthermore, we prove that such assumption holds when the mapping involved in the SVI is locally elicitable monotone or locally monotone. The local convergence of the proposed algorithm is established, and it is shown that the localized PHA has the rate of linear convergence under some mild assumptions. Some numerical experiments, including a two-stage orange market problem and randomly generated two-stage piecewise stochastic linear complementarity problems, indicate that the proposed algorithm is efficient. Funding: This work was supported by the National Natural Science Foundation of China [Grant 12171271].

This paper focuses on the solvability of multistage pseudomonotone stochastic variational inequalities (SVIs). On the one hand, some known solvability results of pseudomonotone deterministic variational inequalities cannot be directly extended to multistage pseudomonotone SVIs, so we construct the isomorphism between both and then establish theoretical results on the existence, convexity, boundedness and compactness of the solution set for multistage pseudomonotone SVIs via such an isomorphism. On the other hand, there does not exist a special algorithm for solving the multistage pseudomonotone SVIs so far, so we propose some sufficient conditions on the elicitability of multistage pseudomonotone SVIs, which opens the door for applying Rockafellar's elicited progressive hedging algorithm to solve such SVIs. Numerical results on solving a two-stage stochastic market optimization problem and randomly generated two-stage pseudomonotone linear complementarity problems are presented.

Stochastic $$R_0$$ matrix, which is a generalization of $$R_0$$ matrix, is instrumental in the study of the stochastic linear complementarity problem (SLCP). In this paper, we focus on the expected residual minimization (ERM) of the SLCP via the Fischer–Burmeister (FB) function, which enjoys better properties of the continuity and differentiability than the min-function-based ERM when the involved matrix is a stochastic $$R_0$$ matrix. First, we prove that the solution set of the FB-function-based ERM is nonempty and bounded if and only if the involved matrix is a stochastic $$R_0$$ matrix. Second, we show that its objective function is continuously differentiable in $${\mathbb {R}}^n$$ , which makes it possible for us to design the gradient-type algorithm. Finally, we implement the numerical experiments generated randomly via sample average approximation, and the numerical results indicate that the optimal solutions of the FB-function-based ERM are better than that of the min-function-based ERM in terms of preserving the nonnegativity of the involved linear function, especially when the involved matrix is a constant matrix.

In this paper, comparison of topological structures between Boolean control networks and nominal Boolean networks is investigated. First, transient periods and attractor basins of Boolean control networks are introduced, and three necessary and sufficient conditions are derived in order to determine transient periods and attractor basins. From two aspects of attractors and transient periods, topological structures of Boolean control networks and nominal Boolean networks are compared, and the existences of different relations are illustrated via some examples. Finally, four necessary and sufficient conditions are proposed for judging relations of topological structures between Boolean control networks and nominal Boolean networks.

Two kinds of optimal control problems of Boolean control networks are investigated in this paper. For these two optimal control problems, existences of optimal control sequences are proved firstly, To find the optimal control sequences, weighted directed graphs are built. Then graph-theoretical approaches are proposed based on the Dijkstra's algorithm for the shortest path problem. Finally, an example is given to show the feasibility of the proposed approaches.

Nominal Boolean network is a kind of Boolean network, which is obtained from a Boolean control network with all controls disconnected. In this paper, the relationship between the transition matrix of the original Boolean control network and the transition matrix of the corresponding nominal Boolean control network is investigated. Due to the fact that transition matrices can be derived from structure matrices, the relationship between structure matrices of the original Boolean control network and structure matrices of the nominal Boolean network is studied. Via constant values corresponding to disconnected controls, structure matrices of the nominal Boolean network can be derived from structure matrices of the original Boolean control network. Finally, a concrete algorithm is presented to derive transition matrices of nominal Boolean networks.