This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem.

Two-stage risk-neutral stochastic optimization problem has been widely studied recently. The goals of our research are to construct a two-stage distributionally robust optimization model with risk aversion and to extend it to multi-stage case. We use a coherent risk measure, Conditional Value-at-Risk, to describe risk. Due to the computational complexity of the nonlinear objective function of the proposed model, two decomposition methods based on cutting planes algorithm are proposed to solve the two-stage and multi-stage distributional robust optimization problems, respectively. To verify the validity of the two models, we give two applications on multi-product assembly problem and portfolio selection problem, respectively. Compared with the risk-neutral stochastic optimization models, the proposed models are more robust.

A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if d _G (u, v) ≠ d _G (u, w) . A subset L _m of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of L _m . The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dim _m (G). In this paper, we studied the mixed metric dimension for three families of graphs D _n , A _n , and R _n , known from the literature. We proved that, for D _n the dim _m (D _n ) = dim _e (D _n ) = dim(D _n ), when n is even, and for An the dim _m (A _n ) = dim _e (A _n ), when n is even and odd. The graph R _n has mixed metric dimension 5.