An alternating direction method of multipliers for elliptic equation constrained optimization problem SCIENCE CHINA Mathematics 60, 361 (2017); A new look at the Lagrange method for continuous-time stochastic optimization SCIENCE CHINA Mathematics 55, 2247 (2012); Continuity for vector optimization problems with equilibrium constraints Science in China Series A-Mathematics 47, 65 (2004); On the existence and regularity of vector solutions for quasilinear systems with linear coupling

By means of the Minkowski-type nonlinear scalarization functional, in this paper, we establish some nonlinear separation theorems via relative algebraic interior and vector closure in a general real linear space, via relative topological interior and topological closure in a real topological linear space and via quasi relative interior and topological closure in a real separated locally convex topological linear space, respectively. These new separation theorems can be applied to study those vector optimization problems with the ordering cones having possibly empty topological interior and even relative topological interior or relative algebraic interior. As their applications, we give some nonlinear scalarization characterizations of the corresponding weak efficient solutions of vector optimization problems. Moreover, we also present some concrete examples to illustrate the main results in some infinite dimensional spaces.

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α, ρ, d)-凸性假设下给出了相应的对偶定理. 本文考虑如下不可微多目标高阶对称对偶问题: