In this paper, we consider approximate solutions (also called $\varepsilon$-solutions) for semi-infinite optimization problems that objective function and constraint functions with uncertainty data are all convex, and establish robust counterpart of convex semi-infinite program and then consider approximate solutions for its. Moreover, the robust necessary condition and robust sufficient theorems are obtained. Then the duality results of the Lagrangian dual approximate solution is given by the robust optimization approach under a cone constraint qualification.

Based on the definition of approximate solution (also known as epsilon -solution), we propose the idea of approximate generalized convexity of semi -infinite optimization in this research. The nonsmooth sufficient and necessary criteria of the resilient approximate solutions are then provided. Additionally, it is demonstrated that the approximative generalized convexity assump-tion holds for both the strong and weak robust dual theorems. The efficient cardinality/mean-variance portfolio is where we finally implement this idea.

In this paper, we propose a new adaptive method for solving nonlinear semi-infinite programming(SIP). In the presented method, the continuous infinite inequality constraints are transformed into equivalent equality constraints in integral form. Based on penalty method and trust region strategy, we propose a modified quadratic subproblem, in which an adaptive parameter is considered. The acceptable criterion of the trial point is adjustable according to the value of this adaptive parameter and the improvements that made by the current iteration. Compared with the existing methods, our method is more flexible. Under some reasonable conditions, the convergent properties of the proposed algorithm are proved. The numerical results are reported in the end.

In this paper, we mainly study the dual problem of the nonconvex semi-infinite programming problem with inequality constraints. It is well known that there is usually a duality gap in constructing duality problem using the ordinary Lagrangian function. To eliminate the duality gap, we construct an augmented Lagrangian function, then discuss its duality. Under reasonable assumptions, the strong duality theorem between the primal problem and the augmented Lagrangian dual problem holds. Finally, an example is given to verify the presented results.

求解极大极小问题的灵活非单调滤子方法与传统的滤子方法相比,对于试探步的可接受性,该方法具有更大的灵活性,而且与单调型方法相比,计算量更小.此外,还利用一个自适应参数来调整接受准则,从而在一定程度上避免了Maratos效应.在合理的假设下,该算法具有全局收敛性,并且通过数值实验验证了该方法的有效性.