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In this paper, we consider an uncertain convex optimization problem with a robust convex feasible set described by locally Lipschitz constraints. Using robust optimization approach, we give some new characterizations of robust solution sets of the problem. Such characterizations are expressed in terms of convex subdifferentails, Clarke subdifferentials, and Lagrange multipliers. In order to characterize the solution set, we first introduce the so-called pseudo Lagrangian function and establish constant pseudo Lagrangian-type property for the robust solution set. We then used to derive Lagrange multiplier-based characterizations of robust solution set. By means of linear scalarization, the results are applied to derive characterizations of weakly and properly robust efficient solution sets of convex multi-objective optimization problems with data uncertainty. Some examples are given to illustrate the significance of the results.

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This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.