讲座内容：A key step in solving minimax distributionally robust optimization (DRO) problems is to reformulate the inner maximization w.r.t. probability measure as a semi-infinite programming problem
through Lagrange dual. Slater type conditions have been widely used for zero dual gap when the ambiguity
set is defined through moments. In this paper, we investigate effective ways for verifying the Slater type
conditions and introduce other conditions which are based on lower semi-continuity of the optimal value
function of the inner maximization problem. Moreover, we apply a random discretization scheme to
approximate the semi-infinite constraints of the dual problem and demonstrate equivalence of the approach to
random discretization of the ambiguity set. Two cutting plane schemes are consequently proposed: one for the
discretized dualized DRO and the other for the minimax DRO with discretized ambiguity set. Convergence
analysis is presented for the approximation schemes in terms of the optimal value, optimal solutions and
stationary points. Numerical results are reported for the resulting algorithms.
