New a priori and a posteriori probabilistic bounds for robust counterpart optimization: I. Unknown probability distributions

• New probabilistic bounds are derived for robust counterpart optimization with bounded uncertainty.
• The new bounds improve upon existing methods and extend the robust counterpart optimization framework.
• The reduction of conservatism is demonstrated through LP and MILP computational case studies.

Optimization problems often have a subset of parameters whose values are not known exactly or have yet to be realized. Nominal solutions to models under uncertainty can be infeasible or yield overly optimistic objective function values given the actual parameter realizations. Worst-case robust optimization guarantees feasibility but yields overly conservative objective function values. The use of probabilistic guarantees greatly improves the performance of robust counterpart optimization. We present new a priori and a posteriori probabilistic bounds which improve upon existing methods applied to models with uncertain parameters whose possible realizations are bounded and subject to unspecified probability distributions. We also provide new a priori and a posteriori bounds which, for the first time, permit robust counterpart optimization of models with parameters whose means are only known to lie within some range of values. The utility of the bounds is demonstrated through computational case studies involving a mixed-integer linear optimization problem and a linear multiperiod planning problem. These bounds reduce the conservatism, improve the performance, and augment the applicability of robust counterpart optimization.