A Set-Valued Optimization Approach for uncertain Decision-Making in Autonomous Engineering Systems

Abstract

Autonomous engineering systems make decisions prior to disturbance, perception error, and model uncertainties becoming known. In such a situation, a decision leads not to a single objective vector but to a whole collection of objective vectors that can be achieved. The current paper presents a mathematically precise set-valued minimax framework for such decisions. A compact uncertainty set is transformed into a closed upper image in a finite dimensional ordered space; robust preferences are defined via the lower set relation generated by a pointed convex cone. The resulting problem minimizes an outcome set rather than a prechosen weighted sum of the objectives. A dual-cone scalarization theorem shows that each weakly robust minimal solution is supported by a scalar minimax problem, and the converse holds, too. Solvability is guaranteed by compactness and Painlevé-Kuratowski outer semicontinuity of the mapping under consideration. Given convex-concave-continuous assumptions, a minimax Sion equality is proven for each scalarization of the robust decision problem. Necessary and sufficient first-order optimality conditions are stated in the form of a set-valued variational inequality involving contingent derivatives and normal cones. A regularized projected primal-dual algorithm is formulated; Fejér monotonicity and an ergodic gap bound of O(1/N) are established for monotone Lipschitz mappings. Robustness of the solution multifunction is characterized in terms of the Hausdorff stability of the uncertainty set. An example of a two-dimensional autonomous navigation problem is provided to illustrate the trade-off between tracking error, energy cost, collision risk, and control effort that can be achieved by the proposed formulation. The computational experiment employs synthetic data for bounded disturbance set and thus is a proof-of-concept for the mathematical approach rather than field validation. Over 200 test cases, the suggested algorithm outperforms both the nominal and the average scenario solutions in terms of worst-case objective and constraint violations, preserving explicit robustness guarantees.