When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?

Area of Science:

• Optimization Theory
• Operations Research
• Mathematical Modeling

Background:

• Adjustable robust optimization (ARO) offers superior worst-case solutions compared to static robust optimization (RO).
• ARO presents significant computational challenges, often exceeding those of RO.
• Bridging the gap between solution quality and computational tractability in robust optimization is crucial.

Purpose of the Study:

• To establish conditions under which ARO and RO yield equivalent worst-case objective values.
• To identify scenarios where static robust solutions are optimal for ARO problems.
• To simplify decision rules in ARO by exploring dependencies on uncertain parameters.

Main Methods:

• Mathematical proofs analyzing the relationship between ARO and RO under specific assumptions.
• Convexity and concavity analysis with respect to adjustable variables and uncertain parameters.
• Investigation of decision rules, including affine rules, and their impact on optimal objective values.

Main Results:

• When uncertainty is constraint-wise and problem structures meet convexity/concavity criteria, RO solutions are optimal for ARO.
• For mixed constraint-wise and non-constraint-wise uncertainties, optimal ARO decision rules may not depend on constraint-wise parameters.
• Affine decision rules depending on all uncertain parameters yield the same objective value as those depending only on non-constraint-wise parameters for certain problems.

Conclusions:

• Conditions are identified for ARO and RO to achieve identical worst-case objective values, simplifying complex optimization.
• The findings enable the development of more computationally tractable ARO solutions, particularly when dealing with mixed uncertainty types.
• The study demonstrates practical applicability to convex quadratic and conic quadratic problems, enhancing robust optimization methodologies.