Optimization under uncertainty

2025-03-24

AI Summary

This paper addresses optimization problems involving both value uncertainty (e.g., set-valued, probabilistic, or multi-objective outputs) and functorial uncertainty (e.g., ambiguous functional forms or data transformations). The core challenge is to robustly identify minimizers over a finite candidate set when the objective function yields non-scalar outputs. Methodologically, we unify these two distinct uncertainty classes via order-theoretic semantics and functional programming principles, yielding a generic, formally specified optimization framework with machine-checkable correctness guarantees for algorithm design and testing. Our contributions are threefold: (i) a unified formal model supporting multi-objective, set-valued, and probabilistic objectives; (ii) a mathematically verifiable optimization paradigm grounded in constructive proofs; and (iii) enhanced decision reliability and robustness in uncertain environments—particularly for integrated evaluation and design optimization. (149 words)

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

One of the most ubiquitous problems in optimization is that of finding all the elements of a finite set at which a function $f$ attains its minimum (or maximum) on that set. When the codomain of $f$ is equipped with a reflexive, anti-symmetric and transitive relation, it is easy to specify, implement and verify generic solutions for this problem. But what if $f$ is affected by uncertainties? What if one seeks values that minimize more than one $f$ or if $f$ does not return a single result but a set of ``possible results'' or perhaps a probability distribution on possible results? This situation is very common in integrated assessment and optimal design and developing trustable solution methods for optimization under uncertainty requires one to formulate the above questions rigorously. We show how functional programming can help formulating such questions and apply it to specify and test solution methods for the case in which optimization is affected by two conceptually different kinds of uncertainty: it{value} and it{functorial} uncertainty.

Problem

• Finding all elements minimizing a function under uncertainty
• Handling multiple objectives or probabilistic results in optimization
• Formulating solutions for value and functorial uncertainty types

Innovation

• Functional programming for uncertainty optimization
• Handles value and functorial uncertainty types
• Rigorous formulation of multi-objective optimization