Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Chance constraints are an important modeling tool in stochastic optimization, providing probabilistic guarantees that a solution “succeeds” in satisfying a given constraint. Although they control the probability of “success,” they provide no control whatsoever in the event of a “failure.” That is, they do not distinguish beten a slight overshoot or undershoot of the bounds and more catastrophic violation. In short, they do not capture the magnitude of violation of the bounds.  addresses precisely this topic, focusing on linear constraints and ellipsoidal (Gaussian-like) uncertainties.  show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi-infinite optimization problem.  provide conditions that guarantee polynomial-time solvability of the resulting semi-infinite formulation.  show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature. As a byproduct,  provide tight probabilistic bounds for comprehensive robust optimization. Thus, analogously to the connection beten chance constraints and robust optimization,  provide a broader connection beten probabilistic envelope constraints and comprehensive robust optimization.
['Huan Xu', 'Constantine Caramanis', 'Shie Mannor']
I am writing a research paper on a idea Optimization Under Probabilistic Envelope Constraints
[{"from": "human", "value": "I am writing a research paper on a idea Optimization Under Probabilistic Envelope Constraints"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Chance constraints are an important modeling tool in stochastic optimization, providing probabilistic guarantees that a solution \u201csucceeds\u201d in satisfying a given constraint. Although they control the probability of \u201csuccess,\u201d they provide no control whatsoever in the event of a \u201cfailure.\u201d That is, they do not distinguish beten a slight overshoot or undershoot of the bounds and more catastrophic violation. In short, they do not capture the magnitude of violation of the bounds.  addresses precisely this topic, focusing on linear constraints and ellipsoidal (Gaussian-like) uncertainties.  show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi-infinite optimization problem.  provide conditions that guarantee polynomial-time solvability of the resulting semi-infinite formulation.  show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature. As a byproduct,  provide tight probabilistic bounds for comprehensive robust optimization. Thus, analogously to the connection beten chance constraints and robust optimization,  provide a broader connection beten probabilistic envelope constraints and comprehensive robust optimization."}]