Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making under severe uncertainty: from worst-case analysis to robust optimization Moshe Sniedovich School of Mathematics and Statistics The University of Melbourne www.moshe-online.com AMSI Optimise 18 June 18-22, 2018 The University of Melbourne, Melbourne, Australia 1/36 A version can be found in the chapter From statistical decision theory to robust optimization: a maximin perspective on robust decision-making (Sniedovich 2016). Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Warning: Mathematical content This is a presentation. 2/36 Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Warning: Mathematical content This is a presentation. A version can be found in the chapter From statistical decision theory to robust optimization: a maximin perspective on robust decision-making (Sniedovich 2016). 2/36 Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme 3/36 Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty 3/36 Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) 3/36 Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm 3/36 Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) 3/36 Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm 3/36 Local vs global WCA Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm 3/36 Conservatism issue Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA 3/36 Summary and conclusions Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue 3/36 Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme Decision making under severe uncertainty Worst-case analysis (WCA) Wald’s maximin paradigm Robust optimization (RO) Relationship between WCA, RO and Wald’s paradigm Role of constraints in RO, WCA and Wald’s paradigm Local vs global WCA Conservatism issue Summary and conclusions 3/36 Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Programme No need to take notes. A copy of the presentation will be available online next week and/or by request. 4/36 Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty 5/36 Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework 5/36 Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. 5/36 Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task 5/36 Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. 5/36 The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty 5/36 Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Conceptual framework Decision apace: D (set of all available decisions, d ∈ D) Uncertainty space: U (set of all values of the uncertainty parameter under consideration, u ∈ U ) Objective functions: O1, O2,..., Ok. Constraints: con1, con2,..., conm. Task Find the best decision d ∈ D given that the ‘true’ value of the uncertainty parameter u ∈ U is unknown. Difficulty The task is ill-defined (except for trivial problems). Note that the decision is made before the ‘true’ value of u is revealed. 5/36 Conventional Linear Programming max {cT x : Ax ≤ b, x ≥ 0} x vs ‘Uncertain’ Linear Programming max {cT (u)x : A(u)x ≤ b(u), x ≥ 0} x If u is unknown then the ‘uncertain’ LP problem is ill-defined. Difficulty Under uncertainty, the set of optimal solutions may depend on the assumed value of the uncertainty parameter. Introduction DMUSU Uncertainty WCA Wald RO Conservatism Summary Decision-making Under Severe Uncertainty Example: linear programming problems 6/36 vs ‘Uncertain’ Linear Programming max {cT (u)x : A(u)x ≤ b(u), x ≥ 0} x If u is unknown then the ‘uncertain’ LP problem is ill-defined. Difficulty Under uncertainty, the set of optimal solutions may depend on the assumed value of the uncertainty