Multicriteria Framework for Robust-Stochastic Formulations of Optimization under Uncertainty

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1 Multicriteria Framework for Robust-Stochastic Formulations of Optimization under Uncertainty Alexander Engau Mathematical and Statistical Sciences University of Colorado Denver th Street, Suite 600 Denver, CO 80202, USA 2013 INFORMS ANNUAL MEETING MINNEAPOLIS CONVENTION CENTER OCTOBER 6-9, 2013 Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 1 of 1

2 Optimization Under Uncertainty (possible interpretations) Optimization problems or methods with elements of uncertainity 1. Stochastic optimization: random input to objective / constraints uses statistical inference to compute statistically optimal solutions assumes a probability distribution and optimizes in expectation chance-constraints, recourse, sampling, scenarios, simulation 2. Robust optimization: optimizes worst-case over uncertainty set 3. Parametric programming: solution stability / sensitivity analysis 4. Probabilistic or heuristic methods and randomized algorithms 5. Fuzzy optimization: imprecise quantification of set membership 6. Multicriteria optimization/decision-making: unknown preference Different concepts use different notions of feasibility or optimality! Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 2 of 1

3 Optimization Under Uncertainty (scope of this talk) Let u U be a continuous or discrete random variable or vector: minimize f (x, u) subject to x X(u) Two possibilities to handle uncertainty in the constraints: Consider the deterministic feasible set: X = u U X(u) Define probabilistic chance constraints: P(x X) 1 ɛ Two possibilities to handle uncertainty in the objective: Robust approach: optimize the worst-case { } minimize max f (x, u): u U subject to x X Stochastic approach: optimize the expectation minimize E U [f (x, u)] = w(u)f (x, u)du subject to x X u U Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 3 of 1

4 Multicriteria Approach to Optimization Under Uncertainty Formulate multiobjective deterministic equivalent program (DEP): { } minimize f (x, u): u U subject to x X Three-way classification based on underlying uncertainty set U finite and discrete: multiple-objective programming (MOP) optimization in R d (outcomes are finite-dimensional real vectors) infinitely discrete: infinitely-many-objective program (IMOP) optimization in l p for p (outcomes are real sequences) continuous: uncountably-many-objective program (UMOP) optimization in L p (outcomes are real-valued functions) Definition (Efficiency under Uncertainty) A point x X is said to be (weakly, properly) efficient under uncertainty if it is (weakly, properly) Pareto optimal for DEP. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 4 of 1

5 Illustration of Efficiency Concepts for a Biobjective Program (BOP) Pareto Points prevent improvement Weak Pareto Points prevent strict improvement Proper Pareto Points prevent unbounded rates of substitution Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 5 of 1

6 IMOPs, UMOPs, and Efficiency Concepts at INFORMS 2013 SA38, Margaret Wiecek, Erin Doolittle, Hervé Kerivin: Preserving Efficiency in Multiobjective Programming under Uncertainty Observation that the addition of (possibly infinite numbers of) new objectives may not alter the Pareto efficient sets. The original objectives and the additional objectives are related through one or more algebraic operations. SA39, MA07, Dan Iancu, Nikos Trichakis Pareto Efficiency in Robust Optimization Observation that robust solutions are only weakly efficient. Suggestion of a direction-type method to test and achieve Pareto efficiency (similar to Benson s method). WA49, Pekka Korhonen (tomorrow morning - you can still make it!) Tutorial: Efficiency Concepts in MCDM, DEA, and Finance Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 6 of 1

7 Finite and Discrete Case: Multiobjective Programming Let I = {1, 2,..., m}, U = {u i : i I}, and f i (x) := f (x, u i ): minimize (f 1 (x), f 2 (x),..., f m (x)) subject to x X Definition (Pareto Optimality and Ideal / Utopia Points) A point x in the set of feasible decisions X is called weakly Pareto optimal: f i (x ) > f i (x) f j (x ) f j (x) Pareto optimal (Pareto 1896): f i (x ) > f i (x) f j (x ) < f j (x) properly Pareto optimal (Geoffrion 1968): there exists M < f i (x ) > f i (x) f i (x ) f i (x) < M(f j (x) f j (x )) A point z or z in the outcome space R m is called ideal point: z := (z1, z 2,..., z m) with zi = min {f i (x): x X} utopia point: z < z or z = z ɛ Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 7 of 1

8 Scalarizations from Compromise Programming (Zeleny 1973) Let w 0 be a vector of weights and z be an ideal or utopia point: ( minimize (w i(f i (x) z i )) p) 1/p subject to x X i I p = 1: weighted sum (Geoffrion 1968) minimize i I w if i (x) subject to x X p = : weighted Tchebycheff norm (Bowman 1976) minimize max {v i (f i (x) z i )} subject to x X i I augmented Tchebycheff norm (Atkins, Choo, Steuer 1983) minimize max{v i (f i (x) z i )} + α w if i (x) subject to x X i I i I Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 8 of 1

9 Characterizations of Pareto Optimality/Efficiency Compromise programming with nonnegative weights: Unique optimal solutions are efficient. Optimal solutions with w > 0 are efficient. Optimal solutions with w 0 are weakly efficient. Tchebycheff norm with nonnegative weights and utopia points: Solutions are weakly efficient if and only if they are optimal (for some suitable weight vector v 0 and a utopia point z). Weighted-sum with nonnegative or strictly positive weights: Optimal solutions with w > 0 are properly efficient. Sufficient conditions are also necessary if problem is convex. Augmented weighted Tchebycheff norm with positive weights: Solutions are properly efficient if and only if they are optimal (for suitable weights v > 0, small α > 0, and a utopia point z). Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 9 of 1

10 Infinite and Discrete Case: Infinitely-Many-Objective Programming Let I = N = {1, 2, 3,...}, U = {u i : i I}, and f i (x) := f (x, u i ): minimize (f 1 (x), f 2 (x), f 3 (x),...) subject to x X Outcomes and utopia points are elements in sequence space l p. Weighted sum becomes a weighted (infinite) series: minimize i=1 w if i (x) = lim k k i=1 w if i (x) subject to x X Tchebycheff (max) norm becomes supremum norm: minimize sup{v i (f i (x) z i )} subject to x X Compromise programming results for (weak) efficiency carry over. Weighted-series results follow from the Hahn-Banach Theorem. Results for proper efficiency do not hold under limiting behavior. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 10 of 1

11 Counterexample: Strictly positive weights give improper solutions Let I = {0, 1, 2...}, X = [0, 1], f 0 (x) = 1 x, f i (x) = x i for i 1. All x [0, 1] are efficient (f 0 is decreasing, f i are increasing). All x can be generated using a weighted series (technical): w 0 = 1 + log(1 r) > 0 w i = r i /i > 0 for i 1 x(r) = 1 + log(1 r) r (1 + log(1 r))r x = 1 is not properly efficient. Consequence for Stochastic Optimization: Efficient solutions in expectation may still allow for unbounded rates of substitution. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 11 of 1

12 Proper Efficiency for Countably Many Objectives Geoffrion: propose a slightly restricted definition of efficiency that (a) eliminates efficient points of a certain anomalous type; and (b) lends itself to more satisfactory characterization. Remark on improper points: because there is but a finite number of criteria, the marginal gain for some criterion can be arbitrarily large relative to each of the marginal losses in other criteria. New Definition of Proper Efficiency in the Sense of Geoffrion Above intention is maintained if for each i, there exist M i < f i (x ) > f i (x) f i (x ) f i (x) < M i (f j (x) f j (x )) If I is finite, M = max{m i : i I} recovers the original definition. If I is infinite, this new definition maintains the characterization of proper efficiency by weighted series and Tchebycheff norms. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 12 of 1

13 Continuous Case: Uncountably-Many-Objective Programming Let U R d be a compact (closed and bounded) uncertainty set: ( ) minimize f (x, u): u U subject to x X Outcomes and utopia points are elements in function space L p. Weighted sum becomes a weighted integral: minimize w(u)f (x, u)du subject to x X u U (here w(u) is measurable Radon-Nikodym derivative / density) Tchebycheff norm becomes uniform (sup/max) norm: { } minimize max v(u)(f (x, u) z(u)) subject to x X u U (here v(u) and z(u) are weight and ideal or utopia functions) Compromise programming results for (weak) efficiency still hold! Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 13 of 1

14 Geoffrion Proper Efficiency for Uncountably Many Objectives Winkler (2004) considers a compact set T X and Y C(T ): minimize y(t) subject to y Y (analogous to UMOP with T = U, Y = {y(t) := f (x, t) = f (x, u)}) ȳ is efficient: ȳ(t) > y(t) ȳ(t 0 ) < y(t 0 ) properly efficient: ȳ(t) > y(t) ȳ(t) y(t) < δ(y(t 0 ) ȳ(t 0 )) All characterizations fail again! X = {x 1}, T = [0, 1], w = 1 { (x + 1) 2 y x (t) = min t 1, 1 } 2x x Then T y x(t)dt = 0 for all y Y but ȳ = 0 is not properly efficient. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 14 of 1

15 Almost Proper Efficiency for Uncountably Many Objectives Solutions generated by weighting methods are proper efficient a.e. New Definition of Proper Efficiency Almost Everywhere Element ȳ is properly efficient a.e. if for every ɛ > 0, there is δ > 0 ȳ(t) > y(t) ȳ(t) y(t) < δ(y(t 0 ) ȳ(t 0 )) for all t T \ V with L(V ) < ɛ (here L is the Lebesgue measure). With the above definition, we could prove the following results: We could also give counterexamples for reversed implications. Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 15 of 1

16 Consequences and Discussion The theory and methods of multicriteria optimization provide a new framework for standard optimization problems under uncertainty. Weighted sums, series, or integrals for stochastic formulations. Tchebycheff and supremum norms for robust formulations. Multicriteria decision-making can help analyze such problems. Stochastic uncertainty corresponds to preferential uncertainty. Statistical inference corresponds to preference articulation. There are interesting new theoretical and practical implications. No characterization of proper efficiency in infinite dimensions. Stochastic and robust solutions may allow unbounded tradeoffs. Questions and comments are always welcome! Alexander Engau University of Colorado Denver Multicriteria Framework for Robust-Stochastic Optimization 16 of 1