Robust Dual-Response Optimization
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1 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 1 / 24 Robust Dual-Response Optimization İhsan Yanıkoğlu, Dick den Hertog, Jack P.C. Kleijnen Özyeğin University, İstanbul, Turkey CMS Conference May 1 June
2 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 2 / 24 Robust Optimization Introduction Robust Optimization Optimization problems often contain uncertain parameters due to errors in estimation, implementation and measurement. The goal of Robust Optimization (RO) is to find solutions that are immune to uncertainty of parameters in a given mathematical optimization problem. Paradigm: use uncertainty set U for a, and solve the robust counterpart problem: min x {f (x) g i (x, a) 0 a U, i}.
3 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 3 / 24 Literature on RO Introduction Robust Optimization Soyster (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. (Oper. Res.) Started in the late 90s with Ben-Tal, El Ghaoui and Nemirovski. Bertsimas and Sim (2004). The price of robustness. (Oper. Res.) Ben-Tal, El Ghaoui and Nemirovski (2009). Robust Optimization. Princeton Press. A Practical Guide to RO: Gorissen, Yanıkoğlu and den Hertog (2015). (Omega) Many practical applications in logistics, finance, engineering, etc.
4 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 4 / 24 Introduction Optimization using metamodels Optimization using metamodels
5 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 5 / 24 Main idea Introduction Main idea Metamodels often contain environmental uncertainties. Idea 1: find optimal solution that is robust against these uncertainties. Idea 2: find adjustable robust optimal solution. Methodology used: (Adjustable) Robust Optimization.
6 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 6 / 24 Introduction Main idea Taguchian regression metamodels We focus on Taguchian regression metamodels: y(e, d) = β 0 + β T d + d T Bd + γ T e + d T e + ɛ Decision factors (d): inputs under control of users (e.g., the number of forklifts, conveyors or shipping doors in a distribution center) Environmental factors (e): inputs not controlled by users (e.g., suppliers production interruption and the quantity variability) Quadratic in d, linear in e Low dimension in e and d However, same methodology can be applied to other metamodels (e.g., Kriging).
7 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 7 / 24 Introduction Main idea Regression output predictors: Expectation and variance Remember: y(e, d) = β 0 + β T d + d T Bd + γ T e + d T e + ɛ E e [y(e, d)] = β 0 + β T d + d T Bd + γ T µ e + d T µ e Var e [y(e, d)] = (γ T + d T )Cov(e)(γ + T d)
8 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 8 / 24 Introduction Main idea Robust Parameter Design vs Dual Response Optimization SNR: E[y]/ Var[y] Mean-Variance: min d E e [y(e, d)] s.t. Var e [y(e, d)] T Assumption 1: known mean and covariance of e Assumption 2: normally distributed e
9 Goal of this research Introduction Goal of this research 1. Develop Robust Optimization approach: that does not need these assumptions; that works with pure historical data; that can be used for many types of metamodels (e.g., polynomial regression, Kriging); 2. Develop Adjustable Robust Optimization approach: where optimal decisions are adjustable according to environmental factor(s) realization(s). Limitation: method is suitable for low dimensions. Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 9 / 24
10 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 10 / 24 Four steps Methodology Step 1: Discretize w.r.t. e; Step 2: Use historical data to define (φ-divergence) uncertainty set for e; Step 3: Reformulate Robust Counterpart problem into tractable one; Step 4: Solve the resulting problem and analyze the robust solution. Limitation: method is suitable for low dimensions.
11 Methodology Step 1 Step 1: Discretization of e Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 11 / 24 y i (d) = y(e i, d) = β 0 + β T d + d T Bd + (γ T + d T )e i Set of all cells: V = {1,..., m}
12 Methodology Step 1 Step 1: Discretization of e Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 12 / 24 Ê e [y(e, d)] = [β 0 + β T d + d T Bd + (γ T + d T )e i ]p i i V Var e [y(e, d)] = ψ i (d) 2 p i [ ψ i (d)p i ] 2 i V i V
13 Methodology Step 2 Preliminary: φ-divergence The φ-divergence ( distance ) between two vectors p = (p 1,..., p m ) 0 and q = (q 1,..., q m ) 0 is I φ (p, q) := m q i φ i=1 ( pi q i ), (1) where φ (t) is convex for t 0, φ (1) = 0, and φ (0/0) = 0. Ben-Tal et al. (2012) use φ-divergence as uncertainty set for uncertain probability vectors. Literature Ben-Tal, den Hertog, De Waegenaere, Melenberg and Rennen Robust solutions of optimization problems affected by uncertain probabilities. Management Science. Yanıkoğlu, İ., D. den Hertog Safe approximations of ambiguous chance constraints using historical data. INFORMS Journal on Computing. Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 13 / 24
14 Methodology Step 2 Preliminary: φ-divergence examples Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 14 / 24 φ-divergence Examples Divergence φ(t), t > 0 I φ (p, q) φ (s) ( ) pi Kullback-Leibler t log t p i log e s 1 q i i ( ) pi Burg entropy log t q i log 1 log( s), s 0 i χ 2 1 (p i q i ) 2 -distance t (t 1) s, s 1 p i i Pearson χ 2 -distance (t 1) 2 (p i q i ) 2 s + s 2 /4, s 2 q i 1, s < 2 i Hellinger distance (1 t) 2 ( p i q i ) 2 s 1 s, s 1 Remark: φ (s) := sup t 0 {st φ (t)} i q i
15 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 15 / 24 Preliminary (cont d) Methodology Step 2 Remember: I φ (p, q) := ( ) m i=1 q pi iφ q i We consider p = (p 1,..., p m ) as the unknown true probability vector of uncertain parameter e R L. q = (q 1,..., q m ) are observed frequencies in the historical data. Remember: We assume there are m cells, V := {1,..., m}.
16 Methodology Step 2 Step 2: φ-divergence uncertainty set Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 16 / 24 Using the chi-squared test statistic 2N φ (1) I φ (p, q), an approximate (1 α)-confidence set (U) for p is { } U := p R m : p 0, p T 1 = 1, I φ (p, q) ρ, where N is the sample size. ρ = ρ φ (N, m 1, α) := φ (1) 2N χ2 m 1,1 α,
17 Methodology Step 3 Step 3: Robust reformulation of the model Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 17 / 24 The robust reformulation of the expectation-variance (E&V) model min d E e [y(e, d)] s.t. Var e [y(e, d)] T is the following semi-infinite optimization problem: min d max Ê e [y(e, d)] p U s.t. Vare [y(e, d)] T p U. Remember: E e [y(e, d)] = β 0 + β T d + d T Bd + γ T µ e + d T µ e, Ê e [y(e, d)] = [β 0 + β T d + d T Bd + (γ T + d T )e i ]p i i V Var e [y(e, d)] = ψ i (d) 2 p i [ ψ i (d)p i ] 2 i V i V
18 Methodology Step 3 Step 3: Robust counterpart of E&V model (cont d) Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 18 / 24 (RC) min d s.t. max p U [β 0 + β T d + d T Bd + (γ T + d T )e i ]p i i V [ ] 2 ψ i (d) 2 p i ψ i (d)p i T p U, i V i V Remember: U := { } p R m : p 0, p T 1 = 1, I φ (p, q) ρ, ψ i (d) := (γ T + d T )e i
19 Methodology Step 3 Tractable robust counterpart of E&V model Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 19 / 24 Theorem The vector d solves (RC) if and only if d, λ, η, and z solve the following problem: min β ( ) 0 + β T d + d T Bd + λ 1 + ρη 1 + η 1 q i φ ψi (d) λ 1 d,λ,η,z i V i V ( s.t. λ 2 + ρη 2 + η 2 q i φ (ψ i (d) + z) 2 λ 2 η 2 Literature η 1, η 2 0 ) T Ben-Tal, A., D. den Hertog, J.-P. Vial Deriving robust counterparts of nonlinear uncertain inequalities. Mathematical Programming. Yanıkoğlu, İ., D. den Hertog Safe approximations of ambiguous chance constraints using historical data. INFORMS Journal on Computing. η 1
20 Methodology Step 4 Step 4: Solve and analyze the robust solution Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 20 / 24 T Robust Average Analysis Nominal Avg(E) Avg(Var) Avg(E) Avg(Var) Remark: max d E e [y(e, d)] s.t. Var e [y(e, d)] T
21 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 21 / 24 Illustrative example Methodology Step 4 General Problem: [ min E e (1 + 5d 1 + 5d 2 + e 1 e 2 ) 2 + (1 + 5d d 2 + e 1 + e 2 ) 2]. d Discretize w.r.t. e: e2 cell(2) (-0.5, 0.5) cell(3) cell(1) (0.5, 0.5) cell(4) e1 (-0.5, -0.5) (0.5, -0.5)
22 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 22 / 24 Illustrative example Methodology Step 4 Nominal Problem: min d 4 i=1 [ (1 + 5d1 + 5d 2 + e1 i e2 i ) 2 ( d1 + 10d 2 + e1 i + e2) i 2 ] q i. q = {0.4, 0.3, 0.2, 0.1} The optimal solution is attained at (d 1, d 2 )=(-0.08, -0.08)
23 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 23 / 24 Illustrative example Methodology Step 4 Robust Counterpart: min max d p P 4 i=1 [ (1 + 5d1 + 5d 2 + e1 i e2 i ) 2 ( d1 + 10d 2 + e1 i + e2) i 2 ] p i, { P := p = (p 1, p 2, p 3, p 4 ) R 4 p1 + p 2 + p 3 + p 4 = 1, 4 (p i q i ) 2 } i=1 0.5, p 0 p i The robust optimal solution is (-0.2, 0) with objective value 1 Remark: The nominal solution (-0.08, -0.08) has objective value 1.2 in the worstcase.
24 Yanıkoğlu, den Hertog, and Kleijnen Robust Dual-Response Optimization 29 May 1 June 24 / 24 Conclusions Conclusions Our approach has no assumptions on type of distribution and the value of its parameter(s). It directly uses historical data via φ-divergence. It leads to tractable robust counterpart problems. Can be extended to alternative metamodels, SNRs, and risk measures. Limitation: We may require a lot of data observations, especially when the number of uncertain parameters is high. Reference: Robust Dual-Response Optimization. IISE Transactions, 48(3), , (Best Paper Award: Operations Engineering & Analytics in 2017)
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