Optimization Problems with Linear Chance Constraints Structure, Numerics and Applications R. Henrion Weierstraß-Institut Berlin Introduction with Examples Structure (convexity, differentiability, existence of solutions) Numerics (singular normal distributions) 75. Sitzung der GOR Arbeitsgruppe Praxis der Mathematischen Optimierung Optimization under Uncertainty Bad Honnef, October 20-21, 2005
Chance Constraints system of stochastic inequalities g x, 0 random parameter (meteorological data, product demands, prices, return rate) 'here-and-now'- decisions: x deterministic reformulations: a) expected-value constraint: Egx, 0 or g x, E 0 b) worst-case constraint: g x, 0 c) chance-constraint: P g x, 0 p p0,1
Portfolio-Optimization (1) Distribute capital K among n assets with random return. simplest model: maximize expected return! x i capital put on asset i, i return rate for asset i random vector, assumption: N, normally distributed linear optimization problem: 0.5 0.5 Example: 1.2,1.3, 0.5 0.5 n max,x i1 x i K, x i 0 Distribution of Return probability of loss: 33.4% expected value
random return:, x Portfolio-Optimization (2) probability of surplus: optimization problem with chance constraints: max,x i1 n x i K, x i 0, P,x K p P, xk p0,1 expected return p=0.9 p=0.8 p=0.7 probability of surplus
A Problem from Chemical Engineering R.H., P. Li, A. Möller, M. Wendt, G. Wozny (Comp.&Math. Appl. 2003) Distillation (Methanole-Water) Modeled by large DAE-System continuous (stochastic) inflow level constraints (as chance constraints)
Optimal Control feed rate F heating Q reflux R p = 0.90/0.99 8 selected state variables in 20 trays of column
Simulated Inflow Profiles Application of optimal feed rate to 100 simulated inflow profiles (Gaussian Process) p = 0.99 optimal feed rate supposing the expected inflow profile Optimal value (OV) of the objective versus safety level (p) OV p
Stochastic Reservoir Constraints 1,..., n x x 1,..., x n l 0 F discretized stochastic inflow to reservoir discretized control of feed extraction from reservoir initial filling level of reservoir upper filling level allowed for reservoir Stochastic reservoir constraints: i l 0 j x j j1 j1 fillinglevelattimei i F i1,...,n Chance Constraints: P L x p regular, triangular
Capacity Expansion in Stochastic Networks Graph with stochastic load in the nodes and with capacities x in the arcs 1 x 1 1 x 1 2 x 2 x 1 + 3 2 x 2 2 1 2 x 3 Chance Constraint Model: P Ax p A not surjective! number of inequalities larger than dimension of random vector
3 Basic Types of Chance Constraints (CC) CC with stochastic coefficient matrix: P xa p (e.g., portfolio optimization, mixture problems) CC with separated randomness: P Ax p a) regular case: A surjective (e.g., stochastic reservoir-constraints) b) singular case: A not surjective occurs when number of inequalities exceeds dimension of random vector (e.g., networks with stochastic demand, robotics: collision avoidance with random obstacles)
Structural Properties of CCs with Stochastic Coefficient Matrix R.H. (SPEPS 2005) Theorem: (Van de Panne/Popp, Kataoka, 1963) Let N,, positive definite. Then, M: x n Px, p is convex for p0.5. Extended Model: M p : x n Pqx, p
An extended convexity result M p : x n Pqx, p Theorem: Let have an ellipticallysymmetric distribution with density and parameters, positive definite.. Under each! of the following assumptions: 1. q is affinelinear or 2. q i 0, convex, i 0, i,j 0 it holds that M p is convex for alle and all p0.5.
Exact domains of convexity, nontriviality and compactness M p : x n Pqx, p, N,, positive definite : 10.5 Theorem: Let q be affine linear a n d regular. Then, domain of convexity domain of nontriviality 0.5 0.5 0.5 p 0.5 0.5 0.5 p domain of compactness 0.5 0.5 0.5
Existence of Solutions to Problems with Stochastic Coefficient Matrix Optimization problem: P min f xp q xa p p0,1 Theorem: f lower semicontinuous i N i, i, i rows of, i positive definite a 0 q homeomorphism e.g., q xx P has a solution for p min i i 1 i 1
Singular Normal Distributions and Quasiconcave Measures CC with separated randomness: P Ax p P Ax p F x p with : A distribution function of Special case: normal distribution: N, N A, A A T regular regular case: A surjective A A T regular singular case: A not surjective A A T singular F is a singular normal distribution
Lipschitz continuity of quasiconcave Measures R.H., W. Römisch (Ann. Oper. Res. 2005) Theorem: Let have a quasiconcave distribution (e.g., regular or singular normal, uniform, Pareto, Dirchlet, lognormal, Gamma etc.). Then, the distribution function of is (Lipschitz) continuous iff Var i 0 i. Example: Singular normal distribution with covariance matrix of rank 1 C 1 0 0 0 C 1 1 1 1
Reduction of singular normal distribution functions to regular ones J. Bukszar, R.H., M. Hujter, T. Szantai (SPEPS 2004) N,, AA T, M xuaux, I u,xi a i, ux i singular Cholesky-decomposition induced polyhedron active index set at u I M x I u M x: I u,xi family of all active index sets x is called nondegenerate, if rank a i ii I I I M x Theorem: If x nondegenerate F x 1 I I M x 1 I F I x I I N I, A I A I T regular singular normal distribution sum of regular normal distributions
If F I Gradient Formula and Algorithm x nondegenerate F x 1 I I M x 1 I F I x I smooth regular normal distribution function x nondegenerate I M x locally constant Corollary: If x nondegenerate F x I I M x 1 I1 F I x I Algorithm: 1. Determine all corners of M x Fukuda, Scdd 2. Compute I Mx as family of subsets of corner index sets 3. Calculate the F by formula from regular normal distributions F I Genz, Szantai 4. Calculate gradients by reduction t o functional values
Numerical Tests Dim: 5 x 10 Prob.: 0.982894 Error: 0.000001 Poly: 1.00 Szantai: 1599.89 Deak: 24579.00 MC: 1715262.63 Genz: 2236403.28 Dim: 5 x 15 Prob.: 0.942901 Error: 0.000002 Poly: 1.00 Deak: 16200.45 Szantai: 105322.83 MC: 720219.79 Genz: 1422991.07 Dim: 5 x 20 Prob.: 0.947281 Error: 0.000002 Poly: 1.00 Deak: 7594.09 Szantai: 27126.86 MC: 360681.66 Genz: 614578.55 Dim: 5 x 25 Prob.: 0.954225 Error: 0.000002 Poly: 1.00 Szantai: 597.93 Deak: 960.76 MC: 48868.45 Genz: 58847.63 Dim: 10 x 20 Prob.: 0.974131 Error: 0.000001 Poly: 1.00 Szantai: 174.41 Deak: 217.08 MC: 1888.59 Genz: 3852.22 Dim: 10 x 25 Prob.: 0.977193 Error: 0.000001 Poly: 1.00 Szantai: 16.08 Deak: 67.87 MC: 601.72 Genz: 1583.00 Dim: 15 x 20 Prob.: 0.940839 Error: 0.000003 Poly: 1.00 Szantai: 2.79 Deak: 4.37 MC: 16.68 Genz: 21.22 Dim: 15 x 25 Prob.: 0.987551 Error: 0.000003 Szantai: 1.00 Deak: 55.42 MC: 204.00 Genz: 373.16 Poly: not available
Stabilität in Problemen mit separiertem Zufall R.H., W. Römisch (Math. Prog. 2004) gegebenes Optimierungsproblem: P min f xx X, F Ax p Verteilung von oft unbekannt. Approximation durch z.b. auf der Basis historischer Daten. Stabilität von Lösungen und Optimalwerten in (P) bei kleinen Störungen? Maß für Störung (Kolmogorov-Abstand): d K, sup z nf z F z Lösungsmengenabbildung: argmin f xx X,F Ax p Optimalwertfunktion: inf f xx X,F Ax p Originalproblem
Qualitative und quantitative Stabilität P min f xx X,F Ax p f konvex, X konvexundabgeschlossen, log F konvex Theorem: für viele multivariate Verteilungen erfüllt (Prekopa) Lösungsmenge von (P) nichtleer und beschränkt x X: F Ax p Slaterpunkt 1. ist oberhalbstetig i n 2. Ld K, lokal Falls zusätzlich: f linearquadratisch, X Polyeder, log F stark konvex 3. d H, Ld K, 12 lokal
Exponentielle Schranken für empirische Approximationen Ausgangsproblem: min f xx X,F Ax p Sei N eine Folge unabhängiger Zufallsvektoren. N Empirische Approximation: N : N 1 i1 i Dvoretzky-Kiefer-Wolfowitz Ungleichung: P d K N,C 1 expc 2 2 N Unter den Voraussetzungen des qualitativen Stabilitätsresultates folgt: 1 0 x N N : Pdx N, N 1. 2 0, 0 : P N 2C 1 expc 2 N 2 L 2 Unter den Voraussetzungen des quantitativen Stabilitätsresultates folgt: 3 0, 0 : Pd H N, 2C 1 expc 2 N 4 L 4