Optimization Problems with Probabilistic Constraints

Transcription

1 Optimization Problems with Probabilistic Constraints R. Henrion Weierstrass Institute Berlin 10 th International Conference on Stochastic Programming University of Arizona, Tucson

2 Recommended Reading A. Prekopa: Stochastic Programming Kluwer, Dordrecht, A. Prekopa: Probabilistic Programming Chapter 5 in A. Ruszczynski and A. Shapiro (eds.): Stochastic Programming. Handbooks in Operations Research and Management Science, Vol. 10, Elsevier, 2003.Amsterdam

3 Overview 1. Example 2. Models 3. Structure 4. Numerics 5. Stability

4 Probabilistic Constraints conventional optimization problem: min f x h j x 0 j 1,...,m In many real life problems from finance, engineering etc., the constraints involve random parameters due to demographic, meteorological, economical etc. uncertainties: h j x, 0 Often, a decision x has to be taken before is observed ('here and now'). No matter, how x is chosen, there is no guarantee that for all possible realizations of. h j x, 0 If the distribution of is known, we may calculate for each x the probability of constraint satisfaction: P h j x, 0 j 1,...,m reasonable: x feasible if this probability is larger than some safety level p. optimization problem with probabilistic constraints:: min f x P h j x 0 j 1,...,m p p 0,1 Applications: finance, power generation, water managment, telecommunication, chemical engineering etc.

5 1. Example

6 The cash matching problem 1 The pension fund of a company has to make payments for the next 15 years. Payments shall be covered by investing an initial capital K in bonds of 3 different types. Decision variables: x 1, x 2, x 3 - number of bonds of each type to be bought. Objective: Maximize final amount of cash (after 15 years) Constraints: cover payments in all years. 1 A. Ruszczynski ( and Dentcheva, Lai, Ruszczynski (2003)

7 Data and liquidity constraints year payments ,000 12,000 14,000 15,000 16,000 18,000 20,000 21,000 22,000 24,000 25,000 30,000 31,000 31,000 31,000 yields per bond of type cost per bond: ij cash available at the end of year j: 3 K i x i ik x i i 1 k 1 i 1 k 1 cash after buying bonds Rearrangement: a ij : k 1 3 i 1 K = 250,000 (initial capital) j j 3 cumulative yields of bonds ik i b j : k 1 liquidity constraints: a ij x i b j j k 0 cumulative payments j k K 0 j 1,...,15 j i

8 Linear optimization problem liquidity in year j: 3 i 1 a ij x i b j 0 final cash: 3 i 1 a i,15 x i b 15 optimization problem: 3 max i 1 3 a i,15 x i i 1 a ij x i b j j 1,...,15

9 Solution of the deterministic cash matching problem x opt 31.1,55.5,147.3 f x opt 127,332 payments cash

10 payments (expected value + 2 scenarios) Random Payments cash (expected value + 2 scenarios) payments (expected value scenarios) cash (expected value scenarios) number of scenarios violating positive cash (empirical probability of constraint violation) 50 %

11 Model with individual probabilistic constraints Now, payments are random variables Assumption: independent, normal distribution, E j j, Var j 500 j j expected payments = deterministic payments variance increasing with time j cumulative payments: j : k k 1 j E j b j, Var j k 1 optimization problem with random parameter: 3 max i 1 3 a i,15 x i i 1 Difficulty: decide on x before normal distribution Var j a ij x i j j 1,...,15 is observed ('here-and-now' decision)

12 3 max i 1 3 a i,15 x i i 1 (continued) expected-value problem ( = our deterministic problem): a ij x i E j b j j 1,...,15 probabilistic constraint: 3 P i 1 a ij x i j p 0,1 probability measure probability level optimization problem with individual probabilistic constraints: 3 max i 1 a i,15 x i P i 1 3 a ij x i j p j 1,...,15

13 (continued) P 3 i 1 a ij x i j P 3 i 1 a ij x i E j Var j j E j Var j P 3 i 1 a ij x i j 3 p i 1 standard normal a ij x i E j q p Var j Linear optimization problem: p-quantile of standard normal 3 max i 1 3 a i,15 x i i 1 a ij x i b j q p Var j j 1,...,15 safety term

14 Solution of the cash matching problem with individual probabilistic constraints (p=0.95) x opt 62.8,72.6,101.1 x opt 31.1,55.5,147.3 f x opt 103,925 recall expected value (or deterministic) solution: f x opt 127,332 cash (indiv. prob. cons.) 100 scenarios cash (expected value solution) 100 scenarios

15 (continued) number of scenarios violating positive cash (empirical probability of constraint violation) expected value solution final cash as a function of the probability level p exp. value prob. constr. final cash worst case indiv. prob. constr. p over a wide range robustness can be increased at moderate costs

16 Model with joint probabilistic constraints individual probabilistic constraints: at each fixed time probability of constraint violation low but: over the whole time interval violation may be likely cash at each t at most 1 out of 5 scenarios violates positive cash 0 time but: all 5 scenarios violate positive cash Therefore, replace the collection P 3 i 1 a ij x i j p j 1,...,15 of individual probabilistic constraints by one single P joint probabilistic constraint: 3 i 1 a ij x i j j 1,...,15 p

17 Solution of the cash matching problem with joint probabilistic constraints (p=0.95) x opt 31.1,55.5,147.3 f x opt 127,332 expected value solution 85 violating scenarios x opt 62.8,72.6,101.1 indiv. prob. constr. 12 violating scenarios f x opt 103,925 x opt 65.8,83.7,86.2 joint prob. constr. 2 violating scenarios f x opt 98,160

18 2. Models

19 Types of Probabilistic Constraints Probabilistic constraints may be given in individual: P h j x, 0 p j 1,...,m or joint form: P h j x, 0 j 1,...,m p decision vector random vector 3 cash matching problem: h j x, i 1 a ij x i j Structural properties and solution methods depend on: Form of the probabilistic constraint (individual or joint) Distribution of the random vector (continuous, discrete, independence) Properties of the constraint function h (linear, convex, separable)

20 Random right-hand side Assume that the constraint function has separable structure: h j x, g j x j individual prob. constr.: p-quantile P h j x, 0 p P g j x j p g j x q p j 1,...,m Probabilistic constraints of same type as deterministic constraints (no additional difficulties by randomness). Cash matching problem: g j linear prob. constr. linear too joint prob. constr.: P h j x, 0 j 1,...,m p P g j x j j 1,...,m p F g x p (no quantile!) multivariate distribution function g is analytically given, in general. Main task: calculate F

21 Independent components joint prob. constr. with random rhs.: F g x p 1,..., s independent F 1 g 1 x F s g s x p 1-dimensional distribution functions (easy to calculate) Independence of components rarely satisfied. Cash matching problem: payments j were assumed to be independent But: cumulative payments j are correlated nevertheless. covariance matrix: Var j j

22 Polyhedral Probabilistic Constraints Let h j x, b j x a j x, b j : n, a j : n s joint prob. constr.: P b j x a j x, j 1,...,m p, 1 Decision and random vector no longer separated. x fixed: probability of a polyhedron Multivariate normal distribution: N,, a j x a j j : a j, 1 : F b x p, N A, A A T normal probability of a polyhedron = normal distribution function value But: If m>s (more inequalities than random components, e.g., networks) singular normal distribution

23 Random Coefficients Let h j x, b j x x, j b j : n, j n Special case of polyhedral prob. constr.: a j x x, T : 1 T,..., m T joint probabilistic constraint: P x b x p random matrix Example: cash matching problem with random yields.

24 3. Structure

25 Structural properties We are interested in properties (continuity, differentiability, convexity) of the constraint function x : P h j x, 0 j 1,...,m and of the feasible set M : x n x p Lemma: h j uppersemicontinuous uppersemicontinuous M closed Example: standardnormal, h 1 x, : x 1, h 2 x, : x 2 discontinuous although all data are smooth h j, distribution of x 1,x 2 x 2 For 0<p<1, the feasible set (blue) 0 is not convex. x 1

26 Properties by composition recall joint prob. constr. with random rhs.: F g x p When is F g continuous, Lipschitzian, differentiable? By composition, all these properties are inherited from F, g. This is obvious, in general, for g. How about F? Lemma: F is continuous if it has a density. F is Lipschitzian if it has bounded marginal densities 1 F is differentiable if the functions t F x x i t are continuous for all x,i. conditional distribution function 1 Römisch/Schultz (1993)

27 Convexity joint prob. constr. with random rhs.: F g x p 2 When is F g concave? convex optimization algorithms components g j concave, A 1 concave and nondecreasing e.g., linear, see cash matching problem automatically satisfied for distribution functions When is F concave? Never! (distribution functions are bounded from below and above but not constant) Is there a nondecreasing function : such that F concave? If so, then 2 F g x p F g concave if components g j concave. increasing and concave Possible candidates: log, n

28 Logconcavity of distribution functions normal uniform F normal uniform exponential Cauchy log F exponential Cauchy normal uniform exponential F 1 Cauchy

29 bivariate normal uniform on unit square Log (bivariate normal) Log (uniform on unit square)

30 Prekopa's Theorem (reduced version) multivariate distribution functions extremely difficult to calculate (no analytic formula) almost impossible to check log-concavity directly Prekopa's Theorem: If the distribution function F has a density f, and if log f is concave, then log F is concave too. Example: multivariate normal distribution f K exp 1 2 x T 1 x log f log K 1 2 x T 1 x concave log F concave Other examples: uniform distribution on convex compact sets, Dirichlet, Gamma, etc.

31 Van de Panne/Popp - Theorem (extended version) Theorem: Let N, have a multivariate normal distribution. Consider a single probabilistic constraint with random coefficients: P x, 0 x p The induced feasible set M x n x p is convex, if p 0, ,1, 1-dim. standard normal distribution function smallest eigenvalue of Theorem generalizes to several individual prob. constr. (intersection of convex sets is convex) but not to joint prob. constr. (see example above).

32 Example: x M x 2 P x, 0 p N 1,0, I 2 x p = 0.8 p = 0.5 convex p = 0.2 convex nonconvex

33 Related Properties constraint set M x P Ax p Theorem: 1 Let have any distribution and let A have positively linear independent rows. Then, M is connected. Example: ~ uniform distribution on (1,-2,0),(-2,1,0),(-2,-2, ), p=0.5 0 A 1 1 M not convex but connected M rows pos. lin. indep. A 1 1 cash matching problem does not satisfy the assumption! 1: H. (2001) 2: H./Römisch (2004) M not connected rows not pos. lin. indep. Theorem: 2 Let ~ nondeg. multiv. normal, independent components. Then, log F is strongly concave on bounded sets. Let ~ uniform on s-dim. interval [a,b]. Then, log F is strongly concave on int[a,b]. Not true for arbitrary polytopes!

34 4. Numerics

35 Random r.h.s, normal distribution joint linear prob. constr. with random rhs.: P Ax p F Ax p Assumptions has a nondegenerate multivariate normal distribution example: cash matching problem with joint probabilistic constraints F A differentiable and concave (recall that F is log-concave) Algorithms from differentiable convex optimization (e.g., supporting hyperplanes) numerical requirements: calculation of F and of F. calculation of gradients may be reduced to the calculation of values 1 (in one dimension less) main challenge: calculation of F 1 Prekopa: Stochastic Programming

36 Calculating the normal distribution function (s = dimension of random vector) 'exact' calculation for s = 1,2 1,2 numerical integration (up to s~15) 3,4 efficient simulation techniques 5,6 bounds (linear programming 7 ; graph-theoretical constructions 8 combined use of bounding and simulation techniques (up to s~50) 1: Donnely (1973) 2: Drezner, Wesolowsky (1990) 3: Schervish (1984) 4: Genz (1992) 5: Deak (1986) 6: Szantai (1985) 7: Prekopa (1990) 8: Bukszar, Prekopa (2001),

37 Bounds Let have a 4-dim. standard normal distribution (independent components) For x:=(2,1,2,1), we want to estimate F x P 1 2, 2 1, 3 2, 4 1 : 1 dim.standardnormaldistr. funct., , exact value: F x trivial upper bound: F x min 1, A i : i x i first order complementary events A ij : i x i j x j second order complementary events S 1 : i P A i, S 2 : i j In the example: S , S P A ij can be calculated exactly (1- and 2-dim. normal distributions) Bonferroni bounds: 1 S 1 F x 1 S 1 S F x improved lower bound: 1 S 1 2 s S 2 F x F x

38 Hunters lower bound Define a graph with nodes = A i, edges = A ij and weights of edges = P(A ij ): A 1 P(A 14 ) = A 4 Find a spanning tree T with maximum P(A 12 ) = A 2 P(A 24 ) = P(A 13 ) = P(A 23 ) = A 3 P(A 34 ) = weight #T. new lower bound: 1 S 1 + #T Here: #T = F 2,1,2, Comparison of bounds: true value: trivial upper bound: Bonferroni lower bound: Bonferroni upper bound: improved lower bound: Hunters lower bound: 0.670

39 Cutting Plane Method min c,x F Ax p c F Ax p F A x p line search: precise values expansive, imprecise values require safe cuts

40 p Line Search (Szantai) acceptance region safety region upper bound true value lower bound repeat bisection until in acceptance region if in safety region increase precision 0 true 1

41 Discrete Distributions min c,x F Ax p 3 finite discrete distribution p level set of F : y s N F y p j 1 z j s p-level efficient points 3 min j J min c,x Ax z j convex hull F y p F y p p-level efficient points conceptual solution method: determine all p-level efficient points (too many!) solve all resulting linear programs select the best solution (out of finitely many) relaxed problem: N min c,x Ax co j 1 min c,x Ax j z j, j j 1 stepwise generation of p-level efficient points 1 1 Dentcheva, Prekopa, Ruszczynski (2000) N z j s N 0, j 1 j 1

42 5. Stability

43 Approximation Original optimization problem with linear joint probabilistic constraint: min f x F Ax p Assumption: f convex, F log-concave (e.g., multivariate normal). Problem: distribution of in general not known Approximation by some (e.g., empirical approximation) F F v : inf f x F Ax p (optimal value) S : x F Ax p, f x v (solution set) How do v and S behave near? sample (N=10)

44 Qualitative Stability min f x F Ax p P Theorem: 1 For the original problem (P) suppose that: S is nonempty and bounded There is some x with F A x p (Slater point). Then, S is upper semicontinuous at converge to a true solution). (approximating solutions will Furthermore, there are L, 0, such that v v L F F : F F 1 H./Römisch (2004)

45 Lipschitz-Continuity of the Value-at-Risk VaR p inf t P t p p 0,1 usually not continuous! Corollary: Let p 0,1 and have a log-concave distribution. Then, there exist L, 0 such that VaR p VaR p F F : F F

46 Quantitative Stability (Hölder type) min c,x x,hx F Ax p H 0 convex quadratic objective Theorem: 1 In addition to the assumptions in the previous Theorem, suppose that F is strongly convex on some open convex set U A S Then, there are L, 0, such that d H S,S L F F : F F 1 H./Römisch (2004)