A stochastic global optimization method for (urban?) multi-parameter systems. Rodolphe Le Riche CNRS and Ecole des Mines de St-Etienne

Transcription

1 A stochastic global optimization method for (urban?) multi-parameter systems Rodolphe Le Riche CNRS and Ecole des Mines de St-Etienne

2 Purpose of this presentation Discussion object : cities as an entity for heating consumption and related public policies. Cities are analyzed, modeled and optimized for heat consumption (including the effects of solar passive input). Cities are objects that are described at different parameterization levels : entire city (coarser level), districts, buildings (finer level). In mechanical engineering, multi-levels of parameters are often associated to composite materials. Talk : describe an idea for multi-level optimization that emerged in composite design. Any use for cities?

3 Notation and application example Composite design at various scales min f ( y ( x)) macro x S y(x) numerical simulator of the structure (stress, strains, strength, mass,...) meso, v ( plate stiffnesses ) v(x) micro, x ( fiber positions ) v(x) is numerically inexpensive. x(v) doesn't exist : there are many fiber positions for one choice of plate stiffnesses.

4 Notation and application example Heating consumption at various scales min f ( y ( x)) macro city x S y(x) numerical simulator of the structure (solar energy, thermal simulation) meso, v «IRIS» (district) micro, x building IRIS : Ilôt regroupé pour l'information statistique

5 Scope of application : global, derivative free problems Here, we focus on f min f x x S S ℝ n or ℕn or { ℝ n1 ℕn2 } i.e., continuous or integer or mixte optimization. local optimum x l V xl x * global optimum

6 Flow chart of a general stochastic optimizer Initialize t and pt(x) Sample Calculate f(xt+1) Update the distribution xt+1 ~ pt(x) pt+1(x) = Update( x1, f(x1),, xt+1, f(xt+1) ) or more often pt+1(x) = Update( pt(x), xt+1, f(xt+1) ) Stop or [ t = t+1 and go back to Sample ] with different p's if x is continuous or discrete or mixed.

7 A simple example in Rn : ES-(1+1) Initializations : x, f(x), m, C, While tmax. t < tmax do, Sample N(m,C) --> x' Calculate f(x'), t = t+1 If f(x')<f(x), x = x', f(x) = f(x') Endif Update m (e.g., m=x) and C End while Normal law N(m,C) [ 2 σ C = 0 σ σ ] for variables seen as independent.

8 Illustration : adaptation of 2D Gaussian with ES-(1+1)

9 Discrete variables : The Univariate Marginal Density Algorithm (UMDA) ( Baluja 1994 as PBIL and Mühlenbein 1996) x S {1,2,, A}n (alphabet of cardinality A ) o o o o n e.g. { 45, 0, 45, 90 } (fiber orientations) n e.g. {matl1,, matla } (material choice) The algorithm is that of a population based stochastic optimization (see before + many x's at each iteration) with different sampling and updating of pt. pt assumes that the variables are independent (drop t ), n p x = p i x i i=1 0.6 pi p2i pia p1i 1 A pij = 1 j= A xi

10 UMDA (2) Sampling : For i=1, n ui ~ U [0,1] If 0 ui p1i x i =1 u 1 0 k 2 p 1 A p +p k If 1 k 1 p ij ui pij j=1 A 1 If j=1 j=1 x i =k j p i ui 1 x i =A Learning : Select the μ best points out of λ, f x 1 : f x 2 : f x : pij is the frequency of j at position i in the bests : j pi = k =1 : k I x i = j 1 p 1 j i, k k I x i = j =1 if x i = j, =0 otherwise (minimum frequency for ergodicity)

11 Application to composite design for frequency density learned by UMDA (2D) contour lines of the penalized objective function Independent densities can neither represent curvatures nor variables' couplings. ( from Grosset, L., Le Riche, R. and Haftka, R.T., A double-distribution statistical algorithm for composite laminate optimization, SMO, 2006 )

12 Stochastic discrete optimization : learning the variables dependencies More sophisticated discrete optimization methods attempt to learn the couplings between variables. For example, with pairwise dependencies : X1:n X2:n... Xn:n p( x ) = p ( x 1 : n ) p( x 2: n x 1: n ) p ( x n : n x n 1: n ) Trade-off : richer probabilistic structures better capture the objective function landscape but they also have more parameters need more f evaluations to be learned. MIMIC ( Mutual Information Maximizing Input Clustering ) algorithm : De Bonnet, Isbell and Viola, BMDA ( Bivariate Marginal Distribution Algorithm ) : Pelikan and Muehlenbein, 1999.

13 Multi-level parameter optimization with DDOA ( from Grosset, L., Le Riche, R. and Haftka, R.T., A double-distribution statistical algorithm for composite laminate optimization, SMO, 2006 ) Mathematical motivation : create couplings between variables using many independent distributions (in x and v spaces). Numerical motivation : take into account expert knowledge in the optimization to improve efficiency. E.g. in composites, the lamination parameters v (the plate stiffnesses) make physical sense.

14 Example in composites Use of the lamination parameters v = lamination parameters = geometric contribution of the plies to the stiffness. Inexpensive to calculate from x (fiber angles). Simplifications : fewer v's than fiber angles. Often, the v's are taken as continuous. But f(v) typically does not exist (e.g., ply failure criterion).

15 Past examples in composites ( Liu, Haftka, and Akgün, «Two-level composite wing structural optimization using response surfaces», Merval, Samuelides and Grihon, «Lagrange-Kuhn-Tucker coordination for multilevel optimization of aeronautical structures», ) Initial problem : Optimize a composite structure made of several assembled panels by changing each ply orientation many discrete variables Decomposed problem : Structure level Optimize a composite structure made of several assembled panels by changing the lamination parameters of each panel few continuous variables optimal v's Laminate level Minimize the distance to target lamination parameters by changing the ply orientations few discrete variables BUT for such a sequential approach to make sense, f (v ) must exist and guide to optimal regions (i.e., prohibits emergence of solutions at finer scales).

16 The DDOA stochastic optimization algorithm objective function If v(x) is costless, it does not cost to learn densities in the x AND v spaces at the same time. does not exist p(x) pddoa ( x) = p X v ( X )=V (x ) x v v(x)

17 The DDOA algorithm : X v(x)=v? Simple mathematical illustration : 1 1 T p X (x ) = exp x x 2π 2 ( ) v = x1+ x pv (v ) = exp (v 1) 2 2 π ( ) Intermediate step for a given v : p X v( X )= v=1 ( x) is a degenerated Gaussian along x1+ x 2 = 1 (cross-section of the 2D bell curve along the blue line + normalization)

18 The DDOA algorithm : X v(x)=v? p X v ( X )=V (x ) = p X v( X )=v pv (v (x )) =... = exp ( x1 x 2 )2 ( x 1 + x 2 1)2 2π 4 2 ( ) p X v( X )=V ( x) is a coupled distribution that merges the effects of X and V Analytical calculation in the Gaussian case. In practice, use simulations...

19 The DDOA algorithm (flow chart) Choose λ, μ, ρ such that ρ>>1 and λ>μ Initialize pv(v)and px(x) For i=1,λ do Sample vtarget from pv(v) Sample ρ>>1 x's from px(x) x(i) = the closest x to vtarget Calculate f(x(i)) sampling of X v(x)=v end For Rank x(1:λ),, x(λ:λ) the proposed points Update pv(v) and px(x) from x(1:λ),, x(μ:λ) Stop? If no, go back to top...

20 Application of DDOA to composite design for frequency px(x) and pv(v) can be simple densities, without variables couplings ( easy to learn), yet pddoa(x) is a coupled density. f(x) and selected points n p X x = i =1 pi x i pddoa(x) One half of the algorithm searches in a low dimension space.

21 additional slides

22 Introduction to stochastic optimization Random numbers are versatile search engines (work both in Rn and / or Nn ). They can also yield efficient methods. Let pt(x) denote the probability density function of x at iteration t (e.g., after t evaluations of f). It represents the belief at t that the optimum x* is at x. How to «sample pt(x)» once (Scilab notation)? if x is uniform between m and M, X ~ U[m,M], call x = m + rand(n,1).*(m-m) if x is (multi-)gaussian with mean m and covariance matrix C, X ~ N(m,C),call x = m + grand(1,'mn',0,c)

23 Application to a laminate frequency problem (1) ( from Grosset, L., Le Riche, R. and Haftka, R.T., A double-distribution statistical algorithm for composite laminate optimization, SMO, 2006 ) max x f 1 x 1,, x 15, the first eigenfreq. of a simply supported plate such that 0.48 eff x 0.52 where x i {0 o, 15o,, 90o } the constraint is enforced by penalty and creates a narrow ridge in the design space

24 Application to a laminate frequency problem (2) Optimum : [90o4 /±75o /±60o2 /±45o5 /±305o ]s Compare UMDA to a GA (genetic algorithm) and SHC (Stochastic Hill Climber) Reliability = probability of finding the optimum at a given cost. UMDA performs fairly well on this problem.

25 Example in composites Use of the lamination parameters Simplifications : fewer v's than fiber angles. Often, the v's are taken as continuous. But f(v) typically does not exist (e.g., ply failure criterion).