Optimization with piecewise-affine cost functions

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1 Optimization with piecewise-affine cost functions G Ferrari Trecate, Paolo Letizia, Matteo Spedicato Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland 21 June Technical Report AUT01-13

2 Abstract This report shows how it is possible to solve the problem of the minimization of an objective function composed by the sum of piece-wise affine functions as a Mixed Integer Linear Programming (MILP) problem The optimization problem min x m i=1 k i p i (x i ) Ax B, x = where p i (x i ) are piece-wise affine functions and A, B are the constraint matrices, will be reformulated as a MILP problem: x 1 x m min V KV F V G where K = [ k 1 k v ], V = and F, G are the constraint matrices In the first chapter of this report a simple case (cost function composed of only one piece-wise affine function) will be considered The generalization to the case of a cost function composed of m piece-wise affine functions will be contained in the second chapter v 1 v v

3 Chapter 1 Simple case Consider the following optimization problem: min x k p(x) Ax B (11) where k is a scalar coefficient, p(x) is a piece-wise affine function defined on the real variable x, A and B are suitable column vectors which define the feasible region for x The piece-wise affine function p(x) is composed by n affne functions, each one defined on a different domain: p(x) = a 1 x + b 1 if x [l 1, u 1 ] a 2 x + b 2 if x ]u 1, u 2 ] a n x + b n if x ]u n 1, u n ] (12) where the lower bound l 1 and the upper bound u n of the variable x are defined by the inequalities Ax B Note that p(x) could have jump discontinuity for x = u i, i = 1,, n 1, as shown in the Figure 11 assuming n = 3 We are interested to reformulate the optimization problem (11) as a Mixed Integer Linear Programming (MILP) problem: min V KV F V G (13) where K is the weights vector; V is the optimization variables (both continuous and logic) vector; F and G are the matrices which define the constraints of the vector V Note that: 1

4 Figure 11: Example of a piece-wise affine function with equation (2) assuming n = 3 K will contain the coefficient k; V will contain the variable x; F and G will contain the matrices A and B, where k, x, A and B are the same defined in (11) To derive a MILP problem we need to introduce some auxiliary logic and real variables First of all introduce n 1 binary variables, each one defined in the following way: δ 1 = 1 x u 1 δ 2 = 1 x u 2 (14) δ n 1 = 1 x u n 1 2

5 So, by giving a binary value to the variables δ i, i = 1,, n 1, it is possible to choose a particular region for the variable x For example, for the piecewiseaffine function reported in Figure 11 we will introduce two binary variables defined as below: δ1 = 1 x 4 δ2 = 1 x 12 and assuming for instance (δ1 = 1) (δ2 = 0), the region 4 x < 12 is defined Note that the variables δ i, i = 1,, n 1, are related by the logical equations δ i = 1 = δ j = 1 j < i, i = 2,, n 1 (15) which imply that not all the combinations of the variables δ i, i = 1,, n 1, are allowed Note also that the number of the relations (15) is equal to (n 2) + (n 3) Define now n 1 real variables z as below: z 1 = z 2 = z n 1 = if δ1 = 1 a 2 x + b 2 (16) else a 1 x + b 1 if δ2 = 1 (a 3 a 2 )x + (b 3 b 2 ) (17) else 0 if δn 1 = 1 (a n a n 1 )x + (b n b n 1 ) (18) else 0 and define the real variable z given by the sum of the variables z i, i = 1,, n 1: z = z 1 + z z n 1 (19) It s easy to verify that, depending on the feasible combination of the variables δ i, i = 1,, n 1, z will be equal to one of the n pieces of the function (12) 3

6 11 Inequalities The definition of the binary variables δ (14) and their relations (15), of the real variables z (16-18) and z (19) can be translated into mixed integer linear inequalities, as shown in [1], [2] and reported in Table 11, 12 inequalities n o ineq (14) D1 δ 1 (u 1 x) + D 1 (d 1 ε)δ 1 (u 1 x) ε D2 δ 2 (u 2 x) + D 2 (d 2 ε)δ 2 (u 2 x) ε Dn 1 δ n 1 (u n 1 x) + D n 1 (d n 1 ε)δ n 1 (u n 1 x) ε 2 (n 1) (15) δ i δ j 0 j < i, i = 2,, n 1 (n 2) + (n 3) Table 11: Translation in inequalities of the definition of the variables δ and of their logical relations In the Table 11 ε is a small tolerance (typically the machine precision); D i and d i, i = 1,, n 1, represent respectively the maximum and the minimum of (u i x), where x [l 1, u n ] Theoretically, an over[under] estimate of the maximum[minimum] suffices for our purpose, so that one can assume i = 1,, n 1 D i =max i (u i x) = max(u i ) min(x) = u n l 1 i = 1,, n 1 d i = min(u i x) = min(u i ) max(x) = u 1 u n In the Table 12 M 1, m 1 are respectively the maximum and the minimum of a 1 x + b 1, where x [l 1, u n ] ; M 2, m 2 are respectively the maximum and the minimum of a 2 x + b 2, where x [l 1, u n ] ; M j, m j, j = 3,, n, are respectively the maximum and the minimum of (a j a j 1 )x + (b j b j 1 ), where x [l 1, u n ] ; 4

7 (16) (17) (18) inequalities (M 2 m 1 )δ 1 z 1 a 2 x + (M 2 m 1 b 2 ) (M 1 m 2 )δ 1 + z 1 a 2 x + (M 1 m 2 + b 2 ) (m 2 M 1 )δ 1 z 1 a 1 x b 1 (m 1 M 2 )δ 1 + z 1 a 1 x + b 1 M 3 δ 2 + z 2 0 m 3 δ 2 z 2 0 m 3 δ 2 + z 2 [(a 3 a 2 )x + (b 3 b 2 )] m 3 M 3 δ 2 z 2 [(a 3 a 2 )x + (b 3 b 2 )] + M 3 M n δ n 1 + z n 1 0 m n δ n 1 z n 1 0 m n δ n 1 + z n 1 [(a n a n 1 )x + (b n b n 1 )] m n M n δ n 1 z n 1 [(a n a n 1 )x + (b n b n 1 )] + M n n o ineq 4 (n 1) (19) z z1 + z z n 1 z z 1 + z z n 1 2 Table 12: Translation in inequalities of the definition of the variables z and z ε is once more the machine precision As explained before, also in this case an over[under] estimate of the maximum[minimum] suffices for our purpose, so that one can assume a global maximum M = max(m i ) and a global minimum m = min(m i ), i = 1,, n and replace all the M i, m i, i = 1,, n listed before with them 12 Derivation of the MILP problem It is now possible to reformulate the initial problem min x k p(x) Ax B as the MILP problem 5

8 min V KV F V G where: V = z z 1 z n 1 x δ 1 δ n 1 ; K = [ k 0 0 ], where the number of 0 is equal to 2(n 1) + 1); F and G are matrices of suitable sizes such that they specify the inequalities Ax B and those contained in Tables 11, 12 6

9 Chapter 2 Generalization Based on the simple case explained before, it s easy to generalize the results to the case of a cost function composed of m piece-wise affine functions The extended optimization problem will be: min x m i=1 k i p i (x i ) Ax B, x = x 1 x m (21) where k i, i = 1,, m, are scalar coefficients, p i (x i ), i = 1,, m, are piecewise affine functions each one defined on the real variable x i, A and B are suitable matrices which define the feasible region for the real variables x i, i = 1, m The i th piece-wise affine function p i (x i ) is composed by n i affine functions: p i (x i ) = a i,1 x i + b i,1 if x i [l i,1, u i,1 ] a i,2 x i + b i,2 if x i ]u i,1, u i,2 ] a i,ni x i + b i,n if x i ]u i,n 1, u i,n ], i = 1,, m (22) Our aim is once more to obtain a MILP formulation for the problem (21) For this purpose we need to introduce auxiliary logic and real variables In particular, analogously to what has just been explained for the simple case, n i 1 logic and n i 1 real variables are needed for each p i (x i ) 7

10 Define the logic variables δ as follow: δ i,1 = 1 x i u i,1 δ i,2 = 1 x i u i,2 δ i,ni 1 = 1 x i u i,ni 1, i = 1,, m (23) The logic variables δ i,j, i = 1,, m, j = 1,, n i 1 are related by the following logical equations: δ i,j = 1 = δ i,s = 1 s < j, j = 2,, n i 1 i = 1,, m (24) The same conclusions of the simple case are valid i, i = 1,, m Define now the real variables z as follows: z i,1 = z i,2 = z i,ni 1 = if δi,1 = 1 a i,2 x i + b i,2 (25) else a i,1 x i + b i,1 if δi,2 = 1 (a i,3 a i,2 )x i + (b i,3 b i,2 ) (26) else 0 if δi,ni 1 = 1 (a i,ni a i,ni 1)x i + (b i,ni b i,ni 1) (27) else 0 and define the real variable z i given by the sum of the variables z i,j, j = 1,, n i 1: z i = z i,1 + z i,2 + + z i,ni 1 (28) 21 Inequalities Analogously to the simple case the definition of the binary variables δ (23) and their relations (24), of the real variables z (25-27) and z (28) can be translated into mixed integer linear inequalities The results of this procedure are shown in the Tables 21, 22 8

11 inequalities n o ineq (23) Di,1 δ i,1 (u i,1 x i ) + D i,1 (d i,1 ε)δ i,1 (u i,1 x i ) ε Di,2 δ i,2 (u i,2 x i ) + D i,2 (d i,2 ε)δ i,2 (u i,2 x i ) ε Di,ni 1δ i,ni 1 (u i,ni 1 x i ) + D i,ni 1 (d i,ni 1 ε)δ i,ni 1 (u i,ni 1 x i ) ε 2 (n i 1) (24) δ i,j δ i,s 0 s < j, j = 2,, n i 1 (n i 2) + (n i 3) Table 21: Translation in inequalities of the definition of the variables δ and of their logical relations (25) (26) (27) (28) inequalities (M i,2 m i,1 )δ i,1 z i,1 a i,2 x i + (M i,2 m i,1 b i,2 ) (M i,1 m i,2 )δ i,1 + z i,1 a i,2 x i + (M i,1 m i,2 + b i,2 ) (m i,2 M i,1 )δ i,1 z i,1 a i,1 x i b i,1 (m i,1 M i,2 )δ i,1 + z i,1 a i,1 x i + b i,1 M i,3 δ i,2 + z i,2 0 m i,3 δ i,2 z i,2 0 m i,3 δ i,2 + z i,2 [(a i,3 a i,2 )x i + (b i,3 b i,2 )] m i,3 M i,3 δ i,2 z i,2 [(a i,3 a i,2 )x i + (b i,3 b i,2 )] + M i,3 M i,ni δ i,ni 1 + z i,ni 1 0 m i,ni δ i,ni 1 z i,ni 1 0 m i,ni δ i,ni 1 + z i,ni 1 [(a i,ni a i,ni 1)x i + (b i,ni b i,ni 1)] m i,ni M i,ni δ i,ni 1 z i,ni 1 [(a i,ni a i,ni 1)x i + (b i,ni b i,ni 1) + M i,ni z i z i,1 + z i,2 + + z i,ni 1 z i z i,1 + z i,2 + + z i,ni 1 Table 22: Translation in inequalities of the definition of the variables z and z 9

12 Fixed i, the number of inequalities which translate the equations (25-27) is 4 (n i 1), and the number of inequalities which translate the equations (28) is 2 Moreover, fixed i, the definitions and the considerations about the coefficients ε, M, m, D, d are the same reported for the simple case 22 Derivation of the MILP problem It is now possible to reformulate the initial problem min x m i=1 k i p i (x i ) Ax B, x = x 1 x m as the MILP problem where: [ ] V V =, V min V KV F V G 10

13 V = z 1 z m z 1,1 z 1,n1 1 z 2,1 z 2,n2 1 z m,1 z m,nm 1 ; V = x 1 x m δ 1,1 δ 1,n1 1 δ 2,1 δ 2,n2 1 δ m,1 δ m,nm 1 ; K = [ k 1 k m 0 0 ], where the number of 0 is equal to m i=1 2(n i 1) + 1; F and G are matrices of suitable sizes such that they specify the inequalities Ax B and those contained in the tables 21 and References [1] Bemporad, A, & Morari, M Control of systems integrating logic, dynamics, and constraints Automatica, 35(3): , 1999 [2] Mignone, D The Really Big Collection Tecnical Report - IfA,