INSTITUTE OF COMPUTER SCIENCE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC
Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization Ladislav Lukˇsan
Jan Vlˇcek
Technical report No. 798 January 2000
Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vodrenskou v 2, 182 07 Prague 8, Czech Republic phone: (+4202) 6884244 fax: (+4202) 8585789 e-mail:
[email protected]
INSTITUTE OF COMPUTER SCIENCE ACADEMY OF SCIENCES OF THE CZECH REPUBLIC
Test Problems for Nonsmooth Unconstrained and Linearly Constrained Optimization1 Ladislav Lukˇsan
Jan Vlˇcek
Technical report No. 798 January 2000
Abstract This report contains a description of subroutines which can be used for testing nonsmooth optimization codes. These subroutines can easily be obtained either by using the anonymous ftp (address ftp://ftp.cs.cas.cz/pub/msdos/opt, files TEST06.FOR, TEST19.FOR, TEST22.FOR or from the homepage http://www.cs.cas.cz/~luksan /test.html. Furthermore, all test problems contained in these subroutines are presented in the analytic form. Keywords nonsmooth optimization, test problems
1
This work was supported under grant No. 201/00/0080 given by the Czech Republic Grant Agency
1
Introduction
This report describes three groups of subroutines which contain problems for testing nonsmooth optimization codes: 1. Unconstrained minimax optimization - subroutines TIUD06, TAFU06, TAGU06, TAHD06. We want to find a minimum of the minimax objective function F (x) = max (fk (x)),
x ∈ Rn ,
F (x) = max |fk (x)|,
x ∈ Rn ,
1≤k≤nA
or 1≤k≤nA
where fk (x), 1 ≤ k ≤ nA , are twice continuously differentiable partial functions. 2. Unconstrained nonsmooth minimization - subroutines TIUD19, TFFU19, TFGU19, TFHD19. We want to find a minimum of the general nonsmooth objective function F (x). 3. Linearly constrained minimax optimization - subroutines TILD22, TAFU22, TAGU22, TAHD22. We want to find a minimum of the minimax objective function F (x) = max (fk (x)),
x ∈ Rn ,
F (x) = max |fk (x)|,
x ∈ Rn ,
1≤k≤nA
or 1≤k≤nA
where fk (x), 1 ≤ k ≤ nA , are twice continuously differentiable partial functions, subject to simple bounds and general linear constraints. Simple bounds are assumed in the form xi − unbounded xli ≤ xi xi ≤ xui xli ≤ xi ≤ xui xi = xli = xui
Iix Iix Iix Iix Iix
, , , , ,
= 0, = 1, = 2, = 3, = 5,
where 1 ≤ i ≤ n. General linear constraints are assumed in the form aTi x − unbounded cli ≤ aTi x aTi x ≤ cui cli ≤ aTi x ≤ cui aTi x = cli = cui
, , , , ,
Iic Iic Iic Iic Iic
= 0, = 1, = 2, = 3, = 5,
where 1 ≤ i ≤ nc and nc is a number of general linear constraints (I x , I c correspond to arays IX, IC in Section 3). 1
All subroutines are writen in the standard Fortran 77 language. Their names are derived from the following rule: • The first letter is T - test subroutines. • The second letter is either I - initiation, or F - objective function, or A - partial function. • The third letter is either U - initiation of unconstrained problem, or L - initiation of linearly constrained problem, or F - computation of the function value, or G computation of the function gradient, or H - computation of the function Hessian matrix. • The fourth letter is either U - universal subroutine, or D - subroutine for dense problems. The last two digits determine a given collection (numbering corresponds to the UFO system [13], which contains similar collections). Initiation subroutines use the following parameters (array dimensions are given in parentheses): N NA NB NC X(N) IX(N) XL(N) XU(N) IC(N) CL(N) CU(N) FMIN XMAX NEXT IEXT IERR
input number of variables. output number of partial functions. output number of box constraints. output number of general linear constraints. output vector of variables. output types of box constraints. output lower bounds in box constraints. output upper bounds in box constraints. output types of general linear constraints. output lower bounds in general linear constraints. output upper bounds in general linear constraints. output lower bound for the objective function value. output maximum stepsize. input number of the problem selected. output type of the objective function: −1 - maximum of values, 0 - maximum of absolute values. output error indicator: 0 - data are correct, 1 - N is too small.
Although N is an input parameter, it can be changed by the initiation subroutine when its value does not satisfy the required conditions. For example, most of the problems require N to be even or a multiple of a positive integer. Evaluation subroutines use the following parameters (array dimensions are given in parentheses):
2
N X(N) F G(N) H(N*(N+1)/2) KA FA GA(N) HA(N*(N+1)/2) NEXT
2
input input output output output input output output output input
number of variables. vector of variables. value of the objective function. gradient of the objective function. Hessian matrix of the objective function. index of the partial function selected. value of the partial function selected. gradient of the partial function selected. Hessian matrix of the partial function selected. number of the problem selected.
Test problems for unconstrained minimax optimization
Calling statements have the form CALL CALL CALL CALL
TIUD06(N,NA,X,FMIN,XMAX,NEXT,IEXT,IERR), TAFU06(N,KA,X,FA,NEXT), TAGU06(N,KA,X,GA,NEXT), TAHD06(N,KA,X,HA,NEXT),
with the following significance: TIUD06 - determination of problem dimension N, NA and initiation of vector of variables X. TAFU06 - evaluation of the KA-th partial function value FA at point X. TAGU06 - evaluation of the KA-th partial function gradient GA at point X. TAHD06 - evaluation of the KA-th partial function Hessian matrix HA at point X. We seek a minimum of a minimax objective function F (x) = max (fk (x)), 1≤k≤nA
x ∈ Rn ,
from the starting point x¯. The description of individual problems follows. Table 2.1 contains problem dimensions and optimum function values.
3
No. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25
Problem CB2 WF SPIRAL EVD52 Rosen-Suzuki Polak 6 PBC3 Bard Kowalik-Osborne Davidon 2 OET5 OET6 GAMMA EXP PBC1 EVD61 Transformer Filter Wong 1 Wong 2 Wong 3 Polak 2 Polak 3 Watson Osborne 2
n nA 2 3 2 3 2 2 3 6 4 4 4 4 3 21 3 15 4 11 4 20 4 21 4 21 4 61 5 21 5 30 6 51 6 11 9 41 7 5 10 9 20 18 10 2 11 10 20 31 11 65
Optimum value 1.9522245 0 0 3.5997193 −44 −44 0.42021427 · 10−2 0.50816327 · 10−1 0.80843684 · 10−2 115.70644 0.26359735 · 10−2 0.20160753 · 10−2 0.12041887 · 10−6 0.12237125 · 10−3 0.22340496 · 10−1 0.34904926 · 10−1 0.19729063 0.61852848 · 10−2 680.63006 24.306209 133.72828 54.598150 261.08258 0.14743027 · 10−7 0.48027401 · 10−1
Table 2.1 Problem 2.1 CB2 [5]. F (x) = max fi (x), f1 (x) f2 (x) f3 (x) x1
1≤i≤3 x21 + x42 ,
= = (2 − x1 )2 + (2 − x2 )2 , = 2 exp(x2 − x1 ), = 2, x2 = 2.
Problem 2.2 WF [16]. F (x) = max fi (x), 1≤i≤3
µ
¶
10x1 1 x1 + + 2x22 , f1 (x) = 2 x1 + 0.1 4
µ
¶
1 10x1 −x1 + + 2x22 , 2 x1 + 0.1 ¶ µ 1 10x1 2 f3 (x) = x1 − + 2x2 , 2 x1 + 0.1 x1 = 3, x2 = 1. f2 (x) =
Problem 2.3 SPIRAL [16]. F (x) = max(f1 (x), f2 (x)), µ
f1 (x) =
µ
f2 (x) =
q
q
x21 + x22 cos
x1 − q
q
x21
x2 −
x21 + x22
x1 = 1.41831,
+
x22
sin
x21
+
x22
¶2
¶2
+ 0.005(x21 + x22 ), + 0.005(x21 + x22 ),
x2 = −4.79462.
Problem 2.4 EVD52 [7]. F (x) = max fi (x), f1 (x) f2 (x) f3 (x) f4 (x) f5 (x) f6 (x) xi
= = = = = = =
1≤i≤6 x21 + x22 x21 + x22
+ x23 − 1, + (x3 − 2)2 , x1 + x2 + x3 − 1, x1 + x2 − x3 + 1, 2x31 + 6x22 + 2(5x3 − x1 + 1)2 , x21 − 9x3 , 1, 1 ≤ i ≤ 3.
Problem 2.5 Rosen-Suzuki [20]. F (x) = max fi (x), f1 (x) f2 (x) f3 (x) f4 (x) x1
= = = = =
1≤i≤4 x21 + x22
+ 2x23 + x24 − 5x1 − 5x2 − 21x3 + 7x4 , f1 (x) + 10(x21 + x22 + x23 + x24 + x1 − x2 + x3 − x4 − 8), f1 (x) + 10(x21 + 2x22 + x23 + 2x24 − x1 − x4 − 10), f1 (x) + 10(2x21 + x22 + x23 + 2x1 − x2 − x4 − 5), 0, x2 = 0, x3 = 0, x4 = 0.
5
Problem 2.6 Polak 6 [17]. F (x) = max fi (x), 1≤i≤4
f1 (x) = (x1 − (x4 + 1)4 )2 + (x2 − (x1 − (x4 + 1)4 )4 )2 + 2x23 + x24 − −5(x1 − (x4 + 1)4 ) − 5(x2 − (x1 − (x4 + 1)4 )4 ) − 21x3 + 7x4 , f2 (x) = f1 (x) + 10((x1 − (x4 + 1)4 )2 + (x2 − (x1 − (x4 + 1)4 )4 )2 + x23 + x24 + +(x1 − (x4 + 1)4 ) − (x2 − (x1 − (x4 + 1)4 )4 ) + x3 − x4 − 8), f3 (x) = f1 (x) + 10((x1 − (x4 + 1)4 )2 + 2(x2 − (x1 − (x4 + 1)4 )4 )2 + x23 + 2x24 − −(x1 − (x4 + 1)4 ) − x4 − 10), f4 (x) = f1 (x) + 10((x1 − (x4 + 1)4 )2 + (x2 − (x1 − (x4 + 1)4 )4 )2 + x23 + +2(x1 − (x4 + 1)4 ) − (x2 − (x1 − (x4 + 1)4 )4 ) − x4 − 5), xi = 0, 1 ≤ i ≤ 4. Problem 2.7 PBC3 [18]. F (x) =
max |fi (x)|,
1≤i≤21
x3 exp(−ti x1 ) sin(ti x2 ) − yi , x2 3 −ti 1 1 = e + e−5ti − e−2ti (3 sin 2ti + 11 cos 2ti ) , 20 52 65 = 10(i − 1)/20, 1 ≤ i ≤ 21, = 1, 1 ≤ i ≤ 3.
fi (x) = yi ti xi
Problem 2.8 Bard [21]. F (x) =
max |fi (x)|,
1≤i≤15
fi (x) = x1 + ui = i, xi = 1,
ui − yi , v i x2 + w i x3 vi = 16 − i, wi = min(ui , vi ), 1 ≤ i ≤ 3.
Problem 2.9 Kowalik-Osborne [21]. F (x) = fi (x) =
max |fi (x)|, 1≤i≤11 x1 (u2i + x2 ui ) u2i + x3 ui + x4 6
− yi ,
1 ≤ i ≤ 15,
i 1 2 3 4 5 6 7 8 9 10 11 x1 = 0.250,
yi 0.1957 0.1947 0.1735 0.1600 0.0844 0.0627 0.0456 0.0342 0.0323 0.0235 0.0246
x2 = 0.390,
ui 4.0000 2.0000 1.0000 0.5000 0.2500 0.1670 0.1250 0.1000 0.0833 0.0714 0.0625 x3 = 0.415
x4 = 0.390.
Problem 2.10 Davidon 2 [21]. F (x) =
max |fi (x)|,
1≤i≤20
fi (x) = (x1 + x2 ti − exp(ti ))2 + (x3 + x4 sin(ti ) − cos(ti ))2 , ti = 0.2i, 1 ≤ i ≤ 20, x1 = 25, x2 = 5, x3 = −5, x4 = −1. Problem 2.11 OET5 [22]. F (x) =
max |fi (x)|,
1≤i≤21
√ fi (x) = x4 − (x1 t2i + x2 ti + x3 )2 − ti , ti = 0.25 + 0.75(i − 1)/20, 1 ≤ i ≤ 21, xi = 1.0, 1 ≤ i ≤ 4. Problem 2.12 OET6 [22]. F (x) =
max |fi (x)|,
1≤i≤21
1 , 1 + ti = −0.5 + (i − 1)/20, 1 ≤ i ≤ 21, = 1.0, x2 = 1.0, x3 = −3.0, = −1.0.
fi (x) = x1 ex3 ti + x2 ex4 ti − ti x1 x4
7
Problem 2.13 GAMMA. F (x) =
max |fi (x)|,
1≤i≤61
³
fi (x) = i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
x1 = 1,
x1 ti + x2 +
1 x3 ti +x4
´(ti + 1 ) 2
− 1,
Γ(ti + 1) exp(ti ) ti 1.000 1.010 1.020 1.030 1.050 1.075 1.100 1.125 1.150 1.200 1.250 1.300 1.350 1.400 1.500 1.600 1.700 1.800 1.900 2.000 2.100
i 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
ti 2.200 2.300 2.500 2.750 3.000 3.250 3.500 4.000 4.500 5.000 5.500 6.000 6.500 7.000 7.500 8.000 8.500 9.000 10.00 11.00 12.00
x2 = 1,
i 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
ti 13.00 15.00 17.50 20.00 22.50 25.00 30.00 35.00 40.00 50.00 60.00 70.00 80.00 100.0 150.0 200.0 300.0 500.0 105
x3 = 10,
x4 = 1.
Problem 2.14 EXP [9]. F (x) =
max fi (x),
1≤i≤21
x1 + x2 ti − exp(ti ), 1 + x3 ti + x4 t2i + x5 t3i = −1 + (i − 1)/10, 1 ≤ i ≤ 21, = 0.5, x2 = 0, x3 = 0,
fi (x) = ti x1
8
x4 = 0,
x5 = 0.
Problem 2.15 PBC1 [18]. F (x) =
max |fi (x)|,
1≤i≤30
q
(8ti − 1)2 + 1 arctan(8ti ) x1 + x2 ti + x3 t2i fi (x) = − , 1 + x4 ti + x5 t2i 8ti ti = −1 + 2(i − 1)/29, 1 ≤ i ≤ 30, x1 = 0, x2 = −1, x3 = 10, x4 = 1, x5 = 10. Problem 2.16 EVD61 [7]. F (x) = fi (x) yi ti x1 x4
= = = = =
max |fi (x)|,
1≤i≤51
x1 exp(−x2 ti ) cos(x3 ti + x4 ) + x5 exp(−x6 ti ) − yi , 0.5e−ti − e−2ti + 0.5e−3ti + 1.5e−1.5ti sin 7ti + e−2.5ti sin 5ti , 0.1(i − 1), 1 ≤ i ≤ 51, 2, x2 = 2, x3 = 7, 0, x5 = −2, x0 = 1.
Problem 2.17 Transformer [2]. F (x) = fi (x) =
max fi (x),
1≤i≤11
¯ ¯ ¯ ¯ v1 (x, ti ) ¯ ¯ ¯, ¯1 − 2 ¯ w1 (x, ti ) + v1 (x, ti ) ¯
where v1 (x, ti ) and w1 (x, ti ) are complex numbers obtained recursively so that v4 (x, ti ) = 1, w4 (x, ti ) = 10 and 1 wk+1 (x, ti ), x2k wk (x, ti ) = cos(ϑi x2k−1 )wk+1 (x, ti ) + j sin(ϑi x2k−1 )x2k vk+1 (x, ti ) vk (x, ti ) = cos(ϑi x2k−1 )vk+1 (x, ti ) + j sin(ϑi x2k−1 )
for k = 3, 2, 1. Here j = ϑi t1 t5 t9 x1 x4
= = = = = =
√
−1 is the imaginary unit and
(π/2) ti , 1 ≤ i ≤ 11, 0.5, t2 = 0.6, t3 = 0.7, t4 = 0.77, 0.9, t6 = 1.0, t7 = 1.1, t8 = 1.23, 1.3, t10 = 1.4, t11 = 1.5, 0.8, x2 = 1.5, x3 = 1.2, 3.0, x5 = 0.8, x6 = 6.0. 9
Problem 2.18 Filter [4]. F (x) =
max |fi (x)|,
1≤i≤41
Ã
fi (x) =
(x1 + (1 + x2 ) cos ϑi )2 + ((1 − x2 ) sin ϑi )2 (x3 + (1 + x4 ) cos ϑi )2 + ((1 − x4 ) sin ϑi )2
= = = = = = = =
2
· !1
Ã
yi ti ti ti ti x1 x5 x9
!1
(x5 + (1 + x6 ) cos ϑi )2 + ((1 − x6 ) sin ϑi )2 2 x9 − yi , (x7 + (1 + x8 ) cos ϑi )2 + ((1 − x8 ) sin ϑi )2 |1 − 2ti |, ϑi = πti 0.01(i − 1), 1 ≤ i ≤ 6, 0.07 + 0.03(i − 7), 7 ≤ i ≤ 20, t21 = 0.50, 0.54 + 0.03(i − 22), 22 ≤ i ≤ 35, 0.95 + 0.01(i − 36), 36 ≤ i ≤ 41, 0.00, x2 = 1.00, x3 = 0.00, x4 = −0.15, 0.00, x6 = −0.68, x7 = 0.00, x8 = −0.72, 0.37.
Problem 2.19 Wong 1 [1]. F (x) = max fi (x), 1≤i≤5
f1 (x) = (x1 − 10)2 + 5(x2 − 12)2 + x43 + 3(x4 − 11)2 + 10x65 + 7x26 + x47 − −4x6 x7 − 10x6 − 8x7 , f2 (x) = f1 (x) + 10(2x21 + 3x42 + x3 + 4x24 + 5x5 − 127), f3 (x) = f1 (x) + 10(7x1 + 3x2 + 10x23 + x4 − x5 − 282), f4 (x) = f1 (x) + 10(23x1 + x22 + 6x26 − 8x7 − 196), f5 (x) = f1 (x) + 10(4x21 + x22 − 3x1 x2 + 2x23 + 5x6 − 11x7 ), x1 = 1, x2 = 2, x3 = 0, x4 = 4, x5 = 0, x6 = 1, x7 = 1. Problem 2.20 Wong 2 [1]. F (x) = max fi (x), f1 (x) = f2 (x) f3 (x) f4 (x) f5 (x)
= = = =
1≤i≤9 x21 + x22
+ x1 x2 − 14x1 − 16x2 + (x3 − 10)2 + 4(x4 − 5)2 + (x5 − 3)2 + +2(x6 − 1)2 + 5x27 + 7(x8 − 11)2 + 2(x9 − 10)2 + (x10 − 7)2 + 45, f1 (x) + 10(3(x1 − 2)2 + 4(x2 − 3)2 + 2x23 − 7x4 − 120), f1 (x) + 10(5x21 + 8x2 + (x3 − 6)2 − 2x4 − 40), f1 (x) + 10(0.5(x1 − 8)2 + 2(x2 − 4)2 + 3x25 − x6 − 30), f1 (x) + 10(x21 + 2(x2 − 2)2 − 2x1 x2 + 14x5 − 6x6 ), 10
f6 (x) f7 (x) f8 (x) f9 (x) x1 x7
= = = = = =
f1 (x) + 10(4x1 + 5x2 − 3x7 + 9x8 − 105), f1 (x) + 10(10x1 − 8x2 − 17x7 + 2x8 ), f1 (x) + 10(−3x1 + 6x2 + 12(x9 − 8)2 − 7x10 ), f1 (x) + 10(−8x1 + 2x2 + 5x9 − 2x10 − 12), 2, x2 = 3, x3 = 5, x4 = 5, x5 = 1, 7, x8 = 3, x9 = 6, x10 = 10.
x6 = 2,
Problem 2.21 Wong 3 [1]. F (x) = f1 (x) =
f2 (x) f3 (x) f4 (x) f5 (x) f6 (x) f7 (x) f8 (x) f9 (x) f10 (x) f11 (x) f12 (x) f13 (x) f14 (x) f15 (x) f16 (x) f17 (x) f18 (x) x1 x8 x15
= = = = = = = = = = = = = = = = = = = =
max fi (x), 1≤i≤18 x21 + x22 + x1 x2 − 14x1 − 16x2 + (x3 − 10)2 + 4(x4 − 5)2 + (x5 − 3)2 + +2(x6 − 1)2 + 5x27 + 7(x8 − 11)2 + 2(x9 − 10)2 + (x10 − 7)2 + (x11 − 9)2 + +10(x12 − 1)2 + 5(x13 − 7)2 + 4(x14 − 14)2 + 27(x15 − 1)2 + x416 + (x17 − 2)2 +13(x18 − 2)2 + (x19 − 3)2 + x220 + 95, f1 (x) + 10(3(x1 − 2)2 + 4(x2 − 3)2 + 2x23 − 7x4 − 120), f1 (x) + 10(5x21 + 8x2 + (x3 − 6)2 − 2x4 − 40), f1 (x) + 10(0.5(x1 − 8)2 + 2(x2 − 4)2 + 3x25 − x6 − 30), f1 (x) + 10(x21 + 2(x2 − 2)2 − 2x1 x2 + 14x5 − 6x6 ),
+
f1 (x) + 10(4x1 + 5x2 − 3x7 + 9x8 − 105), f1 (x) + 10(10x1 − 8x2 − 17x7 + 2x8 ), f1 (x) + 10(−3x1 + 6x2 + 12(x9 − 8)2 − 7x10 ), f1 (x) + 10(−8x1 + 2x2 + 5x9 − 2x10 − 12), f1 (x) + 10(x1 + x2 + 4x11 − 21x12 ), f1 (x) + 10(x21 + 5x11 − 8x12 − 28), f1 (x) + 10(4x1 + 9x2 + 5x213 − 9x14 − 87), f1 (x) + 10(3x1 + 4x2 + 3(x13 − 6)2 − 14x14 − 10), f1 (x) + 10(14x21 + 35x15 − 79x16 − 92), f1 (x) + 10(15x22 + 11x15 − 61x16 − 54), f1 (x) + 10(5x21 + 2x2 + 9x417 − x18 − 68), f1 (x) + 10(x21 − x2 + 19x19 − 20x20 + 19), f1 (x) + 10(7x21 + 5x22 + x219 − 30x20 ), 2, x2 = 3, x3 = 5, x4 = 5, x5 = 1, x6 = 2, x7 = 7, 3, x9 = 6, x10 = 10, x11 = 2, x12 = 2, x13 = 6, x14 = 15, 1, x16 = 2, x17 = 1, x18 = 2, x19 = 1, x20 = 3.
11
Problem 2.22 Polak 2 [17] F (x) f (x) e2 x1
= = = =
max {f (x + 2e2 ), f (x − 2e2 )} , exp(10−8 x21 + x22 + x23 + 4x24 + x25 + x26 + x27 + x28 + x29 + x210 ), second column of the unit matrix, 100, xi = 0.1, 2 ≤ i ≤ 10.
Problem 2.23 Polak 3 [17]. F (x) = fi (x) =
max fi (x),
1≤i≤10
³ ´ 1 exp (xj+1 − sin(i − 1 + 2j))2 , j=0 i + j 10 X
xi = 1,
1 ≤ i ≤ 11.
Problem 2.24 Watson [21]. F (x) =
max |fi (x)|,
1≤i≤31
f1 (x) = x1 , f2 (x) = x2 − x21 − 1, fi (x) =
n X
µ
(j − 1)xj
j=2
xi = 0,
i−2 29
¶j−2
−
n X
j=1
µ
xj
i−2 29
¶j−1 2 ,
3 ≤ j ≤ 31,
1 ≤ i ≤ 20.
Problem 2.25 Osborne 2 [21]. F (x) =
max |fi (x)|,
1≤i≤65
fi (x) = yi − x1 exp(−x5 ti ) − x2 exp(−x6 (ti − x9 )2 ) − x3 exp(−x7 (ti − x10 )2 ) − −x4 exp(−x8 (ti − x11 )2 ), ti = 0.1(i − 1), 1 ≤ i ≤ 65.
12
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
x1 = 1.30, x5 = 0.60, x9 = 2.00,
3
yi 1.366 1.191 1.112 1.013 0.991 0.885 0.831 0.847 0.786 0.725 0.746 0.679 0.608 0.655 0.616 0.606 0.602 0.626 0.651 0.724 0.649 0.649
i 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
yi 0.694 0.644 0.624 0.661 0.612 0.558 0.533 0.495 0.500 0.423 0.395 0.375 0.372 0.391 0.396 0.405 0.428 0.429 0.523 0.562 0.607 0.653
i 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
yi 0.672 0.708 0.633 0.668 0.645 0.632 0.591 0.559 0.597 0.625 0.739 0.710 0.729 0.720 0.636 0.581 0.428 0.292 0.162 0.098 0.054
x2 = 0.65, x3 = 0.65, x4 = 0.70, x6 = 3.00, x7 = 5.00, x8 = 7.00, x10 = 4.50, x11 = 5.50.
Test problems for general nonsmooth unconstrained optimization
Calling statements have the form CALL CALL CALL CALL
TIUD19(N,X,FMIN,XMAX,NEXT,IERR), TFFU19(N,X,F,NEXT), TFGU19(N,X,G,NEXT), TFHD19(N,X,H,NEXT),
with the following significance: TIUD19 TFFU19 TFGU19 TFHD19
-
initiation of vector of variables N, X. evaluation of the objective function value F at point X. evaluation of the objective function gradient G at point X. evaluation of the objective function Hessian matrix H at point X. 13
We seek a local minimum of the objective function F (x) from the starting point x¯. The description of individual problems follows. Table 3.1 contains problem dimensions and optimum function values. No. 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25
Problem Rosenbrock Crescent CB2 CB3 DEM QL LQ Mifflin 1 Mifflin 2 Wolfe Rosen-Suzuki Shor Colville 1 HS78 El-Attar Maxquad Gill Steiner 2 Maxq Maxl TR48 Goffin MXHILB L1HILB Shell Dual
n Optimum value 2 0 2 0 2 1.9522245 2 2 2 −3 2 7.20 2 −1.4142136 2 −1 2 −1 2 −8 4 −44 5 22.600162 5 −32.348679 5 −2.9197004 6 0.5598131 10 −0.8414083 10 9.7857721 12 16.703838 20 0 20 0 48 −638565.0 50 0 50 0 50 0 15 32.348679
Table 3.1 Problem 3.1 Rosenbrock [15]. F (x) = 100(x2 − x21 )2 + (1 − x1 )2 , x1 = −1.2, x2 = 1.0. Problem 3.2 Crescent [15]. o
n
F (x) = max x21 + (x2 − 1)2 + x2 − 1, −x21 − (x2 − 1)2 + x2 + 1 , x1 = −1.5, x2 = 2.0.
14
Problem 3.3 CB2 [15]. n
o
n
o
F (x) = max x21 + x42 , (2 − x1 )2 + (2 − x2 )2 , 2e−x1 +x2 , x1 = 1.0, x2 = −0.1. Problem 3.4 CB3 [15]. F (x) = max x41 + x22 , (2 − x1 )2 + (2 − x2 )2 , 2e−x1 +x2 , x1 = 2, x2 = 2. Problem 3.5 DEM [15]. n
o
F (x) = max 5x1 + x2 , −5x1 + x2 , x21 + x22 + 4x2 , x1 = 1, x2 = 1. Problem 3.6 QL [15]. F (x) = max fi (x), f1 (x) f2 (x) f3 (x) x1
1≤i≤3 x21 + x22 , x21 + x22 + x21 + x22 +
= = = = −1,
10(−4x1 − x2 + 4), 10(−x1 − 2x2 + 6), x2 = 5.
Problem 3.7 LQ [15]. n
o
F (x) = max −x1 − x2 , −x1 − x2 + (x21 + x22 − 1) , x1 = −0.5, x2 = −0.5. Problem 3.8 Mifflin 1 [15]. n
o
F (x) = −x1 + 20 max x21 + x22 − 1, 0 , x1 = 0.8, x2 = 0.6. Problem 3.9 Mifflin 2 [15]. F (x) = −x1 + 2(x21 + x22 − 1) + 1.75|x21 + x22 − 1|, x1 = −1, x2 = −1. 15
Problem 3.10 Wolfe [23]. F (x) = f1 (x) , F (x) = f2 (x) , F (x) = f3 (x) ,
x1 ≥ |x2 |, 0 < x1 < |x2 |, x1 ≤ 0,
q
f1 (x) = 5 (9x21 + 16x22 ), f2 (x) = 9x1 + 16|x2 |, f3 (x) = 9x1 + 16|x2 | − x91 , x1 = 3,
x2 = 2.
Problem 3.11 Rosen-Suzuki [15]. F (x) f1 (x) f2 (x) f3 (x) f4 (x) xi
= = = = = =
max {f1 (x), f1 (x) + 10f2 (x), f1 (x) + 10f3 (x), f1 (x) + 10f4 (x)} , x21 + x22 + 2x23 + x24 − 5x1 − 5x2 − 21x3 + 7x4 , x21 + x22 + x23 + x24 + x1 − x2 + x3 − x4 − 8, x21 + 2x22 + x23 + 2x24 − x1 − x4 − 10, x21 + x22 + x23 + 2x1 − x2 − x4 − 5, 0, 1 ≤ i ≤ 4.
Problem 3.12 Shor [15].
F (x) =
max
1≤i≤10
A
=
x1 = 0,
0 2 1 1 3 0 1 1 0 1
0 1 2 4 2 2 1 0 0 1
bi
5 X
(xj − aij )2 ,
j=1
0 1 1 1 1 1 1 1 2 2 x2
0 1 1 2 0 0 1 2 1 0 =
0 3 2 2 1 , 1 1 1 0 0 0,
b =
x3 = 0,
16
1 5 10 2 4 3 1.7 2.5 6 3.5
,
x4 = 0,
x5 = 1.
Problem 3.13 Colville 1 [3]. F (x) =
5 X
dj x3j
+
j=1
5 X 5 X
cij xi xj +
i=1 j=1
5 X j=1
−16
2
0
1
0
1
1
1
1
1
ej xj + 50 max 0, max bi − 1≤i≤10
C
= "
e
T
=
30 −20 −10 32 −10 −20 39 −6 −31 32 −10 −6 10 −6 −10 , 32 −31 −6 39 −20 −10 32 −10 −20 30 −15 −27 −36 −18 −12
x1 = 0,
x2 = 0,
−40
5 X
aij xj ,
j=1
−2 −0.25 −4 −4 , b = −1 −40 −60 5
0 −2 0 4 2 −3.5 0 2 0 0 0 −2 0 −4 −1 0 −9 −2 1 −2.8 , A = 2 0 −4 0 0 −1 −1 −1 −1 −1 −1 −2 −3 −2 −1 1 2 3 4 5
d=
#
1 4 8 10 6 2
,
,
x3 = 0,
x4 = 0,
x5 = 1.
Problem 3.14 HS78 [10]. F (x) = x1 x2 x3 x4 x5 + 10 f1 (x) f2 (x) f3 (x) x1 x4
= = = = =
3 X
|fi (x)|,
i=1 2 x4 +
x21 + x22 + x23 + x25 − 10, x2 x3 − 5x4 x5 , x31 + x32 + 1, −2.0, x2 = 1.5, x3 = 2.0, −1.0, x5 = −1.0.
Problem 3.15 El-Attar [7]. F (x) =
51 ¯ ¯ X ¯ ¯ ¯x1 e−x2 ti cos(x3 ti + x4 ) + x5 e−x6 ti − yi ¯ , i=1
yi ti x1 x4
= = = =
0.5e−ti − e−2ti + 0.5e−3ti + 1.5e−1.5ti sin 7ti + e−2.5ti sin 5ti , 0.1(i − 1), 1 ≤ i ≤ 51, 2, x2 = 2, x3 = 7, 0, x5 = −2, x6 = 1. 17
Problem 3.16 Maxquad [12] and [15]. ³
´
F (x) = max xT Ai x − xT bi , 1≤i≤5
Aikj Aijj
= Aijk = ej/k cos(jk) sin(i), X j = | sin(i)| + |Aijk |, 10 k6=j
j < k,
bij = ej/i sin(ij), xi = 0, 1 ≤ i ≤ 10. Problem 3.17 Gill [11]. F (x) = max{f1 (x), f2 (x), f3 (x)}, f1 (x) =
10 X
2
(xi − 1) + 10
−3
i=1
f2 (x) =
10 µ X
x2i
i=1
1 − 4
¶2
,
2 ¶j−2 ¶j−1 2 µ µ 10 10 30 X X X i − 1 i − 1 − 1 − xj xj (j − 1)
29
j=2
i=2
29
j=1
+x21 + (x2 − x21 − 1)2 , f3 (x) =
10 h X
i
100(xi − x2i−1 )2 + (1 − xi )2 ,
i=2
xi = −0.1,
1 ≤ i ≤ 10.
Problem 3.18 Steiner 2 [11]. q
q
x21 + x21+m +
F (x) = +
m X
(a21 − xm )2 + (a22 − x2m )2 +
q
pj (aj1 − xj )2 + (aj2 − xj+m )2 +
j=1
+
m−1 X
q
p˜j (xj − xj+1 )2 + (xj+m − xj+m+1 )2 ,
j=1
a21 a11 a21 a31 a41 a51 a61
= = = = = = =
5.5, 0.0, 2.0, 3.0, 4.0, 5.0, 6.0,
a22 a12 a22 a32 a42 a52 a62
= = = = = = =
−1.0, 2.0, 3.0, −1.0, −0.5, 2.0, 2.0, 18
p1 p2 p3 p4 p5 p6
= = = = = =
2, 1, 1, 5, 1, 1,
p˜1 p˜2 p˜3 p˜4 p˜5
= = = = =
1, 1, 2, 3, 2,
m = 6,
x1 = (a11 + a21 ) /3, xj =
³
x1+m = (a12 + a22 ) /3,
´
xj−1 + aj1 + a(j+1)1 /3,
xm = (xm−1 + am1 + a21 ) /3,
xj+m =
³
´
xj−1+m + aj2 + a(j+1)2 /3,
x2m = (x2m−1 + am2 + a22 ) /3.
Problem 3.19 Maxq [15]. F (x) =
max x2i ,
1≤i≤20
xi = i,
i = 1, . . . , 10,
xi = −i,
i = 11, . . . , 20.
xi = −i,
i = 11, . . . , 20.
Problem 3.20 Maxl [15]. F (x) =
max |xi |,
1≤i≤20
xi = i,
i = 1, . . . , 10,
Problem 3.21 TR48 [12] and [15].
F (x) =
48 X j=1
xi = 0,
dj max (xi − aij ) − 1≤i≤48
48 X
s i xi ,
i=1
1 ≤ i ≤ 48.
Coefficients aij , si , dj are given in [12]. Problem 3.22 Goffin [15].
F (x) = 50 max xi − 1≤i≤50
xi = i − 25.5,
50 X
xi ,
i=1
1 ≤ i ≤ 50.
Problem 3.23 MXHILB [11].
F (x) =
¯ ¯ ¯ 50 ¯ ¯X ¯ x j ¯ ¯, max ¯ ¯ 1≤i≤50 ¯ i + j − 1 ¯ j=1
xi = 1,
1 ≤ i ≤ 50.
19
2 ≤ j ≤ m − 1,
Problem 3.24 L1HILB [11].
F (x) =
¯ ¯ ¯ 50 ¯¯X 50 X ¯ x j ¯ ¯, ¯ ¯ i + j − 1 ¯ ¯ i=1 j=1
xi = 1,
1 ≤ i ≤ 50.
Problem 3.25 Shell Dual [8]. ¯ 5 ¯ 5 X 5 ¯X ¯ X ¯ ¯ 3 F (x) = 2 ¯¯ di xi+10 ¯¯ + cij xi+10 xj+10 − i=1 i=1 j=1 à 5 10 X X
−
bi xi + 100
i=1
Pi (x) =
10 X
15 X
max(0, Pi (x)) − Q(x) ,
i=1
aji xj − 2
j=1
Q(x) =
!
5 X
cij xj+10 − 3di x2i+10 − ei ,
1 ≤ i ≤ 5,
j=1
min(0, xi ),
i=1 −4
xi = 10 ,
1 ≤ i ≤ 15,
i 6= 7,
x7 = 60.
Matrices A, b, C, d, e are the same as in Problem 3.13.
4
Test problems for linearly constrained minimax optimization
Calling statements have the form CALL CALL CALL CALL
TILD22(N,NA,NB,NC,X,IX,XL,XU,IC,CL,CU,CG,FMIN,XMAX,NEXT,IERR), TAFU22(N,KA,X,FA,NEXT), TAGU22(N,KA,X,GA,NEXT), TAHD22(N,KA,X,hA,NEXT),
with the following significance: TILD22 - determination of problem dimension N, NA, NB, NC, initiation of vector of variables X, definition of box constraints IX, XL, XU, definition of the general linear constraints IC, CL, CU, CG. TAFU22 - evaluation of the KA-th partial function value FA at point X. TAGU22 - evaluation of the KA-th partial function gradient GA at point X. TAHD22 - evaluation of the KA-th partial function Hessian matrix HA at point X. We seek a minimum of a minimax objective function F (x) = max (fk (x)), 1≤k≤nA
20
x ∈ Rn ,
with simple bounds xi − unbounded xli ≤ xi xi ≤ xui xli ≤ xi ≤ xui xi = xli = xui
Iix Iix Iix Iix Iix
, , , , ,
= 0, = 1, = 2, = 3, =5
and general linear constraints aTi x − unbounded cli ≤ aTi x aTi x ≤ cui cli ≤ aTi x ≤ cui aTi x = cli = cui
, , , , ,
Iic Iic Iic Iic Iic
= 0, = 1, = 2, = 3, =5
from the starting point x¯. The description of individual problems follows. Table 4.1 contains problem dimensions and optimum function values. No. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15
Problem MAD1 MAD2 MAD4 MAD5 PENTAGON MAD6 EQUIL Wong 2 Wong 3 MAD8 BP filter HS114 Dembo 3 Dembo 5 Dembo 7
n nA 2 3 2 3 2 3 2 3 6 3 7 163 8 8 10 6 20 14 20 38 9 124 10 9 7 13 8 4 16 19
nC 1 1 1 1 15 9 1 3 4 0 4 5 2 3 1
Optimum value −0.38965952 −0.33035714 −0.44891079 −0.42928061 −1.85961870 0.10183089 0 24.306209 133.72828 0.50694799 0.27607734 · 10−3 −1768.8070 1227.2260 7049.2480 174.78699
Table 4.1 Problem 4.1 MAD1 [14]. F (x) = max fi (x), 1≤i≤3 x21 + x22
f1 (x) = + x1 x2 − 1, f2 (x) = sin x1 , 21
f3 (x) = − cos x2 , c1 (x) = x1 + x2 − 0.5 ≥ 0 x1 = 1, x2 = 2. Problem 4.2 MAD2 [14]. F (x) = max fi (x), 1≤i≤3
fi (x) − as in Problem 4.1, c1 (x) = −3x1 − x2 − 2.5 ≥ 0, x1 = −2, x2 = −1. Problem 4.3 MAD4 [14]. F (x) = max fi (x), 1≤i≤3
f1 (x) f2 (x) f3 (x) c1 (x) x1
= = = = =
− exp(x1 − x2 ), sinh(x1 − 1) − 1, − log(x2 ) − 1, 0.05x1 − x2 + 0.5 ≥ 0, −1.00, x2 = 0.01.
Problem 4.4 MAD5 [14]. F (x) = max fi (x), 1≤i≤3
fi (x) − as in Problem 4.3, c1 (x) = −0.9x1 + x2 − 1 ≥ 0, x1 = −1, x2 = 3. Problem 4.5 PENTAGON [19]. F (x) = max fi (x), 1≤i≤3
q
f1 (x) = − (x1 − x3 )2 + (x2 − x4 )2 , q
f2 (x) = − (x3 − x5 )2 + (x4 − x6 )2 , q
f3 (x) = − (x5 − x1 )2 + (x6 − x2 )2 , 2πj 2πj + xi+1 sin ≤ 1, xi cos 5 5 i = 1, 3, 5, j = 0, 1, 2, 3, 4.
22
Problem 4.6 MAD6 [14]. F (x) = fi (x) = ϑi = c1 (x) c2 (x) c3 (x) c4 (x) c5 (x) c6 (x) c7 (x) c8 (x) c9 (x) x1 x5
= = = = = = = = = = =
max fi (x),
1≤i≤163
7 1 2 X + cos(2πxj sin ϑi ), 1 ≤ i ≤ 163, 15 15 j=1 π (8.5 + i0.5), 1 ≤ i ≤ 163, 180 x1 ≥ 0.4, −x1 + x2 ≥ 0.4, −x2 + x3 ≥ 0.4, −x3 + x4 ≥ 0.4, −x4 + x5 ≥ 0.4, −x5 + x6 ≥ 0.4, −x6 + x7 ≥ 0.4, −x4 + x6 = 1.0, x7 = 3.5, 0.5, x2 = 1.0, x3 = 1.5, x4 = 2.0, 2.5, x6 = 3.0, x7 = 3.5.
Problem 4.7 EQUIL [12]. F (x) = max fi (x), 1≤i≤8
fi (x) =
wjk xk 5 aji X k=1 − w ji , 8 bj P 1−bj
j=1
c1 (x) =
8 X
xi
k=1
= [wjk ] =
A
ajk xk
xi = 1,
i=1
W
8 P
= [ajk ] =
b = [bj ] =
h
xi ≥ 0,
1 ≤ i ≤ 8,
3 1 0.1 0.1 5 0.1 0.1 10 0.1 0.1 5 0.1 0.1 9 10 0.1 4 0.1 0.1 0.1 0.1 10 0.1 3 0.1 0.1 0.1 0.1 0.1 0.1 1 2 1 2 1.2
0.1 0.1 7 0.1 0.1
6 0.1 0.1 0.1 11
,
1 1 1 1 1 1 1 0.8 1 0.5 1 1 1 1 1.2 0.8 1.2 1.6 2 0.6 0.1 , 0.1 0.6 2 1 1 1 2 1.2 0.8 1 1.2 0.1 3 4 i
0.5 1.2 0.8 2.0 1.5 , 23
xi = 0.125,
1 ≤ i ≤ 8.
Problem 4.8 Wong 2 [1]. F (x) = max fi (x), f1 (x) = f2 (x) f3 (x) f4 (x) f5 (x) f6 (x) c1 (x) c2 (x) c3 (x) x1 x7
= = = = = = = = = =
1≤i≤6 x21 + x22
+ x1 x2 − 14x1 − 16x2 + (x3 − 10)2 + 4(x4 − 5)2 + (x5 − 3)2 + +2(x6 − 1)2 + 5x27 + 7(x8 − 11)2 + 2(x9 − 10)2 + (x10 − 7)2 + 45, f1 (x) + 10(3(x1 − 2)2 + 4(x2 − 3)2 + 2x23 − 7x4 − 120), f1 (x) + 10(5x21 + 8x2 + (x3 − 6)2 − 2x4 − 40), f1 (x) + 10(0.5(x1 − 8)2 + 2(x2 − 4)2 + 3x25 − x6 − 30), f1 (x) + 10(x21 + 2(x2 − 2)2 − 2x1 x2 + 14x5 − 6x6 ), f1 (x) + 10(−3x1 + 6x2 + 12(x9 − 8)2 − 7x10 ), 4x1 + 5x2 − 3x7 + 9x8 ≤ 105, 10x1 − 8x2 − 17x7 + 2x8 ≤ 0, −8x1 + 2x2 + 5x9 − 2x10 ≤ 12, 2, x2 = 3, x3 = 5, x4 = 5, x5 = 1, x6 = 2, 7, x8 = 3, x9 = 6, x10 = 10.
Problem 4.9 Wong 3 [1]. F (x) = f1 (x) =
f2 (x) f3 (x) f4 (x) f5 (x) f6 (x) f7 (x) f8 (x) f9 (x) f10 (x) f11 (x)
= = = = = = = = = =
max fi (x), 1≤i≤14 x21 + x22 + x1 x2 − 14x1 − 16x2 + (x3 − 10)2 + 4(x4 − 5)2 + (x5 − 3)2 + +2(x6 − 1)2 + 5x27 + 7(x8 − 11)2 + 2(x9 − 10)2 + (x10 − 7)2 + (x11 − 9)2 + +10(x12 − 1)2 + 5(x13 − 7)2 + 4(x14 − 14)2 + 27(x15 − 1)2 + x416 + (x17 − 2)2 +13(x18 − 2)2 + (x19 − 3)2 + x220 + 95, f1 (x) + 10(3(x1 − 2)2 + 4(x2 − 3)2 + 2x23 − 7x4 − 120), f1 (x) + 10(5x21 + 8x2 + (x3 − 6)2 − 2x4 − 40), f1 (x) + 10(0.5(x1 − 8)2 + 2(x2 − 4)2 + 3x25 − x6 − 30), f1 (x) + 10(x21 + 2(x2 − 2)2 − 2x1 x2 + 14x5 − 6x6 ), f1 (x) + 10(−3x1 + 6x2 + 12(x9 − 8)2 − 7x10 ), f1 (x) + 10(x21 + 5x11 − 8x12 − 28), f1 (x) + 10(4x1 + 9x2 + 5x213 − 9x14 − 87), f1 (x) + 10(3x1 + 4x2 + 3(x13 − 6)2 − 14x14 − 10), f1 (x) + 10(14x21 + 35x15 − 79x16 − 92), f1 (x) + 10(15x22 + 11x15 − 61x16 − 54), 24
+
f12 (x) f13 (x) f14 (x) c1 (x) c2 (x) c3 (x) c4 (x) x1 x8 x15
= = = = = = = = = =
f1 (x) + 10(5x21 + 2x2 + 9x417 − x18 − 68), f1 (x) + 10(x21 − x9 + 19x19 − 20x20 + 19), f1 (x) + 10(7x21 + 5x22 + x219 − 30x20 ), 4x1 + 5x2 − 3x7 + 9x8 ≤ 105, 10x1 − 8x2 − 17x7 + 2x8 ≤ 0, −8x1 + 2x2 + 5x9 − 2x10 ≤ 12, x1 + x2 + 4x11 − 21x12 ≤ 0, 2, x2 = 3, x3 = 5, x4 = 5, x5 = 1, x6 = 2, x7 = 7, 3, x9 = 6, x10 = 10, x11 = 2, x12 = 2, x13 = 6, x14 = 15, 1, x16 = 2, x17 = 1, x18 = 2, x19 = 1, x20 = 3.
Problem 4.10 MAD8 [14]. F (x) =
max |fi (x)|,
1≤i≤38
f1 (x) = −1 + x21 +
20 X
xj ,
j=2 20 X
fi (x) = −1 + ci x2k +
xj ,
1 < i < 38,
j=1,j6=k
f38 (x) = −1 + x220 +
19 X
xj ,
j=1
k k xj xj
= = ≥ =
(i + 2)/2, ci = 1, (i + 1)/2, ci = 2, 0.5, 1 ≤ j ≤ 10, 100, 1 ≤ j ≤ 20.
i = 2, 4, . . . , 36, i = 3, 5, . . . , 37,
Problem 4.11 BP filter. F (x) = f1 (x) f2j (x) f2j+1 (x) f124 (x)
= = = =
max fi (x),
1≤i≤124
ϕ(x, t0 ) − x9 + a, ϕ(x, tj ) − x9 , 1 ≤ j ≤ 61, x9 − ϕ(x, tj ), 1 ≤ j ≤ 61, ϕ(x, t62 ) − x9 + a,
4 ³ ´ 1X log (x2i+4 − t2 )2 + t2 x2i − 4 log(x), 2 i=1 = 0.967320, t1 = 0.987423, = t1 + (j − 1)(t61 − t1 ), 1 ≤ j ≤ 61,
ϕ(x, t) = t0 tj
25
a = 3.0164,
t61 ci (x) x1 x4 x8
= = = = =
1.012577, t62 = 1.032680, xi+4 − 10000xi ≤ 0, 1 ≤ i ≤ 4, 0.398 · 10−1 , x2 = 0.968 · 10−4 , x3 = 0.103 · 10−3 , 0.389 · 10−1 , x5 = 1.010, x6 = 0.968, x7 = 1.030, 0.981, x9 = −11.6.
Problem 4.12 HS114 [10] F (x) = max fi (x), 1≤i≤9
f1 (x) = 5.04x1 + 0.035x2 + 10x3 + 3.36x5 − 0.063x4 x7 , µ
f2 (x) = f3 (x) = f4 (x) = f5 (x) = f6 (x) = f7 (x) = f8 (x) = f9 (x) = c1 (x) = c2 (x) = c3 (x) = c4 (x) = c5 (x) = a =
¶
1 f1 (x) + 500 1.12x1 + 0.13167x1 x8 − − x4 , a ´ ³ 2 f1 (x) − 500 1.12x1 + 0.13167x1 x8 − 0.00667x1 x8 − ax4 , µ ¶ 1 2 f1 (x) + 500 1.098x8 − 0.038x8 + 0.325x6 − x7 + 57.425 , a ³ ´ 2 f1 (x) − 500 1.098x8 − 0.038x8 + 0.325x6 − ax7 + 57.425 , µ ¶ 98000x3 f1 (x) + 500 − x6 , x4 x9 + 1000x3 µ ¶ 98000x3 − x6 , f1 (x) − 500 x x + 1000x3 µ 4 9 ¶ x2 + x5 − x8 , f1 (x) + 500 x1 µ ¶ x2 + x5 − x8 , f1 (x) − 500 x1 0.222x10 + bx9 ≤ 35.82, 1 0.222x10 + x9 ≥ 35.82, b 3x7 − ax10 ≥ 133, 1 3x7 − x10 ≤ 133, a 1.22x4 − x1 − x5 = 0, 0.99, b = 0.90, 0.00667x1 x28
26
10−5 10−5 10−5 10−5 10−5 85 90 3 1.2 145
≤ x1 ≤ x2 ≤ x3 ≤ x4 ≤ x5 ≤ x6 ≤ x7 ≤ x8 ≤ x9 ≤ x10
≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤
2000, 16000, 120, 5000, 2000, 93, 95, 12, 4, 162,
x1 = 1745, x2 = 12000, x3 = 110, x4 = 3048, x5 = 1974, x6 = 89.2, x7 = 92.8, x8 = 8.0, x9 = 3.6, x10 = 145.
Problem 4.13 Dembo 3 [6]. F (x) =
max fi (x),
1≤i≤13
f1 (x) = a1 x1 + a2 x1 x6 + a3 x3 + a4 x2 + a5 + a6 x3 x5 , ³
´
f2 (x) = f1 (x) + 105 a7 x26 + a8 x−1 1 x3 + a 9 x6 − 1 , ³
´
−1 −1 2 f3 (x) = f1 (x) + 105 a10 x1 x−1 3 + a11 x1 x3 x6 + a12 x1 x3 x6 − 1 ,
³
´
f4 (x) = f1 (x) + 105 a13 x26 + a14 x5 + a15 x4 + a16 x6 − 1 , ³
´
−1 −1 −1 2 f5 (x) = f1 (x) + 105 a17 x−1 5 + a18 x5 x6 + a19 x4 x5 + a20 x5 x6 − 1 ,
³
´
−1 −1 f6 (x) = f1 (x) + 105 a21 x7 + a22 x2 x−1 3 x4 + a23 x2 x3 − 1 ,
³
´
−1 −1 −1 −1 −1 f7 (x) = f1 (x) + 105 a24 x−1 7 + a25 x2 x3 x7 + a26 x2 x3 x4 x7 − 1 ,
³
´
³
´
−1 f8 (x) = f1 (x) + 105 a27 x−1 5 + a28 x5 x7 − 1 , −1 f9 (x) = f1 (x) + 105 a33 x1 x−1 3 + a34 x3 − 1 ,
³
´
−1 −1 f10 (x) = f1 (x) + 105 a35 x2 x−1 3 x4 + a36 x2 x3 − 1 ,
³
´
f11 (x) = f1 (x) + 105 a37 x4 + a38 x−1 2 x3 x4 − 1 , f12 (x) = f1 (x) + 105 (a39 x1 x6 + a40 x1 + a41 x3 − 1) , ³
´
−1 f13 (x) = f1 (x) + 105 a42 x−1 1 x3 + a43 x1 + a44 x6 − 1 ,
c1 (x) = a29 x5 + a30 x7 ≤ 1, c2 (x) = a31 x3 + a32 x1 ≤ 1,
27
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
ai 1.715 0.035 4.0565 10.0 3000.0 −0.063 0.59553571 · 10−2 0.88392857 −0.11756250 1.10880000 0.13035330 −0.00660330 0.66173269 · 10−3 0.17239878 · 10−1 −0.56595559 · 10−2 1 1 1 85 90 3 145
i 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ≤ ≤ ≤ ≤ ≤ ≤ ≤
x1 x2 x3 x4 x5 x6 x7
ai −0.19120592 · 10−1 0.56850750 · 102 1.08702000 0.32175000 −0.03762000 0.00619800 0.24623121 · 104 −0.25125634 · 102 0.16118996 · 103 5000.0 −0.48951000 · 106 0.44333333 · 102 0.33000000 0.02255600 −0.00759500
i 31 32 33 34 35 36 37 38 39 40 41 42 43 44
ai 0.00061000 −0.0005 0.81967200 0.81967200 24500.0 −250.0 0.10204082 · 10−1 0.12244898 · 10−4 0.00006250 0.00006250 −0.00007625 1.22 1.0 −1.0
≤ 2000 , x1 = 1745, ≤ 120 , x2 = 110, ≤ 5000 , x3 = 3048, ≤ 93 , x4 = 89, ≤ 95 , x3 = 92, ≤ 12 , x4 = 8, ≤ 162 , x5 = 145.
Problem 4.14 Dembo 5 [6]. F (x) = max fi (x), 1≤i≤4
f1 (x) = x1 + x2 + x3 , ³
´
−1 −1 −1 −1 f2 (x) = f1 (x) + 105 ax−1 1 x4 x6 + 100x6 + bx1 x6 − 1 ,
³
´
−1 −1 f3 (x) = f1 (x) + 105 x4 x−1 7 + 1250(x5 − x4 )x2 x7 − 1 ,
³
´
−1 −1 −1 −1 f4 (x) = f1 (x) + 105 cx−1 3 x8 + x5 x8 − 2500x3 x5 x8 − 1 ,
c1 (x) c2 (x) c3 (x) a
= = = =
0.0025(x4 + x6 ) ≤ 1, 0.0025(x5 + x7 − x4 ) ≤ 1, 0.01(x8 − x5 ) ≤ 1, 833.33252, b = −83333.333,
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c = 1250000.0.
100 1000 1000 10 10 10 10 10
≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤
x1 x2 x3 x4 x5 x6 x7 x8
≤ 10000 , x1 = 5000, ≤ 10000 , x2 = 5000, ≤ 10000 , x3 = 5000, ≤ 1000 , x4 = 200, ≤ 1000 , x5 = 350, ≤ 1000 , x6 = 150, ≤ 1000 , x7 = 225, ≤ 1000 , x8 = 425.
Problem 4.15 Dembo 7 [6]. F (x) =
max fi (x),
1≤i≤19
f1 (x) = a(x12 + x13 + x14 + x15 + x16 ) + b(x1 x12 + x2 x13 + x3 x14 + x4 x15 + x5 x16 ), ³
´
³
´
³
´
³
´
³
´
2 −1 f2 (x) = f1 (x) + 103 cx1 x−1 6 + 100dx1 − dx1 x6 − 1 , 2 −1 f3 (x) = f1 (x) + 103 cx2 x−1 7 + 100dx2 − dx2 x7 − 1 , 2 −1 f4 (x) = f1 (x) + 103 cx3 x−1 8 + 100dx3 − dx3 x8 − 1 , 2 −1 f5 (x) = f1 (x) + 103 cx4 x−1 9 + 100dx4 − dx4 x9 − 1 , 2 −1 f6 (x) = f1 (x) + 103 cx5 x−1 10 + 100dx5 − dx5 x10 − 1 ,
³
´
−1 −1 −1 −1 f7 (x) = f1 (x) + 103 x6 x−1 7 + x1 x7 x11 x12 − x6 x7 x11 x12 − 1 ,
³
´
−1 −1 f8 (x) = f1 (x) + 103 x7 x−1 8 + 0.002(x7 − x1 )x8 x12 + 0.002(x2 x8 − 1)x13 − 1 ,
f9 (x) = f1 (x) + 103 (x8 + 0.002(x8 − x2 )x13 + 0.002(x3 − x9 )x14 + x9 − 1) , ³
´
−1 −1 −1 −1 f10 (x) = f1 (x) + 103 x−1 3 x9 + (x4 − x8 )x3 x14 x15 + 500(x10 − x9 )x3 x14 − 1 ,
³
´
−1 −1 −1 −1 f11 (x) = f1 (x) + 103 (x−1 4 x5 − 1)x15 x16 + x4 x10 + 500(1 − x4 x10 )x15 − 1 ,
³
´
−1 f12 (x) = f1 (x) + 103 0.9x−1 4 + 0.002(1 − x4 x5 )x16 − 1 ,
³
´
f13 (x) = f1 (x) + 103 x−1 11 x12 − 1 , ³
´
³
´
³
´
³
´
³
´
³
´
f14 (x) = f1 (x) + 103 x4 x−1 5 −1 , f15 (x) = f1 (x) + 103 x3 x−1 4 −1 , f16 (x) = f1 (x) + 103 x2 x−1 3 −1 , f17 (x) = f1 (x) + 103 x1 x−1 2 −1 , f18 (x) = f1 (x) + 103 x9 x−1 10 − 1 , f19 (x) = f1 (x) + 103 x8 x−1 9 −1 , c1 (x) = 0.002(x11 − x12 ) ≤ 1, a = 1.262626, b = −1.231060,
c = 0.034750,
29
d = 0.009750.
0.1 0.1 0.1 0.1 0.9 10−4 0.1 0.1 0.1 0.1 1 −6 10 1 500 500 10−6
≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
≤ 0.9 , x1 = 0.80, ≤ 0.9 , x2 = 0.83, ≤ 0.9 , x3 = 0.85, ≤ 0.9 , x4 = 0.87, ≤ 1.0 , x5 = 0.90, ≤ 0.1 , x6 = 0.10, ≤ 0.9 , x7 = 0.12, ≤ 0.9 , x8 = 0.19, ≤ 0.9 , x9 = 0.25, ≤ 0.9 , x10 = 0.29, ≤ 1000 , x11 = 512, ≤ 500 , x12 = 13.1, ≤ 500 , x13 = 71.8, ≤ 1000 , x14 = 640, ≤ 1000 , x15 = 650, ≤ 500 , x16 = 5.7.
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