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Further investigation on feasibility of mathematical programs with equilibrium constraints
An International Journal
computers & mathematics with applioations
PERGAMON www.elsevier.com/locate/camwa
Further Investigation on Feasibility of M a t h e m a t i c a l P r o g r a m s with Equilibrium Constraints ZHONG WAN I n s t i t u t e of A p p l i e d M a t h e m a t i c s , t t u n a n U n i v e r s i t y C h a n g s h a 410082, P.R. C h i n a
(Received January 2001; revised and accepted October 2001) Abstract--In this paper, we present feasibility conditions for m a t h e m a t i c a l programs with affine equilibrium constraints (MPECs) where additional joint constraints are present t h a t m u s t be satisfied by the state and design variables of the problems. We show t h a t these conditions are also sufficient for quadratic p r o g r a m m i n g subproblems arising from t h e penalty interior point algorithm (PIPA) and t h e s m o o t h sequential quadratic p r o g r a m m i n g (SQP) algorithm for solving M P E C s to be consistent. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords--Mathematical
programs with equilibrium constraints, Feasibility conditions.
1. I N T R O D U C T I O N Let f
: R n+rn --~ R be a continuously differentiable function, A E R px'~, B ~ / ~ p x m , N ¢ 17~rnxn, and M E R mx'~ be given matrices, b E/~P and q E R m be given vectors, and C be a polyhedral cone in R n. Consider the following mathematical program with affine equilibrium constraints where additional joint linear constraints are presents besides the equilibrium constraints:
minimize
f(x, y)
subject to
A x + B y = b,
(1.1)
w = Nx + MywTy=O,
q,
x¢C,
y > O , w>_O.
In (1.1), the variable x is called the design variable or the first-level variable and the variables y and w are called the state variables or the second-level variables. The equilibrium constraints (also called state constraints) are expressed in the form of a parametric linear complementarity problem (LCP) with x as the parameter and y as the primary variable. The additional joint constraint between the design variable x and the state variable y is assumed to be modeled by the linear equation A x + B y = b. T h e a u t h o r would like to t h a n k one a n o n y m o u s referee whose incisive c o m m e n t s helped me improve t h e result of t h e paper. T h e a u t h o r would also like to t h a n k Professors S. Z, Zhou and D. H. Li for their e n c o u r a g e m e n t a n d helpful c o m m e n t s on t h e paper. 0898-1221/02/$ - see front m a t t e r © 2002 Elsevier Science Ltd. All rights reserved, PII: S0898-1221(02)00126-8
Typeset by A2MS-TF/X
8
Z. ~VAN
The purpose of this paper is to further study the feasibility of problem (1.1) on the basis of the work by Fukushima and Pang [1], where the matrix B admits being nonzero. In the case where B = 0, the feasibility of (1.1) is relatively easy, we refer to Section 1 of [1] for some explanations. In the case where B ¢ 0, the feasibility of M P E C (1.1) has been the major bottleneck for a complete treatment of problem (1.1) [1]. In this paper, we present sufficient conditions for problem (1.1) to be feasible. We show that these conditions are also sufficient to guarantee the consistence of the subproblems arising from the penalty interior point algorithm (PIPA) [2] and the smooth sequential quadratic programming (SQP) algorithm [3] for solving MPECs.
2. F E A S I B I L I T Y
CONDITIONS
FOR
MPECS
In this section, we introduce sufficient conditions to ensure the feasibility of problem (1.1). We first rewrite the constraint system of (1.1) as follows.
A x + B y = b, y>O, N x + M y - q >_ O,
(2.1)
XEC, y T ( N x + M y - q) = O. Then the M P E C (1.1) is feasible if and only if (2.2) is consistent. To study the consistence of system (2.2), we propose the following two assumptions. (At) The linear system
N x + M y - q > O,
y > O, xEC,
(2.2)
is consistent. (A2) For any (u, v, s) satisfying
A T u + N T v E C*, V o (BTu q- M T v ) < O~
(2.s)
s o (BT~ + ~ F ~ ) > 0, ( v + s ) os>_O,
s>_O,
we have (v + s ) T ( B T u + M T v ) > (1/2) uTu. Conditions (A1) and (A2) are similar to the conditions given by Fukushima and Pang [1]. But both do not imply each other. As follows, we first provide some sufficient conditions for Assumption (A2) to hold. PROPOSITION 2.1. Let
PI ~
(,
BT
M+M
T
)
'
P2~
0)
2
5i r
,
and ]C denote the cone {(u,v) E R p+m : ATu + N T v C C*}. i f [ o r any (u,v,s) E K x R~', P1 and P2 satisfy the [ollowing inequality:
(uT vT) . l
(:)
> (sTsT) p,2
(0
'
where R~_~ denotes the nonnegative orthant of R m. Then Assumption (A2) holds. It is easy to prove the above proposition, we omit it. Next, we state the main theorem in this paper.
F~arther Investigation
9
THEOREM 2.1. Under Assumptions (A1) and (A2), system (2.1) is consistent.
PROOF. We only require to prove t h a t the following quadratic p r o g r a m respect to t h e variable (x, y) E R n+m has an o p t i m a l solution, and t h a t the m i n i m u m value of the objective function is zero.
minimize
yT ( N x + M y - q) + l ( A x + B y - b)T(Ax + B y - b)
subject to
(2.2).
.
(2.4)
Let (~, 9) be a solution of (2.4). Then, there exist some multiplier vector # c R "~ such t h a t 0 < 9_1_ ( N 2 + M 9 + MT!I - q + B r B 9 + B T A 2 - B T b -
M T # ) > 0,
C D 2 2 ( N T 9 + A T A 2 + A T B 9 -- ATb -- NTIt) E C*,
O < itJ_(N2 + M g - q )
(2.5)
>_O.
Let ¢ = N 2 + M 9 - q and ~ = A2 + B 9 - b. T h e n (2.5) can be w r i t t e n as:
0 < 9_[_(¢ 4- BT¢ 4- M T (9 - It)) >- 0, C ~ 22 ( A T e + N T (9 -- It)) e C*, 0 _< ~ 2 ¢ _> 0. It t h e n follows t h a t
t,o¢=0,
9 o (¢ + B T e + M r (9 - , ) ) = O,
Ito ( ¢ 4 - B T e 4 - M
T (Y--tt)) ~ 0,
(2.6)
90¢>0, 9o/,>0. So, we can deduce
(Y--it)° ( B T e + M T ( y - I t ) ) % ( Y - i t ) ° ( ¢ + B T ~ + M
T(y-It))
--/~ o ((;b 4- B T e 4- M T (Y - It))
<0 and
It o ( £ %
+ M = (9 - It)) = It o (¢ + B r e
+ M r (9 - It)) -> 0.
Let u = e , v = 9 - It, and s = It. T h e n (u, v, s) satisfies (2.3). So, it follows from (A2) t h a t
(v + s) T (BTu + M r v ) > luTu, --2
(2.7)
F r o m the second equation in (2.3), we have
(9) r
+
=-(9)T
(BT
+ MT ( 9 - - . ) ) +
= - ( v + s) r ( B r .
+ M%)
+ !~-%
2
<_0, where the last inequality comes from (2.7). Also from the fourth inequality in (2.3), we have (9)T¢ + ( 1 / 2 ) e T e > o, so we o b t a i n t h a t
(9) r
'4- l"t/)Tl/) = O.
Since t h e objective function in (2.4) can be rewritten as (9)T(~) 4- ( 1 / 2 ) ~ T ~ , we have proved t h a t it a t t a i n s m i n i m u m value zero at (2, Y), thus, the desired result holds. II
10
Z. WAN
3. F E A S I B I L I T Y
OF S U B P R O B L E M
OF P I P A O R S Q P
In this section, we show that Conditions (Ai) and (A2) are also sufficient to guarantee the consistence of the snbproblems arising from PIPA [2] and the smooth SQP algorithm [3] for solving M P E C (1.1). The subproblem of the PIPA is the following quadratic program
( d x ) + l ( d x T , d y T , d w T ) Qk
minimize
v f (x
subject to
A d z + B dy = O,
dy
(3.1)
N dx + M dy - dw = O, W k dy + y k dw = - - Y k w k + (Tkpke, x k + dx E C, where Qk ~ R ('~+2~)x(n+2m) is a s y m m e t r i c positive definite matrix, W k = diag(w~), y k = diag(y~), (7k E (0, 1) is the centerizing stepsize that control the direction (dx ~, dy k, dw k) between the pure Newton direction ((7 = 0) and the perfectly central direction ((7 = 1), and #k is the positive scalar given by #k - ( w k ) T Y k / m . We refer to Chapter 6 of [2] for details about PIPA. The following theorem shows that Conditions (A1) and (A2) are sufficient to ensure the consistence of (3.1). THEOREM 3.1. Under Assumptions (A1) and (A2), the quadratic program (3.1) is consistent for all k. PROOF. Let t = v x k + dx. It then follows from (3.1) that At + B d y = A x k, Nt + (M + (Yk)-'wk')
dy = s k, tEC,
where s k - N x k - w k + (Tkpk(Ya)-le. By a generalized Farkas lemma [4], it is clear that for an arbitrary s k, system (3.2) is consistent if and only if the implication A T u + N T v E C* BTu+(MT+(y~)-'W~) T v=0
"1 ~ ~(Ax~)Tu+(s~)T
v>0
(3.3)
holds. T h e arbitrariness of s k implies that the right-hand side of implication (3.3) is equivalent to (Axk) T u > 0,
v = 0.
(3.4)
Let (u, v) satisfy the left-hand side of implication (3.3). Then B T u + M T v = - ( Y k ) - l W k v . Therefore, we have v o ( B T u -~- ]~/Tv) = - - v o ((Yk) -1 ~/-kv) < 0,
(3.5)
because yk abd W k are diagonal matrices with positive entities. Under Assumptions (A1) and (A2), we get from (3.5) and Theorem 2.1 that u = 0 and v T M r v > 0 by letting s = 0 in (2.3). Consequently, (3.5) implies v T ( Y k ) - l W k v = 0. By the positive definiteness of ( Y k ) - l W k , we obtain v = 0. Thus, the implication (3.3) holds. This completes the proof. |
Further Investigation
11
Now, consider the subproblem of the smooth SQP algorithm for M P E C (1.1) (see [3]):
minimize subject to
V f (xk , yk ) r
dy
-2
dw
A dx + B dy = O, N dx + M dy - dw = O, x a + dx E C,
where Qk ¢ R (n+2m)x(n+2'n) is a symmetric positive definite matrix,
D~a = diag ( O¢ (y~'qw~'#k) ) Oa
Db~" = diag
k wk ) 0¢ (Yi, i , P k )
Ob
q~(y,w,p) =-
and
¢ (Ym,Wm,.) / with ¢(a, b, it) -= a + b - v/a 2 + b2 + p. In a way similar to Theorem 3.1, it is not difficult to prove the following theorem. THEOREM 3.2. Under Assumptions (A1) and (A2), the quadratic program (3.6) is feasible for
all k. 4.
CONCLUSIONS
The results presented in this paper show that Conditions (A1) and (A2) not only guarantee the feasibility of M P E C (1.1), but also can be used as the conditions to extend the convergence results for PIPA [2] and the smooth SQP method [3] such that these algorithms are applicable for the case that B ¢ 0. However, it still deserves further investigation to find some more easily verified conditions for Assumption (A2) to hold in practical applications (see Proposition 2.1).
REFERENCES 1. M. Fukushima and J.S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints, SIAM Journal on Optimization 8, 673-681, (1998). 2. Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, (1996). 3. M. Fukushima, Z.Q. Luo and J.S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Computational Optimization and Applications 10, 5-34, (1998). 4. I~.W. Cottle, J.S. Pang and K.E. Stone, The Linear Complementarity Problem, Academic Press, (1992).
computers & mathematics with applioations
PERGAMON www.elsevier.com/locate/camwa
Further Investigation on Feasibility of M a t h e m a t i c a l P r o g r a m s with Equilibrium Constraints ZHONG WAN I n s t i t u t e of A p p l i e d M a t h e m a t i c s , t t u n a n U n i v e r s i t y C h a n g s h a 410082, P.R. C h i n a
(Received January 2001; revised and accepted October 2001) Abstract--In this paper, we present feasibility conditions for m a t h e m a t i c a l programs with affine equilibrium constraints (MPECs) where additional joint constraints are present t h a t m u s t be satisfied by the state and design variables of the problems. We show t h a t these conditions are also sufficient for quadratic p r o g r a m m i n g subproblems arising from t h e penalty interior point algorithm (PIPA) and t h e s m o o t h sequential quadratic p r o g r a m m i n g (SQP) algorithm for solving M P E C s to be consistent. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords--Mathematical
programs with equilibrium constraints, Feasibility conditions.
1. I N T R O D U C T I O N Let f
: R n+rn --~ R be a continuously differentiable function, A E R px'~, B ~ / ~ p x m , N ¢ 17~rnxn, and M E R mx'~ be given matrices, b E/~P and q E R m be given vectors, and C be a polyhedral cone in R n. Consider the following mathematical program with affine equilibrium constraints where additional joint linear constraints are presents besides the equilibrium constraints:
minimize
f(x, y)
subject to
A x + B y = b,
(1.1)
w = Nx + MywTy=O,
q,
x¢C,
y > O , w>_O.
In (1.1), the variable x is called the design variable or the first-level variable and the variables y and w are called the state variables or the second-level variables. The equilibrium constraints (also called state constraints) are expressed in the form of a parametric linear complementarity problem (LCP) with x as the parameter and y as the primary variable. The additional joint constraint between the design variable x and the state variable y is assumed to be modeled by the linear equation A x + B y = b. T h e a u t h o r would like to t h a n k one a n o n y m o u s referee whose incisive c o m m e n t s helped me improve t h e result of t h e paper. T h e a u t h o r would also like to t h a n k Professors S. Z, Zhou and D. H. Li for their e n c o u r a g e m e n t a n d helpful c o m m e n t s on t h e paper. 0898-1221/02/$ - see front m a t t e r © 2002 Elsevier Science Ltd. All rights reserved, PII: S0898-1221(02)00126-8
Typeset by A2MS-TF/X
8
Z. ~VAN
The purpose of this paper is to further study the feasibility of problem (1.1) on the basis of the work by Fukushima and Pang [1], where the matrix B admits being nonzero. In the case where B = 0, the feasibility of (1.1) is relatively easy, we refer to Section 1 of [1] for some explanations. In the case where B ¢ 0, the feasibility of M P E C (1.1) has been the major bottleneck for a complete treatment of problem (1.1) [1]. In this paper, we present sufficient conditions for problem (1.1) to be feasible. We show that these conditions are also sufficient to guarantee the consistence of the subproblems arising from the penalty interior point algorithm (PIPA) [2] and the smooth sequential quadratic programming (SQP) algorithm [3] for solving MPECs.
2. F E A S I B I L I T Y
CONDITIONS
FOR
MPECS
In this section, we introduce sufficient conditions to ensure the feasibility of problem (1.1). We first rewrite the constraint system of (1.1) as follows.
A x + B y = b, y>O, N x + M y - q >_ O,
(2.1)
XEC, y T ( N x + M y - q) = O. Then the M P E C (1.1) is feasible if and only if (2.2) is consistent. To study the consistence of system (2.2), we propose the following two assumptions. (At) The linear system
N x + M y - q > O,
y > O, xEC,
(2.2)
is consistent. (A2) For any (u, v, s) satisfying
A T u + N T v E C*, V o (BTu q- M T v ) < O~
(2.s)
s o (BT~ + ~ F ~ ) > 0, ( v + s ) os>_O,
s>_O,
we have (v + s ) T ( B T u + M T v ) > (1/2) uTu. Conditions (A1) and (A2) are similar to the conditions given by Fukushima and Pang [1]. But both do not imply each other. As follows, we first provide some sufficient conditions for Assumption (A2) to hold. PROPOSITION 2.1. Let
PI ~
(,
BT
M+M
T
)
'
P2~
0)
2
5i r
,
and ]C denote the cone {(u,v) E R p+m : ATu + N T v C C*}. i f [ o r any (u,v,s) E K x R~', P1 and P2 satisfy the [ollowing inequality:
(uT vT) . l
(:)
> (sTsT) p,2
(0
'
where R~_~ denotes the nonnegative orthant of R m. Then Assumption (A2) holds. It is easy to prove the above proposition, we omit it. Next, we state the main theorem in this paper.
F~arther Investigation
9
THEOREM 2.1. Under Assumptions (A1) and (A2), system (2.1) is consistent.
PROOF. We only require to prove t h a t the following quadratic p r o g r a m respect to t h e variable (x, y) E R n+m has an o p t i m a l solution, and t h a t the m i n i m u m value of the objective function is zero.
minimize
yT ( N x + M y - q) + l ( A x + B y - b)T(Ax + B y - b)
subject to
(2.2).
.
(2.4)
Let (~, 9) be a solution of (2.4). Then, there exist some multiplier vector # c R "~ such t h a t 0 < 9_1_ ( N 2 + M 9 + MT!I - q + B r B 9 + B T A 2 - B T b -
M T # ) > 0,
C D 2 2 ( N T 9 + A T A 2 + A T B 9 -- ATb -- NTIt) E C*,
O < itJ_(N2 + M g - q )
(2.5)
>_O.
Let ¢ = N 2 + M 9 - q and ~ = A2 + B 9 - b. T h e n (2.5) can be w r i t t e n as:
0 < 9_[_(¢ 4- BT¢ 4- M T (9 - It)) >- 0, C ~ 22 ( A T e + N T (9 -- It)) e C*, 0 _< ~ 2 ¢ _> 0. It t h e n follows t h a t
t,o¢=0,
9 o (¢ + B T e + M r (9 - , ) ) = O,
Ito ( ¢ 4 - B T e 4 - M
T (Y--tt)) ~ 0,
(2.6)
90¢>0, 9o/,>0. So, we can deduce
(Y--it)° ( B T e + M T ( y - I t ) ) % ( Y - i t ) ° ( ¢ + B T ~ + M
T(y-It))
--/~ o ((;b 4- B T e 4- M T (Y - It))
<0 and
It o ( £ %
+ M = (9 - It)) = It o (¢ + B r e
+ M r (9 - It)) -> 0.
Let u = e , v = 9 - It, and s = It. T h e n (u, v, s) satisfies (2.3). So, it follows from (A2) t h a t
(v + s) T (BTu + M r v ) > luTu, --2
(2.7)
F r o m the second equation in (2.3), we have
(9) r
+
=-(9)T
(BT
+ MT ( 9 - - . ) ) +
= - ( v + s) r ( B r .
+ M%)
+ !~-%
2
<_0, where the last inequality comes from (2.7). Also from the fourth inequality in (2.3), we have (9)T¢ + ( 1 / 2 ) e T e > o, so we o b t a i n t h a t
(9) r
'4- l"t/)Tl/) = O.
Since t h e objective function in (2.4) can be rewritten as (9)T(~) 4- ( 1 / 2 ) ~ T ~ , we have proved t h a t it a t t a i n s m i n i m u m value zero at (2, Y), thus, the desired result holds. II
10
Z. WAN
3. F E A S I B I L I T Y
OF S U B P R O B L E M
OF P I P A O R S Q P
In this section, we show that Conditions (Ai) and (A2) are also sufficient to guarantee the consistence of the snbproblems arising from PIPA [2] and the smooth SQP algorithm [3] for solving M P E C (1.1). The subproblem of the PIPA is the following quadratic program
( d x ) + l ( d x T , d y T , d w T ) Qk
minimize
v f (x
subject to
A d z + B dy = O,
dy
(3.1)
N dx + M dy - dw = O, W k dy + y k dw = - - Y k w k + (Tkpke, x k + dx E C, where Qk ~ R ('~+2~)x(n+2m) is a s y m m e t r i c positive definite matrix, W k = diag(w~), y k = diag(y~), (7k E (0, 1) is the centerizing stepsize that control the direction (dx ~, dy k, dw k) between the pure Newton direction ((7 = 0) and the perfectly central direction ((7 = 1), and #k is the positive scalar given by #k - ( w k ) T Y k / m . We refer to Chapter 6 of [2] for details about PIPA. The following theorem shows that Conditions (A1) and (A2) are sufficient to ensure the consistence of (3.1). THEOREM 3.1. Under Assumptions (A1) and (A2), the quadratic program (3.1) is consistent for all k. PROOF. Let t = v x k + dx. It then follows from (3.1) that At + B d y = A x k, Nt + (M + (Yk)-'wk')
dy = s k, tEC,
where s k - N x k - w k + (Tkpk(Ya)-le. By a generalized Farkas lemma [4], it is clear that for an arbitrary s k, system (3.2) is consistent if and only if the implication A T u + N T v E C* BTu+(MT+(y~)-'W~) T v=0
"1 ~ ~(Ax~)Tu+(s~)T
v>0
(3.3)
holds. T h e arbitrariness of s k implies that the right-hand side of implication (3.3) is equivalent to (Axk) T u > 0,
v = 0.
(3.4)
Let (u, v) satisfy the left-hand side of implication (3.3). Then B T u + M T v = - ( Y k ) - l W k v . Therefore, we have v o ( B T u -~- ]~/Tv) = - - v o ((Yk) -1 ~/-kv) < 0,
(3.5)
because yk abd W k are diagonal matrices with positive entities. Under Assumptions (A1) and (A2), we get from (3.5) and Theorem 2.1 that u = 0 and v T M r v > 0 by letting s = 0 in (2.3). Consequently, (3.5) implies v T ( Y k ) - l W k v = 0. By the positive definiteness of ( Y k ) - l W k , we obtain v = 0. Thus, the implication (3.3) holds. This completes the proof. |
Further Investigation
11
Now, consider the subproblem of the smooth SQP algorithm for M P E C (1.1) (see [3]):
minimize subject to
V f (xk , yk ) r
dy
-2
dw
A dx + B dy = O, N dx + M dy - dw = O, x a + dx E C,
where Qk ¢ R (n+2m)x(n+2'n) is a symmetric positive definite matrix,
D~a = diag ( O¢ (y~'qw~'#k) ) Oa
Db~" = diag
k wk ) 0¢ (Yi, i , P k )
Ob
q~(y,w,p) =-
and
¢ (Ym,Wm,.) / with ¢(a, b, it) -= a + b - v/a 2 + b2 + p. In a way similar to Theorem 3.1, it is not difficult to prove the following theorem. THEOREM 3.2. Under Assumptions (A1) and (A2), the quadratic program (3.6) is feasible for
all k. 4.
CONCLUSIONS
The results presented in this paper show that Conditions (A1) and (A2) not only guarantee the feasibility of M P E C (1.1), but also can be used as the conditions to extend the convergence results for PIPA [2] and the smooth SQP method [3] such that these algorithms are applicable for the case that B ¢ 0. However, it still deserves further investigation to find some more easily verified conditions for Assumption (A2) to hold in practical applications (see Proposition 2.1).
REFERENCES 1. M. Fukushima and J.S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints, SIAM Journal on Optimization 8, 673-681, (1998). 2. Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, (1996). 3. M. Fukushima, Z.Q. Luo and J.S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Computational Optimization and Applications 10, 5-34, (1998). 4. I~.W. Cottle, J.S. Pang and K.E. Stone, The Linear Complementarity Problem, Academic Press, (1992).