Download Multi-Criteria Linear-Quadratic Control Problem by KSH

Applied Mathematical Sciences, Vol. 5, 2011, no. 2, 91 - 101
Multi-Criteria Linear-Quadratic Control Problem
by KSH-Direction Interior-Point Method
Kanuengkid, W. and Mouktonglang, T.1
Department of Mathematics, Faculty of Science
Chiang Mai University, Chiang Mai, Thailand
Abstract
We describe an implementation of an infinite dimension interiorpoint method for solving multi-criteria linear -quadratic control problem
based on KSH-direction. It is shown that such a descent direction is
independent on the scaling point. Each iteration requires solving finite
rank perturbation of linear-quadratic control problems and a system of
algebraic equations. The numerical result shows rapid convergence to
the optimal solution.
Mathematics Subject Classification: 47H09; 47H10
Keywords: Infinite dimension interior point method; Multi-criteria linearquadratic control; KSH-direction
1
Introduction
In recent years, there are a lot of developments on generalizations of interiorpoint algorithms for linear programming (LP). The theories have been extended to the context of semidefinite programming (SDP) and more on several
second-order cone problem. Recently, this class of problems gets more attention because of its wide applications. Now we study the infinite-dimensional
optimization problem and a primal-dual algorithm based on Kojima-ShindolHara direction (KSH direction). We implement the algorithm for solving multi
- criteria linear-quadratic control problem, that is a min-max optimization
problem with linear constraints in the Hilbert space. This problem was recently implemented by using NT-Direction. For more detail, see [1]. It is well
known that the NT-Direction is a member of both the KSH and MZ family.
See [4] for more details. However, the NT-direction requires the, so called,
scaling point on each iteration. The KSH-direction which was developed by
1
Corresponding author: [email protected]
92
Kanuengkid, W. and Mouktonglang, T.
Kojima, Shindol and Hare is, on the other hand, independent on a scaling
point. The existence and uniqueness of the direction is shown in [3]. This
paper is organized as follows. In section 2, we introduce the general primal
- dual optimization and Jordan algebras of finite rank for the analysis of an
infinite-dimensional situations and describe the multi- criteria linear-quadratic
control problem. In section 3, We state and prove the technical results by using
KSH- direction. Finally, numerical results and conclusion are given in section
4.
2
Primal-dual interior-point algorithms
In this section following [1], we describe a general set up for a class of
infinite-dimension optimization problems. We then proceed with the optimality criterion and describe a primal-dual algorithm based on KSH direction.
Let H be the Hilbert space. Consider the following optimization problem:
t −→ min
Wi (y − yi )H ≤ t i = 1, . . . , m,
y ∈ c + Z.
(1)
Here .H is the norm induced by the scalar product |H .yi ∈ H are fixed. Z
is a closed vector subspace in H, and Wi : H → H is a bounded linear operator for i = 1, . . . , m. The problem (1) is an example of infinite-dimensional
second-order cone programming problem which has been analyzed in detail in
[1].
The problem(1) can be easily rewritten in the conic form by following
the substitution. Let V1 be a vector space. V1 = R × H. Let, further,
V = V1 × · · · × V1 (m-times), X ⊂ V be a closed vector subspace in V .
Ω1 = {(s, y) ∈ R × H : s ≥ yH } is the second-order cone which is known
to be an open convex set in V1 . Ω = Ω1 × Ω1 × · · · × Ω1 (m − times).
Let, further, Λ : V1 → V be a linear operator such that Λ(s, y) = {(s, W1 y), . . . , (s, Wm y)}.
Then we define constants elements a and b in the vector space V and a closed
vector space X ∈ V . a = ((1, 0), (0, 0), . . . , (0, 0)) ∈ V, b = ((0, W1 (c −
y1 )), . . . , (0, Wm(c − ym ))), and X = Λ(R × Z).
Naturally, the scalar product <, >V in V is defined as follows:
< (t1 , x1 ), . . . , (tm , xm ), (s1 , y1), . . . , (sm , ym ) >V =
m
(si ti + xi , yiH ).
i=1
With these notations, we can rewrite the problem (1) in the general optimiza-
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KSH LQ-control problem
tion problem
< a, z >V → min,
(2)
z ∈ (b + X) ∩ Ω,
and its dual
< b, w >V → min,
(3)
∗
w ∈ (a + X ⊥ ) ∩ Ω ,
Proposition 2.1. We have
m
Wi∗ ui ∈ Z ⊥
X = (r1 , u1), . . . , (rm , um ) ∈ V ; r1 + . . . , +rm = 0,
⊥
i=1
(4)
where Z ⊥ is the orthogonal complement of Z and Wi∗ is the adjoint of Wi for
each i.
Consequently, a conic dual to (2) will have the following form [1]:
m
m
< Wi (c − yi), ui >H → min
(5)
i=1
Wi∗ ui ∈ Z ⊥ , ui ≤ ri i = 1, . . . , m,
i=1
m
ri = 1.
i=1
Here Wi∗ is the adjoint of Wi . The following theorem is a particular case of
the result in [1]
∗
Theorem 2.2. Let int(F) = [(x + X) ∩ int(Ω)] × [(a + X ⊥ ) ∩ int(Ω )] = 0.
Then both equatiom (2) have optimal solutions. Moreover, z ∗ is an optimal
solution to equation (3) andw ∗ is an optimal solution to equation (2) and (3)
if and only if
< z ∗ , w ∗ >V = 0
One of the most important steps in implementation of primal-dual algorithm is a computation of a descent direction which drives duality gap μ to
zero.
One of these direction (The so-called KSH direction) is described later.
94
2.1
Kanuengkid, W. and Mouktonglang, T.
Some Jordan-algebraic properties of spin-factors
In what follows we restrict ourselves to the analysis of problem (2) and (3)
for the case where V is a JB-algebra of a finite rank. Then we describe some
Jordan-algebraic aspects of a spin-factor R × H essential for future considerations. (See [1] for more detail)
Let V be the real commutative algebra with the unit element e. Given
z ∈ V, consider the multiplication operator L(z) : V → V ,
L(z)z1 = z ◦ z1 , z1 ∈ V,
Definition 2.3. We say that V is a Jordan algebra if the identity
[L(z), L(z 2 )] = L(z)L(z 2 ) − L(z 2 )L(z) = 0 ∈ V,
hold for anyz ∈ V.
We can introduce the so-call quadratic representation in an arbitrary Jordan algebraV Given z ∈ V,
P (z) = 2L(z)2 − L(z 2 ) = 0 ∈ V,
s
Let z = (s, y) ∈ R × H we write (s, y) ∈ V as a column vector
. Then
y
each linear operator on R × H admits the following block partition:
α A
,
B C
where α ∈ R, A : H → R, B : R → H, A : H → H. Then
α A
s
αs + Ay
.
=
B C
y
Bs + Cy
Given y ∈ H, introduce notation:
ly : H → R, ly (y1 ) = (y|y1), y1 ∈ H.
Observe that lyT : R → H, has the form:
lyT (s) = sy, s ∈ R
Here lyT is the transpose of ly with respect to the given scalar product (.|.) on
Y and the standard scalar product on R i.e.,
sly (y1 ) = (lyT (s)|y1), s ∈ R, y ∈ H.
With this notation, we have the following properties.
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KSH LQ-control problem
Proposition 2.4. Let z = (s, y) ∈ R × H. Then a multiplication linear operator L(z) is defined follow:
s ly
,
L(z) = T
ly sIH
Here IH is the identity operator on H.
Proposition 2.5. Let z = (s, y) ∈ R × H, y = 0. Consider
y y 1
1
1,
, e2 =
1, −
2
y
2
y
λ1 = s + y, λ2 = s − y, y = (y|y).
e1 =
Then
(s, y) = λ1 e1 + λ2 e2
is a spectral decomposition of z.
And we define a determinant of z as follows:
det(z) = s2 − y2.
Then if det(z) = 0 the inverse z −1 of z is given by the formula
z −1 =
1
(s, −y) = λ1 (z)−1 e1 + λ2 (z)−1 e2 ,
det(z)
Next, we consider a function
f (z) = −lndet(z) = +ln(s2 − y2 ).
We next describe the quadratic representation in a spin-factorR × H. Given
y ∈ H, we introduce a linear operator y ⊗ y as follows:
y ⊗ y(y1) = (y|y1)y, y1 ∈ H
Proposition 2.6. Let z = (s, y) ∈ R × H. Then a quadratic representation
P (s, y) is defined follow:
(y|y) sly
,
P (s, y) = det(z)IV + 2
slyT y ⊗ y
Here IV is the identity operator on V = R × H
For proof of the previous proposition see [1].
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Kanuengkid, W. and Mouktonglang, T.
2.2
Multi-criteria linear-quadratic control problem
Denote by Ln2 [0, T ] the Hilbert space of square integrable function f : [0, T ] →
Rn , T > 0. Let (x, u) ∈ H = Ln2 [0, T ] × Li2 [0, T ]. Consider the following
optimization problem:
T
((x − xi )T Qi (x − xi ) + (u − ui )T Ri (u − ui ))dt → min,
J(x, u) = max
i∈[1,m]
0
(6)
ẋ(t) = Ax(t) + Bu(t),
x(0) = x0 .
where (xi , ui) ∈ H, i = 1, . . . , m, A, B, vQi , Ri , i = 1, . . . , m are given matrices and Qi , Ri are symmetric positive definite matrices. We can rewrite the
problem (6) as a second order cone programming problem in(1).
Here .H is the norm induced by the scalar product < | >H , yi =
(xi , yi), i = 1, . . . , m; Z is the closed vector subspace in H described follows:
Z = {(x, u); ẋ(t) = Ax(t) + Bu(t),
T
LQi 0
Wi =
, i = 1, . . . , m,
0 LTRi
x(0) = x0 }
(7)
where LTQi , LTRi are the lower-triangular matrices obtained by Cholesky factorization of Qi and Ri respectively. For more detail see [1,2]
3
KSH- direction
The implementation of an infinite dimension interior point method for
solving multi-criteria linear quadratic control problem described in the previous section can also be applied to other well known descent directions for
instance KSH or KSH-dual [4]. The computation of a searching direction can
be done in a similar way which requires solving finite rank perturbation of
linear-quadratic control problem. This section devotes to the implementation
of infinite dimension interior point method for solving multi-criteria linear
quadratic control problem via KSH-direction.
Given (n, d) ∈ Ω × Ω a pair of feasible solutions to the problem (2) and
(3). Let (ξ, η) ∈ X × X ⊥ be KSH-direction. Hence (ξ, η) satisfies the following
equation [4].
1
1
1
1
1
η + P (n) 2 L(d)−1 P (n) 2 ξ = γP (n) 2 L(d)−1 P (n) 2 n−1 − P (n) 2 L(d)−1 d
1
= γP (n) 2 d−1 − n
1
where d = P (n)− 2 n , L(.) is a multiplication operator and P (.) is a quadratic
representation. For simplicity let
1
Δ = γP (n) 2 d−1 − n.
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KSH LQ-control problem
Then the above equation is equivalent to
1
1
P (n)) 2 L(d)−1 P (n) 2 ξ − Δ ∈ X ⊥
which also equivalent to the following optimization problem:
1
1
< P (n) 2 L(d)−1 P (n) 2 ξ, ξ >
− < Δ, ξ >→ min
2
ξ∈X
(8)
1
We rewrite the equation (8) by using description of L(.)−1 and P (.) 2 suggested
1
in [1]. Note that each L(.)−1 and P (.) 2 can be written as a linear combina(1)
(1)
(m)
(m)
tion of rank one operators. Let Δ = (Δ0 , Δ1 ), . . . , (Δ0 , Δ1 ), then by
combining all above terms, we obtain the following:
1
1
< P (s) 2 L(w)−1 P (s) 2 ξ, ξ >
− < Δ, ξ >
ρ(μ, ζ) =
2
2
m
m
m
1 γ (i)
(i)
2
(i)
(i)
2
=
Wi ζ +
β < a , Wi ζ > +
ω (i) < u1 , Wi ζ >2
(i)
2 i=1 d0
i=1
i=1
+
m
(i)
α(i) < a(i) , Wi ζ >< u1 , Wi ζ > −
i=1
m
(i)
< Δ1 , Wi ζ > +μν2 +
i=1
where:
2
β (i) =
γ (i)
(i)
2d0 (1 − a(i) 2 )
2
ω
(i)
=
γ (i)
(i)
2d0
(2 +
θ(i)
α
(i)
(i)
γ (i) (u0 − < a(i) , u1 >)
(i)
d0 (1 − a(i) 2 )
2 (i)
(i)
(i) (i)
(i)
(i) (i)
2
−
<
u
,
a
>
−4(u
−
<
u
,
a
>)
γ (i) u0
u
(i)
1
0
1
=
1 + u1 2 + 0
(i)
(i) 2 )
(1
−
a
d0
(i)
2
(i)
(i)
(u − < a(i) , u1 >)2
)
+ 0
(1 − a(i) 2 )
(i)
2
α(i) =
(i)
u1 2
=
φ(i) =
ν1 =
(i)
(i)
γ (i) (1 − u0 (u0 − < u1 , a(i) >))
γ
2
(i)
(i)
d0
m
i=1
(i)
d0 (1 − a(i) 2 )
(i)2
(i)
u0 u1 2 +
(i) (i)
(u0 (u0 −
(i)
< u1 , a(i) >)
(1 − a(i) 2 )
− 1)
2
φ(i)
m
(i)
(i)
(i) ν1 =
θ < u1 , Wi ζ > +λ(i) < a(i) , Wi ζ > −Δ0
i=1
ν1 μ2
2
98
Kanuengkid, W. and Mouktonglang, T.
Then we apply the first optimality of ρ(μ, ζ) to obtain the following:
∂ρ
ν2
=0⇔μ=−
∂μ
ν1
Consequently, we have
ψ(ζ) = min{ρ(μ, ζ) : μ ∈ R}
=
(9)
2m+2
ζ, Mζ 1
lij (vi ⊗ vj )ζ > + < v0 , ζ > −
+ < ζ,
2
2
i,j=1
(i) 2
(Σm
i=1 Δ0 )
2ν1
where
2
M =
γ (i)
(i)
w0
Wi∗ Wi
√
vi = 2Wi∗ a(i) , f or i = 1, . . . , m
√
(i)
vm+i = 2Wi∗ u1 , f or i = 1, . . . , m
m
1 (i) ∗ (i)
v2m+1 = √
λ Wi a
ν1 i=1
m
1 (i) ∗ (i)
v2m+2 = √
θ Wi u1
ν1 i=1
m
m
1 (i) (i)
v0 = √
Δ0 (v2m+1 + v2m+2 ) −
Wi∗ Δ1 ,
ν1 i=1
i=1
L =lij , where lij = 0 except
lij =β (i) ,
for i = 1, . . . , m,
(i)
l(m+i)(m+i) =ω , for i = 1, . . . , m,
l(2m+1)2(m+1) = − 1 and l(2m+2)2(m+2) = −1,
l(i)(m+i) =l(m+i)(i) =
α(i)
,
2
f or i = 1, . . . , m.
From procedure above we have the following theorem.
Theorem 3.1. Problem (8) is equivalent to following problem
2m+2
< ζ, Mζ > 1 ψ(ζ) =
lij < vi , ζ >< vj , ζ > + < v0 , ζ >→ min (10)
+
2
2 i,j=1
ζ∈Z
Proof. By a direct calculation of the procedure above .
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KSH LQ-control problem
Then to obtain KSH-direction, it suffices to solve the problem (10). The
following theorem further simplifies the problem(10).
Theorem 3.2. Let ζ0 be the optimal solution to the problem :
< ζ, Mζ >
+ < v0 , ζ >
2
ζ∈Z
(11)
and ζi i = 1, . . . , 2m + 2 be the optimal solution to the problem :
< ζ, Mζ >
+ < Vi , ζ >
2
ζ∈Z
(12)
Let S = (sij ), sij =< vi , ζj >, i , j = 1, . . . , 2m+2.
where Vi = m
j=1 lij vj
Then the set of optimal solution to the problem (10). is in one to one
correspondence with the set of solution of the system of linear equations
⎡
⎤ ⎡
⎤
δ1
< v1 , ζ0 >
⎢
⎥ ⎢
⎥
..
(I − S) ⎣ ... ⎦ = ⎣
(13)
⎦
.
δ2m+2
< v2m+2 , ζ0 >
More precisely if (δ1 , . . . , δ2m+2 ) is a solution to equation (13), then
ζ(δ) = ζ0 +
2m+2
δi ζi
(14)
i,=1
is the optimal solution to the problem (10)
For a proof see [1,5].
∂ρ
= 0, μ = − νν21 .
Let ζ be an optimal solution to equation(10) , ∂μ
Therefore we obtain the primal descent direction
ξ = Λ(μ, ζ) = ((μ, W1 ζ), . . . , (μ, Wmζ)),
and a dual descent direction
1
1
η = P (n) 2 L(d)−1 P (n) 2 ξ − Δ.
Note that the problem (10) is the a finite-rank perturbation of the linearquadratic control problem which can be handled by a similar technique as in
NT-direction case. Nevertheless, one of the drawbacks of KSH-direction is
100
Kanuengkid, W. and Mouktonglang, T.
that in order to solve (10), it is require to solve 2m + 3 linear quadratic control
problems and a (2m + 2) × (2m + 2) linear algebraic equations. By comparing
to NT-direction case, solving the arose finite-rank perturbation of the linearquadratic control problem can be solved by solving m + 2 linear quadratic
control problems and a (m + 1) × (m + 1) linear algebraic equations. However,
the scaling points are not required for KSH-direction.
4
Numerical results
We use the same test problems as in [2]. As it is shown in the table below,
the duality gap decreases as the number of iterations increases. Our approximated solution approaches to the optimal solution according to the theory of
interior-point method. However, when compare to NT-direction, the number of
iterations for NT-direction is less than KSH-direction for the same tolerance.
As for average time per iteration, NT-direction gains advantage for a large
number of targets and that is because the system of algebraic equations obtained in KSH-direction is much larger than the system of algebraic equations
in NT-direction. Due to the independence of scaling point for KSH-direction,
however, our algorithm still works even when the scaling point reaches the
boundary.
T
1
2
3
4
5
Opt V alue
2035 2071 2075 2076 2077
Number of Iterations via NT-direction
6
6
6
6
6
Number of Iterations via KSH-direction 10
11
12
12
12
T
6
7
8
9
10
Opt V alue
2077 2077 2077 2077 2077
Number of Iterations via NT-direction
6
6
6
6
6
Number of Iterations via KSH-direction 13
13
13
13
13
5
Conclusion
We describe an implementation of an infinite dimension interior point method
for solving multi-criteria linear -quadratic control problem based on KSHdirection. It is shown that such a descent direction is independent on the
scaling point. Each iteration requires solving finite rank perturbation of linearquadratic control problems and a system of algebraic equations. It is require
KSH LQ-control problem
101
to solve 2m + 3 linear quadratic problems and a (2m + 2) × (2m + 2) linear algebraic equations. However, scaling points are not required for KSH-direction.
Numerical results show rapid convergence to the optimal solution.
Acknowledgement: This research was supported by the Commission for
Higher Education, CHE, RG2550 Methods of Nonlinear Analysis.
References
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of infinite-dimensional interior-point method for solving multi-criteria
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[3] Tsuchiya, T., Aconvergence analysis of the Scaling-invariant PrimaldualPath-following Algorithms for Second-order cone Programming. Optimization Method and Software,10/10,141-182, 1999.
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Optimization Method and Software,11/12, 47-52, 1999.
[5] Faybusovich, L. and Mouktonglang, T., Finite rank perturbation of the
linear-quadratic control problem. Proceeding of the American Control
Conference, Denver, Colorado, June 4-6, pp. 5347-5350, 2003.
Received: August, 2010